ELEMENTARY LESSONS IN PHYSICS MECHANICS {INCLUDING HYDROSTATICS) AND LIGHT BV . EDWIN H. HALL, Ph.D. Assistant Professor of Physics in Harvard College NEW YORK HENRY HOLT AND COMPANY 1900 TWO COPIES RECEIVED, Library of Congre«% Office of the FEB 7 - 1900 Ktglttar of Copyright* 54196 Copyright, 1900 BY Henry Holt & Co t SECOND COPY, ^ O O (o ROBERT DRUMMOND, ELECTROTYPER AND PRINTER, NEW YORK. INTRODUCTION. This volume, which is the First Part of Hall and Ber- gen's revised Text-booh of Physics (1897), may be regarded as the second edition of Hair's Elementary Lessons in Physics. It is intended for the use of pupils in the early years of a high-school course or even the last year of a grammar-school course, and it assumes no previous system- atic study of physics. The course of study here given includes laboratory work, to be done by the pupils, combined with a considerable amount of general 'instruction, to be illustrated by lec- tures given by the teacher. The laboratory work is mainly or wholly quantitative, as it must be for large classes, quali- tative laboratory work in such classes making impossible demands upon the time and energy of the teacher. A First Course made up, as this one is, of simple experi- ments in mechanics (including hydrostatics) and optics, more difficult matters in mechanics, together with heat, sound, electricity, and magnetism, being deferred, is un- usual; but it is here proposed as better suited to many pu- pils and to many schools than the more familiar practice of going through the whole of mechanics before entering upon any other part of physics, and putting light, or optics, after heat. The laboratory outfit required for these early exer- cises is much less complicated and expensive than that required for much of the later laboratory work ; so that many schools which would be quite unable to offer a labo- iii iv INTRODUCTION. ratory course extending over the whole field of elementary- physics will find it possible to do what this book requires. Moreover, those teachers w r ho have ample means and facil- ities, and who intend to take their pupils through the whole range of elementary laboratory work, will find it advantageous to interrupt the course in mechanics lest their classes grow tired of what is, for many young pupils, the least interesting part of the study. The book follows, as a rule, the method of leading up to the statement of laws by means of carefully chosen experi- ments, rather than the opposite one of giving experiments as illustrations or proofs of laws already stated. It can hardly be said for the former method that it teaches the art of making discoveries, — that art is as difficult to teach as the art of getting rich, — but it has a tendency to keep the pupil in a more active, self-dependent state of mind than the latter method, and in particular it prevents iu a large measure that state of bias, or preconception, in the performance of experiments, which is so dangerous not merely to accuracy of observation but to mental rectitude. On the other hand, the teacher using the method of this book must not allow his pupils to think that their experi- ments, even when most satisfactory, really demonstrate the rigid accuracy of any numerical law, — the law of a balanced lever, for instance. He should ask of them, " What law do your experiments indicate as true ?" and after their answer he should tell them whether their inference is or is not in accordance with the opinion held by those best qualified to judge of the matter in question. It is the firm conviction of the writer that class labora- tory work not accompanied by persistent, energetic teach- ing is sure to be a failure. We are often told that the favorite method of the elder Agassiz with a new pupil was INTROD UCTION. V to set him to gaze in solitude at a single fish for two or three days. Those who would make this the model for science-teaching in general forget that pure observation of numerous, minute, varied details plays a much more important part in natural history than in physics. The teacher of physics who would produce good and lasting results must see to it not merely that the laboratory work shall be carefully done, but that the proper lessons shall be drawn from it and the proper applications made. In fact, the young pupil should give as much time to the study of physics in the lecture- or recitation-room as in the laboratory proper. The course of study described in this book is intended to run through a school year and to occupy the pupil at least two school-periods, each forty minutes long, or more, per week; one usually in the laboratory, and the other in the lecture- or recitation-room. The number of laboratory Exercises is considerably less than the number of school- weeks in the year, but some of them may prove to be too long for a single school-period, and teachers will welcome an occasional opportunity for repetition or review. The amount of time required for the course will depend some- what upon the age of the pupils taking it, and classes in the first year of a high-school course may find three school-periods a week for one year none too much time for doing the work well. It is highly desirable that pupils whose laboratory work is confined to that described in this book should have lec- ture-room illustrations of many things not here dealt with, elementary facts and principles in heat, sound, electricity, and magnetism. It is therefore recommended that every teacher of such pupils be provided with the means neces- sary for such illustrations, for example much of the appa- vi INTRODUCTION. rat us designated by Roman numerals in Hall and Bergen's Physics. The following estimates of cost for apparatus and ma- terials are only approximate. It is hardly possible to make an accurate estimate, as prices will vary from time to time, and different dealers have somewhat different grades of ap- paratus. The cheapest is not necessarily the best to buy. FOR THIS BOOK. Teacher's apparatus and supplies, pp. 174-178 .... $78.00 Students' apparatus, pp. 170-173, for each member of a laboratory squad 6.00 Table, accommodating six workers, p. 178 25.00 Total for all Exercises and Experiments of this book, with laboratory squads limited to twelve.. . 200.00 FOR THE SECOND PART OF HALL AND BERGEN'S PHYSICS. Teacher's apparatus, pp. 581-586 350.00 By omittiug the thermopile and accompanying apparatus and the Roentgen-ray apparatus this ex- pense can be reduced about $100. Apparatus for the course can be obtained from the fol- lowing well-known manufacturers: The Chicago Laboratory Supply and Scale Company, 39 West Randolph Street, Chicago. The Franklin Educational Company, Harcourt Street, Boston. The Knott Company, 16 Ashburton Place, Boston. The Ritchie Company, Brookline, Mass. The Ziegler Electric Company, 141 Franklin Street, Boston. E. II. H. January 16, 1900 TABLE OF CONTENTS. PAGE Introduction iii CHAPTER I. INTRODUCTORY. Definition of Physics. — Use of Physics. — Qualitative and Quantitative Knowledge. — Object of this Course.— Preliminary Exercises: Measurement of Distance, of Area, of Volume, — with Estimation of Errors 1 MECHANICS. CHAPTER II. DENSITY AND SPECIFIC GRAVITY. Density.— EXERCISE 1: Weight of Unit Volume of a Sub- stance.— Density of Water.— Weight.— Mass.— EXERCISE 2: Lifting Effect of Water.— EXERCISE 3: Specific Gravity of a Solid that will Sink in Water.— EXERCISE 4: Specific Gravity of Wood by Use of Sinker.— EXERCISE 5: Weight of Water Dis- placed by a Floating Body. — EXERCISE 6: Specific Gravity by Floating Method.— EXERCISE 7: Specific Gravity of a Liquid by two Methods 15 CHAPTER III. FLUID-PRESSURE. Fluids: Liquids and Gases— Experiments with Pressure- gauge in Water. — Torricelli's Experiment. — Atmospheric Pres- vii viii TABLE OF CONTENTS. PAGE sure. —Barometer. — Boyle's Law.— Hydraulic Press. — Water- pumps. — Siphon. — Balancing Columns 28 CHAPTER IV. THE LEVER, EXERCISE 8 : The Straight Lever, First Class, — Circular Lever.— EXERCISE 9: Centre of Gravity and Weight of Lever. — EXERCISE 10: Levers of the Second and Third Class: s.— EXER- CISE 11: Force Exerted at the Fulcrum. — Pulleys. — General Law for Relation of Power to Weight 41 CHAPTER Y. THREE FORCES ACTING THROUGH ONE POINT — THE PARALLELO- GRAM OF FORCES. Introductory. — EXERCISE 12: Errors of a Spring-balance. — EXERCISE 13: Parallelogram of Forces.— The Inclined Plane: Wedge, Screw. — Equilib i ant aDd Resultant 61 CHAPTER VI. FRICTION. EXERCISE 14: Friction between Solid Bodies.— EXERCISE 15: Coefficient of Friction.— Friction in Applied Mechanics.— Rolling Friction. — Friction between Solids and Fluids , 78 CHAPTER VII. THE PENDULUM. Use in Clocks.— Experiments.— Springs in Place of Pendu- lums 86 LIGHT. CHAPTER VIII. Nature of Light— Visibility of Objects. Light is Something that Travels. — Velocity.— A Wave- motion.— Pencils and Rays.— Shadows.— EXERCISE 16: Use of TABLE OF CONTEXTS. ix PAGE Rumford Photometer. — Bunsen's Photometer. — Effect of Body on which Light Falls. — Visibility of Objects 90 CHAPTER IX. REGULAR REFLECTION OF LIGHT. EXERCISE 17: Images in a Plane Mirror. — Images of Images, Kaleidoscope. — EXERCISE 18: Images Formed by a Convex Cylindrical Mirror. — EXERCISE 19 : Images Formed by a Concave Cylindrical Mirror. — Relation of Cylindrical to Spherical Mir- rors. — The Ophthalmoscope. — Formulas relating to Curved Mirrors 103 CHAPTER X. REFRACTION OF LIGHT. Introductory Experiments. — Angles of Incidence and Re- fraction.— EXERCISE 20: Index of Refraction of Glass.— EX- ERCISE 21 : Index of Refraction of Water.— Index Different for Different Colors. — Index of Refraction and Velocity. — Internal Reflection, Critical Angle. — Transparent Plates and Prisms. — Dispersion, the Spectrum.— Lenses.— EXERCISE 22; Focal Length of Converging Lens.— EXERCISE 23 : Conjugate Foci of a Lens. — EXERCISE 24 : Shape and Size of Real Image Formed by a Lens. — EXERCISE 25 : Virtual Image Formed by a Lens. — Spherical and Chromatic Aberration in Lenses. — Achromatic Lenses. ..,,,,,,,, 123 CHAPTER XL THE EYE : SIGHT AND COLOR. Description of the Eye.— Accommodation, etc. — The Color- sense. — Mixing Color Impressions.;— Complementary Colors. — Fatigue of Retina. — After-images 152 CHAPTER XII. OPTICAL INSTRUMENTS. Photographer's Camera. — Magic Lantern. — Projecting a Spec- trum. — Simple Microscope. — Compound Micro-cope. — Tele- scope 156 X TABLE OF CONTENTS. APPENDIX I. PAGE Focal Length, etc., of Lenses and Combinations 168 APPENDIX II. Indices of Refraction e . 169 APPENDIX III. List of Apparatus, etc., for the Exercises and Experiments of the Preceding Pages 170 ELEMENTS OF PHYSICS. CHAPTER I. INTRODUCTORY. 1. Definition of Physics. — Physics is the science of. mechanics, heat, sound, light, electricity, and magnetism,. Everybody knows something about these things before he begins to study them in a regular way, but sometimes he does not know them by the names which are given to them in books. 2. Use of Physics. — In sailing boats or flying kites, in walking or swimming, in almost any kind of bodily work or play, we have to do with physics, that part of physics which is called mechanics. We learn to do many mechanical acts very well indeed by observation and experience, without thinking very much about them or knowing exactly how we do them ; but when w T e have to do something that we have never done before and have never seen any one else do, something, perhaps, that nobody ever did before, we must think and study. 3. Illustrations. — Thus no man who has practiced swim- ming need study mechanics to improve himself in that art; but if he would build a ship and make it swim through all kinds of weather and water, he must study mechanics a good deal in order to know what size and shape to give the various parts, how best to put them together, and how to 2 PHYSICS. balance the whole. If it is to be a steamship, some one must know a good deal about heat, to make the furnaces and boilers right. We must know about magnetism to make and use the ship's compass. We may use electricity to furnish light on board at night. We must study sound in order to make the best fog-signals to guard against colli- sions and shipwreck in thick weather. In short, the science of physics in all its main divisions is not only a very interesting study to many minds, but it is of great use to civilized mankind. Man has become civilized, indeed, not by merely imitating what his fathers have done, but by studying, that is, observing and thinking, and gradually improving upon the work of those who have gone before him. 4. Qualitative Knowledge. — Everybody knows that a piece of wood will float in water, and that a stone will sink. Everybody knows that if a stick and a stone are tied together and put into water, the stone tends to sink the stick, and the stick tends to float the stone. This kind of knowledge is called qualitative. It tells in a general way how the stick and the stone act toward each other. 5. Quantitative Knowledge. — Some people know enough about the laws of flotation to calculate with accuracy how large a stick of a known kind of wood will be needed to float a stone of known size and weight. They have what is called quantitative knowledge of the matter. They can tell how much the stone will pull down on the wood, and the wood pull up on the stone, when the two are together in water. Everybody knows that a beam has a quality which we call strength — it can bear a load. This is qualitative knowl- edge. Everybody knows that a thick beam is stronger than a thin beam. This is quantitative knowledge of a kind, a rather indefinite kind. Some people know how much INTROD UCTOR Y. 3 stronger the thick beam is than the thin beam. They have a more complete quantitative knowledge. 6. Comparison of the Two Kinds of Knowledge. — It is evident that quantitative knowledge is more useful than mere qualitative knowledge. The former includes the latter. The qualitative man says, " I want to build a house. I shall need some land to put it on, and some beams and boards and bricks," etc. The quantitative man says, " Yes, you will need ail these things, but if you don't make your ideas more precise you will have a lumber-yard when you have done, not a house." 7. Object of this Course in Physics. — The knowledge of physics which children and older people get by merely knocking about in the world is mostly qualitative. The course of work laid out in this book is intended to add greatly to the pupil's stock of this kind of knowledge, and to do something more. It aims to make the pupil familiar with quantitative work, and to give him a considerable amount of quantitative knowledge. We shall begin at once with simple introductory measure- ments. All the exercises of this chapter are called Prelim- inary Exercises. Measurement of Distance. 8. The Straight Line. — The line to be measured may be along the edge of a table (or sheet of paper) from one fine scratch to another, a distance of about 15 inches. It is a great convenience to have all the pupils measure equal distances; accordingly, the teacher is advised to lay off these distances by some method like the fol- lowing: A carpenter's square is placed a~ along the edge of the table as in Fig. 1, FlG - 1 - and while it is held firmly in place a fine light scratch is 4 PHYSICS. made with the point of a sharp knife-blade at right angles with the edge of the table at the points a and b. The dis- tance from a to i is the one to be measured by the pupil. The first-described method of using the measuring-stick in the following Exercise is not a good method, but it is one that many will use if they are not properly instructed. The second method is a good one, and the two are here brought together in order that the pupil may see at once the right way and the wrong way to use such an instrument. Much of the interest and profit of this Exercise will come from the opportunity given each pupil to compare his own work with that of others. EXERCISE A. MEASUREMENT OF A STRAIGHT LINE. Apparatus : A short measuring-stick (No. 1) * and a meter-rod (No. 2). To each pupil is given a measuring-stick about one-fourth as long as the distance from a to b. We will suppose that these sticks are made by sawing a meter-rod, graduated to millimeters, into ten equal parts. The saw-cut will usually leave the divisions at the very ends of the sticks imperfect, and these divisions should not be used in the measurements. Let each pupil measure his distance at least twice carefully, with his measuring-stick laid flat upon the table, the marks upon the stick being thus horizontal, and let him write upon the blackboard the results of his two measurements. Then let each pupil measure his distance twice again, this time placing his measuring-stick upon its edge, so that the marks upon it will be vertical, making a light, fine mark upon the table with a sharp pencil to set the stick by, whenever it is moved forward a length. These new measurements are also to be placed upon the blackboard under the first ones. Finally let each pupil measure his whole distance at once with his meter-rod and write this last measurement with the others. * Any piece of apparatus to be used in the Exercises will usually be referred to by the number it bears in the list of apparatus given at the end of the book. INTBODUCTOBT. 5 9. Errors. — To judge of the accuracy of a set of measure- ments it is not enough to know how much these differ among themselves, for the importance of the difference usually depends upon the ratio which the difference bears to the whole quantity measured. A thousandth part of an inch might be a very serious difference to a watchmaker in the measurement of some small cylinder, while a difference of several inches in the measurement from one mile-post to another would be of little consequence. The pupil should therefore form the habit of comparing his errors, or the differences of his measurements, with the whole quantity that he had to measure. Let us suppose, for instance, that in Exercise A the measurements made by one pupil are 37.30 cm., 37.00 cm., and 37.10 cm. The greatest difference is found between the first and second. It is 0.3 cm., and its ratio to 37.15 cm., which is midway between 37.30 cm. and 37.00 cm., is 0.0081 — . We see, then, that the difference between the two measurements of the line is about eight one-thousandths, not quite one per cent, of the length of the line. Each pupil should make a similar calculation from his own measurements in Exercise A. 10. Units and Standards of Measurement. — The impor- tance of having definite units of length, of weight, etc., so that any man in dealing with his neighbor may know just how much is meant by the words foot, pound, and the like, is so great that in all civilized countries the exact meaning of such words is fixed by law, and very great care is taken to make and preserve government standards, as they are called, standard yard-sticks, standard pound-weights, for instance, with which as patterns the measuring instruments used in business are compared and tested. Interesting accounts of the foot, the yard, the meter, etc., can be found in almost any encyclopedia. 6 PHYSICS. Meter-rods for school use are in many cases marked off in inches on one side. With the information given by such a rod, the class can find how many centimeters are equal to one inch. This number carried to two places of decimals is accurate enough for most purposes. 11. The Right Triangle. — Eight triangles, that is, tri- angles having one right angle (see Fig. 2), are much used in the study and application of physics. In such triangles Fig. 2. there is a simple and important relation between the length of the longest side and the length of the other two sides. Part 1 of the following Exercise B is intended to show this relation and at the same time to give practice in measure- ment. 12. Circles. — The relation between the length of the diameter of a circle and the length of its circumference is also very frequently used in physics. Part 2 of Exercise B has to do with this relation. EXERCISE B. THE LINES OF THE RIGHT TRIANGLE AND THE CIRCLE. Apparatus : A 30-cm. measuring- stick (No. 3). A sheet of paper upon which is drawn carefully a right triangle no side of which is less than 10 cm. long. (No two pupils should use exactly similar INTRODUCTORY. 7 triangles.) A cylinder of wood 4 or 5 cm. in diameter (No. 4). A narrow straight-edge strip of thin paper. Part 1. Measurement of the Sides of a Right Triangle. — Let each pupil measure very carefully all the sides of his triangle, not being content to read to the nearest 0.1 cm., but striving to note and measure 0.05 cm. distances, if he can do so without hurting his eyes. After the measurements are made, square the length of each side and compare the greatest square with the sum of the other two squares. The conclusion drawn from this comparison must not be extended to triangles which are not right-angled. Part 2. Measurement of the Circumference and Diam- eter of A Circle. — Measure carefully the diameter of one end of the cylinder. Then wrap the strip of paper around the curved sur- face of the cylinder at the same end, and mark upon the edge of the strip the point where the second winding of the paper begins to overlap the first. Then unfold the paper and measure upon it that distance which extended once around the cylinder. Then divide this distance, which of course is equal to the circumference of the circle, by the length of the diameter. The ratio thus obtained is one which it is important to know, although we shall not have much occasion to use it in this book. Mathematicians, physicists, and engineers use it so much that thay have a particular sign, it, to denote it. This sign is a Greek letter and is called pe by students of Greek, but when used as just described it is often called pi to distinguish it from p. 13. Discussion of Exercise B. — The measurements of Exercise B may be discussed somewhat as follows: The square of the longest side of the triangle is found by one pupil to be 404.01, and the sum of the squares of the other two sides 406.05. If the two short sides were measured correctly, how large an error in the measurement of the longest side would cause the disagreement here found ? The long side was measured as 20.10 cm. If it had been called 20.20 cm., its square would have been 408.04, which is about as much too large as the square actually found is too small. If the distance had been measured as 20.15 cm., the square would have been 406.02, a quantity very close 8 physics. indeed to the sum of the other two squares. If, therefore, the original error lay entirely in the measurement of the longest side, this error must have been very nearly 0.05 cm. Of course the error may have been made in measuring the other sides, or in drawing the triangle, or in all parts of the work. An error which mistakes 20.15 for 20.10, or 201.5 for 201.0, or 2015 for 2010, is called in each case an error of 5 parts in 2015, or 1 part in 403, or an error of about i per cent (see remarks following Exercise A). QUESTION. In the case of the circle, which would make the greater difference in the result (circumference -5- diameter), an error of 0.05 cm. in the measurement of the diameter or an error of 0.10 cm. in the measure- ment of the circumference ? Measurement of Area. 14. Unit of Area. — Thus far we have been measuring lines. To measure a line, as we see, is merely to find out by trial that it is so many centimeters or inches long. A line 10.6 cm. long is one that could be divided into ten full centimeters and six tenths of another centimeter. We here call the centimeter our unit of length. If we have to measure a surface, the whole table-top, for instance, our task is to find the number of square centi- meters, or square inches, or square feet, that would be required to cover it, or that it would make if it were cut up without waste into squares. In this case the square centimeter, or square inch, or whatever square we choose to take, is the unit of area. We might set about to measure surfaces by actually placing a little square, a square centi- meter, for instance, on the given surface, marking a line* close around it, then moving it to a new place, marking around it, and so on till we had marked off the whole sur- face \n\o little squares, with perhaps some fractions of INTRODUCTORY. 9 squares. But this is not the common or the best way of measuring surfaces. The common way is to measure the length of certain lines on the surface and from the lengths of these lines to calculate the extent of the surface. 15. Measurement of Rectangles. — If the surface is in the form of a rectangle, like Fig. 3, it is plain that we have merely to multiply the number of units, centimeters let us say, in the length by the number of centime- ters in the width, and the result, 8x1 = 32 in this figure, is the number of square centimeters into which the surface can be divided. This is called the extent or area of the surface. In the next Exercise we shall undertake to find rules for the measurement of surfaces not quite so simple in shape as the rectangle shown in Fig. 3. These will be of the class called parallelograms. Fig. 3. 16. Parallelograms. — A parallelogram is a flat figure bounded by four straight linos, each line being parallel to the line opposite. Thus A and B in Fig. 4 are parallelo- B Fig. 4. grams. A is what we have just called a rectangle, and we have seen how to find the area of any rectangle, but B is not quite so simple at first sight. A parallelogram like B, which contains no right angle, is called an oblique parallel- ogram , 10 PHYSICS. EXERCISE C. AREA OF AN OBLIQUE PARALLELOGRAM. Apparatus: The 30-ciu. measuring-stick (No. 3). An oblique parallelogram of paper about 20 cm. long and 10 cm. wide. (One of the straight-edged rulers (No. 24) may prove useful in this Exercise.) Draw upon the paper figure a line like c in Fig. 5, taking care to make a right angle with the top line and the bottom line, and then cut or tear the paper along the line c. Take the small piece thus removed and join it to the larger piece, in such a way as to Fig. 5. make a figure that you know how to measure. Measure the length and width of the figure thus formed and calculate the extent of its surface. Then put the two pieces together as they were at first and ask yourself whether you could not, if another oblique parallelogram were given you, find the extent of its surface without cutting it. For the Class-room. Estimate without measurement the length and width of some visible and convenient rectangles, a book-cover, a table-top, a win- dow, etc., and calculate the areas from these estimated dimensions. Then take the true dimensions and calculate the true areas. Measurement of Volume. 17. Unit of Volume. — We have now to speak of the measurement of volume. The unit of volume may be the cubic centimeter, or the cubic inch, or the cubic foot, etc, We shall generally use the cubic centimeter as our unit. We mean, then, by the volume of a body the number of cubic centimeters that could be made of that body if it were cut up without waste, as one might cut up a large piece of clay or putty. 18. Rectangular Bodies. — In the case of a body whose surface is made up of rectangles, a brick, for instance, it is INTBODUCTORY. 11 easy to see how the volume may be calculated, if we know the length and the width and the thickness. We have volume = length X width X thickness, 19. Irregular Bodies.— If the body is of less regular shape, like an ordinary stone or a lump of coal, it is not so easy to calculate its volume from measurements of length, width, and thickness. There is, however, a very easy way of finding the volume of such a body by the use of water, as will presently be seen. 20. Volume of Water.— It is easy to find the volume of a quantity of water in several ways. One way is to pour the water into a rectangular box. Then we can measure its length and width and depth and calculate its volume. Another way is to pour it into a glass measuring-dish having marks upon it to tell the number of cubic centimeters required to fill it to certain depths. Another method is to weigh the water, for i t is known that one cubic centimeter of water weighs one gram. Indeed this is the definition of one gram, the to eight of a cubic centimeter of tvater.* If the balance which we use for weighing reads in ounces instead of grams, we shall have to remember that 1 oz. = about 28.3 gm., so that 1 oz. of water will be 28.3 cubic centimeters. We shall commonly find the volume of a body of water by weighing. 21. The Water Method. — We will now try the water method of finding the volume of a body, a rectangular solid. We shall find its volume by the water method and also by direct measurement and calculation, and then see how well the two results agree. This will test the water method, and if we find it to work well, we can use it with irregular solids which we cannot measure directly. * To be exact one must add at 4° of the centigrade scale of tempera- ture. For the purpose of this book such exactness is unnecessary. 12 PHYSICS. EXERCISE D. VOLUME OF A RECTANGULAR BODY BY DISPLACEMENT OF WATER, Apparatus : A brass can (No. 5) called G in Fig. 6 A small catch- bucket (No. 6) called p in Fig. 6. A spring-balance (No. 7). A rectangular block of wood (No. 8) so loaded as to sink ia water. Closing the overflow tube t of the can G, pour water into G until it is filled nearly to the brim. Then open the tube and let all the water flow out that will do so, catching it in the small can p, The large can should rest steadily upon the table, but the small one is better held in the hand when the flow begins, otherwise some water may be spilled. The flow should stop rather suddenly at last, with little or no drip. Throw away all the water thus caught in p and then weigh p on the spring-balance to the nearest gram or the nearest twentieth of an ounce, according to the graduation of the balance.* Then, closing the tube t as before, lower into the can G the wooden block until it rests upon the bottom. Then, or sooner if the can G seems Fig. 6. likely to be overflowed, open the tube t, and as before catch the water that runs out in the small can p. The water, Fig. 7, now * Ordinary small spring-balances now in the market are often marked off in half-ounce divisions, which are about £ inch long. The pupil will learn to estimate the position of the pointer when it falls between two lines, so as to read to about ^ of au ounce. INTRODUCTORY. 13 stands just as high in G as it did just before the block was put into it. The block has crowded out into the can p just its own bulk of Fig. 7. water. If, then, we can find the volume of the water that the block drove over into p, we shall have the volume of the block itself. Weigh p and the water it contains. Weight of small can and water = 11 " " " empty = " " water alone = If the weight as thus found is in grams, it is equal to the number of cubic centimeters in the block. If the weight as thus found is in ounces, we must multiply the number of ounces by 28.3 in order to find the number of cubic centimeters in the block. Now measure carefully the length, width, and thickness of the block and calculate the number of cubic centimeters it contains from these measurements. (Experiments for finding the volumes of irregular bodies by the water method may well be postponed till the next Exercise, which would otherwise be a very brief one. Potatoes, stones, lumps of coal, etc., of suitable size may be used for these further experiments. Practice for the Eye. A line 10 inches long is drawn on a blackboard with a cross-line at any point, and the members of the class estimate the distance from either end to the cross-line. Practice like this helps toward accurate reading of the spring-balance. 14 PHYSICS. QUESTIONS. (1) The true length of a certain line is 16.4 cm. One person meas- ures it as 16.6 cm., another as 16.3 cm. How great is the error of each in per cents of the true length ? (2) A certain rectangle is 50 cm. long and 20 cm. wide. It is meas- ured by one person as 50 cm. long and 20.2 cm. wide, and by another person as 50.2 cm. long and 20 cm. wide. If the area is calculated from each set of measurements, how great (in per cents) will the error be in each case ? (3) A certain rectangle has a base 100 cm. long and an altitude of 40 cm. Which will cause the greater error in the estimated area, an error of 2 cm. in the base or an error of 1 cm. in the altitude ? (4) A rectangular solid is 40 cm. long, 30 cm. wide, and 20 cm. thick. How great (in per cents) is the error made by calculating the volume from measurements which give 41 cm. for the length, 31 cm. for the width, and 19 cm. for the thickness ? CHAPTEE II. DENSITY AND SPECIFIC GRAVITY. 22. Definition of Density. — The weight of unit volume of a substance is called the density of the substance. If we know the density of a substance we can calculate the weight of any volume of that substance. Engineers and other scientific men often have to find by this method the weight of objects which it would be inconvenient to weigh. The weights of buildings and bridges, for instance, are found in this way. Books used by scientific men contain tables giving the densities of many different substances. The density of a substance may be expressed as the weight in grams of one cubic centimeter, or as the weight in pounds of one cubic foot, or in any one of many other ways. For brevity, we call the first method of expression just given the density in grams and cubic centimeters, and the second, the density in pounds and cubic feet. The following Exercise will make the matter plainer, and will give good practice in measuring and weighing. EXERCISE I. WEIGHT OF UNIT VOLUME OF A SUBSTANCE. Apparatus: A block of wood (No. 9). A spring balance (No. 7). A measuring-stick (No. 3). Thread for suspending the block. Find the weight of the block in grams and also in ounces. Measure the length of each of the four edges which are parallel to the grain of the wood, take the average of these measurements and call it the length of the block. 15 16 P&TS1C3. Measure the length of each of the four long edges which are cross- wise to the grain of the wood, and call the average of these four measurements the icidth of the block. Measure the length of each of the four short edges and call the average of these four measurements the thickness of the block. The weight in ounces is to be turned into pounds. From the length, width, and thickne-s in centimeters the length, width, and thickness in feet may be found by the rule that 1 ft. = 30.5 cm., but it is shorter to find the volume in feet from the volume in cubic centimeters by the rule that 1 cu. ft. = 28300 cu. cm. Calculate, 1st, how many grams, or what part of a gram, 1 cu. cm. of the block weighs ; 2d, how many pounds, or what part of a pound, 1 cu. ft. of such wood weighs. 23. Density of Water. — The density of water in grams and cubic centimeters is 1; that is, 1 cu. cm. of water weighs 1 gm. (see § 20). The density of water in pounds and cubic feet is very nearly 62.4; that is, 1 cu. ft. of water weighs 62.4 lbs. These numbers for water should be com- mitted to memory. QUESTIONS. 1. What ratio is found from the results* of Exercise 1 between the density of wood in grams and cubic centimeters and its density in pounds and cubic feet? 2. How does this compare with the ratio of the two densities of water, as given above ? 3. If the ratio is the same for the wood as for water, is this a mere coincidence, or is the same thing true in the case of other substances ? PROBLEMS. (1) If a piece of iron 10 cm. long, 8 cm. wide, and 7 cm. thick weighs 4000 gm., what is its density in gm. and cu. cm, ? What is its density in lbs. and cu. ft. ? (2) The density of mercury in gm. and cu. cm. is about 13.6. How many lbs. would 1 cu. ft. of it weigh ? * It is well to take the average of the results found by the various members of the class. DENSITY AND SPECIFIC GRAVITY. 17 24. Weight. — Before going farther we need to think carefully about the meaning of the word weight, which we have already used a number of times, and shall have to use very often. The word has two meanings. Sometimes when we speak of the weight of a body we mean the amount of the body, as when we speak of 10 lbs. of butter or 100 lbs. of iron. At other times we mean by the weight of a body the amount of the earth's downward pull upon that body, as shown by the spring-balance, for instance. It is somewhat hard to remember this distinction, because the units in which we tell the amount of a body have the same name as the units in which we tell the pull which the earth exerts upon the body. For instance, we say that the earth exerts a pull, ov force, of 5 lbs. upon 5 lbs. of wood, or 5 lbs. of coal, or anything which consists of, or is, 5 lbs. of substance. Often when we use the word weight it makes no difference which of its two meanings we have in mind, but sometimes it does make a difference. Thus, when we put a body under water, as we shall do in the next Exercise, and say that it appears to lose weight in going from air to water, we do not mean that there appears to be any less of the body in water than there was in air. We mean that it requires a smaller pull of the spring-balance to keep the body from sinking in water than it does to keep it from sinking in air. 25. Mass. — In strict scientific language, the word mass is commonly used in speaking of the amount of a substance, and the word iveight in speaking of the earth's pull upon that substance. For example, a piece of iron the mass of which is 50 lbs. is subject to a iveight * of 50 lbs. exerted by the earth. * Such distinct ions, which use words in a scientific sense different from the popular everyday sense, are often necessary in science, but it would be rather absurd to try to make the popular use of the words agree with the scientific use in all cases. 18 PHYSICS. Specific Gravity. 26, Definition. — It is often convenient to know the ratio which the weight of a body dears to the tueight of an equal bulk of ivater. This ratio is called the specific gravity of the body. Gravity comes from a Latin word gravis, mean- ing heavy. Specific here means distinctive, or particular. The specific gravity of a body is its particular heaviness — - the degree of heaviness which distinguishes this body from other bodies of the same size but different weight. 27. Loss of Weight in Water. — In finding specific gravities it is a common practice to weigh bodies under water. The use of this practice will be made plain by Exercise 3. The loss of apparent weight suffered by a body in going from air to water is shown in Exercise 2. EXERCISE 2. LIFTING EFFECT OF WATER UPON A BODY ENTIRELY IMMERSED IN IT. Apparatus: Overflow-can (No. 5). Catcli- bucket (No. 6) Spring- balance (No. 7). Loaded block (No. 8). Thread. Fill the can and let it overflow and drip as in Exercise D. Catch this overflow in the small bucket and throw it away. Then weigh the empty bucket in grams. Weigh the block in grams before immersing it in the water. Lower the block, still suspended from the balance, into the over- flow-can till it is entirely covered, catching the overflow and saving it. Weigh the block in the water, the balance being entirely above the water. Weigh the bucket with the overflowed water. Subtract the (apparent) weight of the block in water from its weight in air, and call the difference the loss of weight of the block in water, or the buoyant force exerted upon the block by the water. Find weight of the water in the small bucket, and compare this with the loss of weight of the block in water. DENSITY AND SPECIFIC GRAVITY. 19 If there is time, make a similar experiment with other "bodies. The law illustrated in this Exercise is called from its discoverer the law, or principle, of Archimedes. (See any encyclopedia for an account of Archimedes.) PROBLEMS, (1) A certain body weighs 100 gm. out of water and 50 gm. in water. How great is the volume of the body? (2) A certain body 5 cm. long, 3 cm. wide, and 2 cm. thick weighs 200 gm. in water. How much does it weigh out of water ? EXERCISE 3. SPECIFIC GRAVITY OF A SOLID BODY THAT WILL SINK IN WATER. Apparatus : The spring-balance (No. 7). The gallon jar (No. 10) nearJy filled with water. A lump of sulphur (No. 11). Thread. Weigh the sulphur out of water ; then in water. We know from Exercise 2 that a body immersed in water loses in apparent weight an amount equal to the weight of the water whose place it has taken. It is easy, therefore, to get from the two weigh- ings just made the ratio which we have undertaken to find in this Exercise. If time permits, find in this Exercise, by the same method that is used for the sulphur, the specific gravity of other solids that will sink in water — such as glass, coal, etc. QUESTIONS. 1. If the specific gravity of 1 cu. cm. of iron is 7, what is the specific gravity of 50 cu. cm. of the same kind of iron ? Of 1 cu. ft. of the same kind of iron ? 2. If the sp. gr. of lead is 11.3, what is the weight in grams of 1 cu. cm. of lead? What, then, is the density of lead in grams and cubic centimeters (see § 22)? 3. If the sp. gr. of a certain kind of wood is 0.7, what is the weight in lbs. of 1 cu. ft. of this wood? What, then, is its density in lbs. and cu. ft. ? 4. A certain body weighs 7 lbs. out of water and 4 lbs. in water. What is its specific gravity ? 20 PHYSICS. 28, Various Expressions for Specific Gravity. — By defi- nition we have ~ /. 7 7 Wt. of the body bp. grav. of a body = Wt. of an equal volume of water It is evident that the quantity written below the line in this definition may be expressed in other ways. We may write Sp. grav. of a body Wt. of the body Wt. of water displaced by the body when immersed? or ~ a _ Wt. of the body 1 ' Loss of weight of the body token immersed 9 or Sp. grav. Wt. of the body Lifting effect of water upon the body when immersed' These expressions all mean the same thing, bat some- times one of them is more convenient than the others. In the Exercise next before us we shall use the last form. EXERCISE 4. SPECIFIC GRAVITY OF A BLOCK OF WOOD BY USE OF A SINKER. Apparatus]: A rectangular block of wood (No. 9). The spring- balance (No. 7). The gallon jar (No. 10) nearly filled with water. A lead sinker (No. 12). Thread. We have to find two quantities, by experiment : 1st, the weight of the body ; 2d, the lifting effect of water upon it ichen immersed. Weigh the wood in air and record its weight. Now put the block into water, Jou see tliat it floats, fq mtik$ it DENSITY AND SPECIFIC GRA VITT. 21 stay under water you must hold it down. Try this, putting your fingers on the block. In this case, you see, the lifting effect of the water, when the block is wholly beneath its surface, is greater than the weight of the block. We must find out how much it is. We shalL use the lead sinker to hold the block under water, and we need to know the weight of the sinker alone under water. Weigh it in this position and record the weight. Now suspend the block from the balance * and the lead sinker from the thread under the block, and consider how much the two, block and sinker, would weigh in the position shown by Fig. 8, the block out of water and the sinker in water. You can tell this from the weighings already made. Write it down. Wt. of block in air -f Wt. of sinker in water =....+.... Now lower the block and sinker till both are cov- Fig. 8. F*g. 9. * The success of a difficult experiment like this depends greatly upon the care with which the details of the work are thought out by the teacher. The following method of attaching the block to the bal- ance is recommended : Take a thread two feet long and tie the ends together. Then make of it a slip- noose by passing one end, I (Fig. 9i, through the other end, k. The block may then be placed in the noose and the loop I slipped upon the hook of the balance, but to prevent slipping when the lead weight is to be suspended from the loop below the block it is well to pass tlie loop I twice through at z- 22 PHYSICS. ered by the water, and weigh the two together in this position and record : Wt. of block and sinker together in water = . . . . Just before the block entered the water, the sinker being already in, the weight was .... Just as soon as the block also was cov- ered the weight was only .... The difference is the lifting effect of the water upon the block. We have now all that we need for cal- culating the specific gravity of the block by means of the formula, _ Wt. of block y " ~~ Lifting effect of water upon block immersed' QUESTIONS. (1) A brick-shaped body 20 cm. long, 10 cm. wide, and 5 cm. thick weighs 1500 grams. What is its density in gram and centi- meter units ? What would be the weight of an equal bulk of water ? What, then, is the specific gravity of this body? (2) A body whose volume is 700 cu. cm. has the density 8 in gram and centimeter units. How much does it weigh? What is its specific gravity ? (3) A body 20 ft. long, 10 ft. wide, and 5 ft. thick weighs 93,600 lbs. What is its density in pound and foot units ? What would be weight of an equal bulk of water, one cu. ft. of water weighing 62.4 lbs ? What, then, is the specific gravity of the body ? (4) A body whose volume is 700 cu. ft. has the density 499.2 in pound and foot units. How much does it weigh ? What is its specific gravity ? (5) What numerical relation do we find in these problems, and in those of page 19, between density in gram and centimeter units and specific gravity ? (6) What relation do we find in the same problems between density, in pound and foot units, and specific gravity? 29. Flotation. — Thus far we have been considering the action of water upon bodies entirely immersed in it. We shall now have to do with floating bodies. DENSITY AND SPECIFIC GRA VII Y. 23 EXERCISE 5. WEIGHT OF WATER DISPLACED BY A FLOATING BODY. Apparatus: The same as in Exercise 2, with the exception of the sinking body, which is here replaced by one that floats (No. 4). Weigh the cylinder, in grams, in air. Find, in grams, the weight of water which it displaces from the overflow-can. Compare these two weights. It will be well to repeat the overflow operation carefully a number of times. The fact shown in this Exercise concerning the relation between the weight of a floating body and the weight of water displaced by it should be firmly fixed in the experi- menter's mind. It leads to a method of finding the specific gravity of floating bodies. EXERCISE 6. SPECIFIC GRAVITY BY FLOATING METHOD. Apparatus : The gallon jar (No. 10) nearly filled with water. A slender wooden cylinder (No. 13). A support for holding this cylin- der upright in water (No. 14). A measuring- stick (No. 3). If a cylinder floated upright with its top just level with the top of the water, we should at once know its specific gravity to be 1. If it floated just half in and half out of water, we should know its specific gravity to be 0.5. The cylinder that we have to use will not float all in water or exactly half in water, but if we float it, and find the length of tlie part then in the water, we shall, by comparing this with the length of the whole cylinder, find some way of ascertaining the specific gravity of the cylinder, Measure the length of the whole cylinder. Float the cylinder in the jar (Fig. 10), keeping it upright by 24 PHYSICS. means of the holder, which is attached to the side of the jar. Jeggle the cylinder to make sure that it is free to take its proper posi- tion. After each joggling it should come to rest at the same depth as before. The ruigsof the holder must not grip the cylin- der at all. When sure that the cylinder floats as it should, meas- ure the length of the submerged part, from the bottom of the cylinder up to the flat surface of the water. To find the specific gravity from the two mi asurements now made, beg n by recalling the fact (see Exercise 5) that the water displaced by the float- ing cylinder weighs just as much as the cylinder itself. How many times is the length of the submerged part of the cylinder contained in the whole length ? How ma iy times the weight of the cylinder would be the weight of a like cylinder of water ? How great, then, do you find Fig. 10. the specific gravity of the wooden cylinder to be? PROBLEMS AND QUESTIONS. (1) A block whose specific gravity is 0.6 floats in water. How much of it is below the surface ? (2) A block whose volume is 1000 cu. cm, and whose specific gravity is 0.4, floats in water. How many cu. cm. of the block are below the surface ? (3) A block that weighs 4 oz. in air is fastened to a sinker that weighs 6 oz. in water, and the two together weigh 3 oz. in water. What is the specific gravity of the block ? DENSITY AND SPECIFIC GRAVITY. 25 (4) A block whose specific gravity is 0.5, and which weighs 100 gm. alone in air, is fastened to a sinker that weighs 150 gin. alone in water. How much will both together weigh in water? (5) A certain body has the density 187.2 in pound and foot units. What is its specific gravity ? (6) Is the specific gravity of the human body much greater or much less than 1 ? (7) Why doei filling the lungs w r ith air help one to float in water ? EXERCISE 7. SPECIFIC GRAVITY OF A LIQUID: TWO METHODS. Apparatus: The gallon jar (No. 10) nearly filled with water, and the smaller jar (No. 15) nearly filled with a solution of sulphate of copper.* The small glass bottle (No. 16). The spring-balance (No. 7). Thread. First Method. Weigh the bottle empty. Dip the bottle into the jar of sulphate of copper and let it fill with the liquid. Holding the bottle over the jar, put the stopper in place, thus crowding out the excess of liquid, then wipe the outside of the bottle and weigh it carefully with its contents. Pour the sulphate of copper back into its jar, then fill the bottle with water, just as it was before filled with the other liquid, and again w r eigh the bottle and its contents. From the three weighings now made the specific gravity of sul- phate of copper can easily be found. Second Method. We found in Exercise 2 that a body going from air into water lost in apparent w r eight an amount equal to the weight of its own bulk of water. So a body going from air into a solution of sulphate of copper will lose in apparent weight an amount equal to the weight of its own bulk of the solution. This gives a method of finding the specific gravity of the solution. As a body to be weighed first in air, then in water, then in the solution, we will use the bottle with enough water *This solution may be made by putting 2 lbs. of sulphate of cop- per crystals into about 3 qts. of warm water in a glass vessel and stirring occasionally till the crystals are dissolved, 26 PHYSICS. in it to make it sink in either liquid. We may, indeed, use the bottle full of water, just as it was left at the end of the first part of this Exercise. EXPERIMENTS. 1. Exhibit and show in operation two graduated glass hydrometers — one for determining the specific gravity of liquids less dense than water (App. No. XI), the other for use with liquids more dense than water (App. No. XII. \ 2. Show in a bottle together several liquids of different specific gravities that do not tend to mix with each other ; for instance, mer- cury, chloroform, water, and kerosene. 3. Take a small tumbler containing some mercury and drop into it a piece of iron. Do not put into it gold or silver, as mercury at- tacks these metals. 4. Place a dry sponge on water. It floats lightly, but is the spe- cific gravity of the fibres of the sponge greater or less than that of water ? To answer this question push the sponge beneath the sur- face. What rises from it ? Squeeze the sponge very hard till noth- ing more seems to come from it. Now will it rise to the surface when released ? QTJESTIONS. 1. A glass sphere which weighs 100 gm. in air weighs 60 gm. in water and 40 gm. in sulphuric acid of a certain strength. What is the specific gravity of the glass ? What is the specific gravity of the sulphuric acid? 2. A vessel contains a layer of water 10 cm. deep and above this a layer of kerosene (sp. gr. 0.8) 10 cm. deep. What is the weight of a cube, each edge of which is 10 cm. long, that, if placed in this ves- sel, will sink till one-half its volume is in the water and one-half in the kerosene? Ans. 900 gm. What is its specific gravity? Arts. 0.9. 3. A certain ship weighs with its cargo 10,000 tons. (a) How many cubic feet of fresh water would it displace ? (b) How many cubic feet of sea- water of specific gravity 1.026 would it displace ? 4. If 1 cu. cm. of mercury weighs 13.6 gm. and 1 cu. cm. of cork weighs 0.25 gm., how deep will a cylinder of cork 20 cm. long sink, when placed on end in mercury ? DENSITY AND SPECIFIC GBAVITT. 27 5. Which has the greater specific gravity, cream or skimmed milk? 6. A piece of cloth thrown upon water will float at first and after- ward sink. Why does it not sink at once ? 7. Can the pupil tell from his own observation which of the follow- ing substances are more dense and which are less dense than water : kerosene-oil, ordinary lubricating- oil, butter, cheese, potatoes, eggs, meat, ice, india-rubber ? CHAPTEE III. FLUID-PRESSURE. 30. Fluids. — Water and air readily floiv from one position or shape to another. They are examples of that class of substances called fluids. Fine sand and other like sub- stances flow in a certain way, but examination shows them to consist of little hard or tough particles very different from equally small particles of water. Fluids are divided into liquids and gases. Water is an example of the liquids ; air an example of the gases. 31. Fluid-pressure. — Fluids settle snugly around solid, that is, non-fluid, bodies placed in them and act upon these bodies with a peculiarly even pressure. We shall now make some experiments with liquid-pressure and later with gas- pressure. EXPERIMENTS WITH PRESSURE-GAUGE. Fill the gallon glass jar (No. 10) with water to a level about one inch from the top. Close the smaller end of a student- lamp chim- ney tight with a good cork stopper. Make the pressure-gauge (No. I.) ready for use by the following operation, having first put on a fresh rubber diaphragm if necessary : Release the glass tube from the rubber tube and wet the whole length of the glass tube inside with water, leaving within it a column of water about one-half inch long to serve as an index.* Hold the gauge itself under water for a little time before reconnecting the glass tube with the rubber tube, *It may be necessary to use water colored by some aniline dye be- fore a large class. 28 FLUID-PRESSURE. 29 in order to allow the air within the gauge to come to the temperature of the water. On reconnecting the glass tube leave the water-index near the rubber tube. Different Letels. — Xow push the gauge down into the jar and raise and lower it repeatedly in the water, keeping the glass tube with the water-index horizontal, and let the class determine from the movements of this index whether the pressure of the water against the rubber diaphragm increases or decreases when the gauge is pushed deeper in the water. Different Directions. — Rest the bottom of the supporting pillar of the gauge upon the bottom of the jar, and, still keeping the glass tube horizontal, turn the upper pulley so that by means of the rubber band the lower pulley will be turned and the rubber diaphragm will face downward, sidewise, and upward in succession, its centre re- maining practically unchanged in position. Let the class determine by watching the water-index whether the pressure upon the rubber diaphragm is any greater when it faces upward than when it face? downward or sidewise. Different Points on the S^me Leyel. — Push the closed end of the lamp-chimney down into the water till it is near the bottom of the jar. Move the gauge face about, without changing its level, so as to bring it under this closed end. Move it now out of and now into this position, thus changing the depth of water immediately above it from one-half inch or less to several inches. Let the class determine by watching the index whether such changes of position, without change of level, make any difference in the pressure against the gauge- face. We shall make considerable use farther on of the facts brought out by these experiments. Just here we can see that they explain, at least in a general way, why a body immersed in water weighs, or appears to weigh, less than when in the air. For we see that there is an upward pressure of the water against the under side of the body, and that this upward pressure is greater than the down- ward pressure against the upper side of the body. 32. Slight Effect of Pressure upon the Density of Water. — Having seen that there is greater pressure on low levels 30 PHYSICS. than on high levels in water, we may well ask whether this greater pressure crowds the particles of water closer together on the low levels, thus making the water denser than on high levels. In fact there is an effect of this kind, but it is so slight that we need take no account of it in any ordi- nary case. It is very difficult to compress water much. EXPERIMENT. Fill a bottle with water and close it with, a rubber stopper having one hole through it. Then, holding the stopper firmly in place, push down into the hole a solid brass rod of a size to fit rather closely. The bottle will probably be broken by this effort to compress the water within it. (App. No. II.) 33. Uniform Increase of Pressure with Depth. — We have not made, and cannot well make with the gauge used, any accurate measurement of the rate at which pressure changes with change of level in water. The fact is, however, that if we place a surface of 1 sq. cm. horizontal at any depth in water the column of water just above it is resting upon the given surface.* If we carry the given surface down 1 cm. farther, we now have resting upon it a load somewhat greater than before, greater by the weight of the additional 1 cu. cm. of water which is now above it. As 1 cu. cm. of water weighs 1 gm., the pressure upon a surface of 1 sq. cm. changes by 1 gm. for each 1 cm. change of level in the water. QUESTIONS. A cubical box, 10 cm. along each edge, has extending from its top, as in Fig. 11, a tube 15 cm. tall and 1 sq. cm. in cross-section (inside). (1) If the box, but not the tube, is full of water, how great is the water-pressure on the whole of the bottom ? * The pressure upon the given surface may be greater than the weight of the column of water resting upon it, for there may be, and usually is, a downward pressure of air or something else upon the top of the water-column. FLUIb-PRBSStmE. 31 (2) If the tube as well as the box is full of water, how great is the pressure upon that one sq. cm. of the bot- tom which lies just beneath the tube ? (3j Is the pressure equally great per sq. cm. at other parts of the bottom ? (4) How much is the total pressure now on the bottom of the box? (5) How great is the pressure per sq. cm. at the top of the box just at the bottom of the tube ? (6) How great is the total upward pressure of the water against the top of the box ? (Disregard the atmospheric pressure upon the top of the water-column in all these questions at first. Afterward call this at- mospheric pressure 1000 gm. per sq. cm., and ask the same questions as before.) 34. Gas-pressure. — We have made some experiments with liquid-pressure. We must now begin to learn some- thing about air-pressure, which in many practical matters of e very-day life has a very important connection with water-pressure. We will at the start repeat in a slightly varied form a famous experiment first made by Torricelli, an Italian, about the middle of the seventeenth century. It is intended to show the pressure of the air about us, which is called atmospheric pressure. Fig. 11. EXPERIMENT.* Take two pieces of strong glass tubing about 0.7 cm. in inside di- ameter, one of them, about 1 m. long, closed at one end, and the other, about 20 cm. long, open at both ends, and connect them by means of a thick- walled piece of rubber tubing about 25 cm. long. The rubber tube should fit tight upon the glass tubes, and for greater security should be fastened on by means of wire or string. * This experiment can be more conveniently performed with a single straight glass tube if a mercury-well is available. 32 PHYSIOS. Holding the tubes thus connected (A pp. No. III.) by the free end of the short glass tube, the closed end of the long glass tube hanging down, pour mercury by means of a small funnel of glass or paper into the tubes, tapping or shaking them occasionally to dislodge air-bubbles, until the top of the mercury-column reaches the rubber tube. Then gently raise the closed end of the long glass tube until this tube points straight upward (Fig. 12), meanwhile holding the other glass tube upright and taking care that no mercury is spilled. During the latter part of this operation it will be noticed that the mercury begins to fall away from the closed end of the long glass tube, and finally several inches of this tube will be apparently empty.* But the mercury contin- ues to stand very much higher in the long glass tube than in the short one. y Fig. 12. 35. Explanation. — It was known before the time of Torricelli that if air was drawn from the upper part of a tube the lower end of which rested in water the water would rise in the tube, but the true reason for this was not known. Torricelli maintained, and Pascal, a Frenchman, showed by experimenting at different heights in the air, that the pressure of the atmosphere, due to its weight, accounted for the rise of liquids in a vacuum. We have only to think of the fact that the air, although its density is very small compared with that of water, has, because of its great quantity, a great weight, and we see that the air, pressing upon the mercury-surface in the shorter tube, balances the column of mercury in the long tube. 36. Amount of the Atmospheric Pressure — Barometer. — By measuring the difference in height of the two mercury- * Really this space contains a very little air, from the bubbles that were in the mercury-column before it was inverted, but so little that we may at present disregard it and consider the space above the mer- cury as empty. Such a space is called a vacuum, from a Latin word meaning empty. FL TJID-PBESS USE u surfaces we can get a measure of the atmospheric pressure. We find that the atmospheric pressure is about as great upon the surface of the earth as would be the pressure of a layer of mercury 76 cm. deep, or a layer of water about 10.3 m. deep, over the whole earth. The pressure per square centimeter at any given part of the earth's surface varies somewhat from day to day, and even from hour to hour. If we fasten the apparatus that has just been used to a suitable support, it will serve permanently as a rude barometer, indicating the variations of the atmospheric pressure. 37. Pressure in Different Directions. — Air-pressure, like liquid-pressure, is at any given point equal in all direc- tions, if the air is at rest. EXPERIMENT. J- -J Take a strong thistle-tube (No. IV.) of the shape shown in Fig. 13 and tie a piece of thick sheet rubber across the mouth, which may be about 1 inch in diameter. Make the covering air-tight by means of some cement, melted beeswax and rosin, for instance, poured in at the point J. Con- nect this thistle-tube by means of a thick- walled rubber tube to an air-pump (No. V.), and exhaust the air. The rubber cap, not being supported by air-pressure beneath, will now be pushed down by the atmos- pheric pressure into a deep cup-shape Pinch the rubber tube so that no air shall leak back into the thistle-tube, and then turn the mouth of the latter in all directions, sidewise, downward, and oblique. Observe whether the depth of the rubber cup changes during this operation, as it would do if the pressure upon it changed. Fig. 13. 34 PHYSIOS. 38. Air-pressure at Different Levels. — We should find by proper experiments that in air at rest, as in water at rest, pressure is equally great at all points on the same level. We should find, also, that the air-pressure diminishes with increase of height from the earth's surface, but, as the density of air is very little compared with that of water, it requires a considerable change of level to make much difference in the air-pressure. The rate at which atmospheric pressure decreases with increase of height being well known, it is a common prac- tice to estimate the height of mountains by noting the difference of atmospheric pressure at the summit and base. "Aneroid " barometers are frequently used for such work. Aneroid means without liquid. An aneroid barometer contains no mercury nor other liquid. It is an air-tight metal box with a flexible metal cover. The middle of the cover moves in or out slightly with changes of pressure, and its slight motions are magnified to the eye by various mechanical contrivances. Some aneroid barometers are about as large as ordinary watches and look much like them. 39. Difference between Liquids and Gases. — Liquids are much heavier than gases, in most cases. Most liquids are easily seen. Most gases are practically invisible. But per- haps the most striking difference between liquids and gases is a difference in compressibility. We have seen that it is difficult to compress water much, but it is very easy to com- press air. EXPERIMENT. Take the bent glass tube (No. VI.), closed at one end, and pour into it a little mercury, enough to fill the bend. At first the mercury will stand a little higher in the long arm, but by tipping the tube and letting ont a little of the air imprisoned in the short arm the level can be made nearly the same in both arms, as in Fig. 14. Now FLUID-PRESSURE. 35 measure the length of the imprisoned air-column, and write it under the letter V* on the blackboard. P. FXP. Fig. 14. The pressure upon this air is now, if the mercury- level is the same in both arms, equal to that upon the unim pris- oned air. It is as great a pressure as would be exerted by the weight of a column of mercury as tall as that in the barometer (Fig. 12). Take, then, a reading of this barom- eter and record this reading under the letter P. Pour in more mercury till the difference of level in the two arms is about 20 cm., then measure again the length of the inclosed air-column. Record this length under V, and record under P the present difference of mercury level plus the height of the barometer column. Proceed by stages in this way till the volume of the inclosed air- column is about one-half what it was at first. Multiply each number under Fby the corresponding number under P, and write the prod- ucts in the column headed VX P. 40. Boyle's Law. — An examination of the last column in the table of the preceding section will probably indicate a very simple law connecting pressure and volume in the case of a given body of air. This law is important, and should be remembered by the pupil. It is sometimes called Boyle's law and sometimes Mariotte's law. "We shall call it by the shorter name, Boyle* s lata. Illustrations and Applications of Fluid-pressure. 41. Principle of the Hydraulic Press. — The questions on pp. 30 and 31 have brought out the fact that pressure trans- * The length of the air-column is the same as its volume, if we take for our unit of volume the space contained in unit length of the tube. 36 physics. mitted through a small tube may extend to a broad surface beyond the tube so as to make the total pressure on this surface very great. The following experiment will show that similar effects can be produced with air-pressure. EXPERIMENT.* Take a common rubber football and blow air into it till it is about half filled, connecting a rubber tube with the key for greater con- venience in blowing (App. No. VII). Then rest one end of a board, Fig, 15. f S in Fig. 1 5, on the football and the other end upon a box or block of about the same he : ght. Then place a weight of 25 lbs. or more on the board nearly over the ball, holding the rubber tube attached to the key in such a way that the air cannot escape from the ball. Then blow through the tube into the ball and observe that you can in this way lift the weight. 42. Hydrostatic Press. — The preceding experiment illustrates the operation of the hydrostatic press, a machine in which a very great force is obtained, for lifting or compressing bodies, by pumping water through a small tube into a large cylinder, one end of which is closed by a movable stopper called & piston. (See § 204.) EXPERIMENTS. 1. Take again the pressure-gauge and the accompanying apparatus, § 31. Fill the lamp-chimney with water, and then, holding a card across the open end, invert the chimney, lower the end covered by the card into the water, and then remove the card. Most of the water will now remain in the chimney, although its upper end is nine or ten inches above the surface of the water in the jar. * An experiment with Gage's piston and cylinder apparatus may be substituted for this to show the same effect. FL UID-PRESSURE. 37 How does tlie pressure per sq. cm. inside the chimney on a level with the outside water surface compare with the pressure per sq. cm. at this outer surface, that is, the atmospheric pressure ? How, then, will the pressure per sq. cm. at points higher in the chimney compare with the atmospheric pressure ? After these questions have been answered by the aid of what the class already knows about liquid-pressure, test the correctness of the answer by means of the gauge. 2. Take a long narrow glass tube open at both ends, and dip one end into a vessel of water. Apply the lips to the other end and draw the water up till the tube is filled. In what sense is the water drawn up ? (The operation begins with an expansion of the lungs which les- sens the air-pressure within them. Then air runs from the place of high pressure, the tube, to the place of low pressure, the lungs. So the air-pressure within the tube is lessened.) 3. After nearly filling the tube as in Experiment 2 quickly close the top with a finger and then lift the lower end from the water. Uncover the top of the tube for an instant, then cover it again. Explain the behavior of the water during these operations. 4. Fill or nearly fill a tumbler or broad-mouthed bottle with water and then cover it with a sheet of thick paper. Hold the paper firmly in place with the hand and invert the tumbler ; then take away the hand that holds the paper. (As accidents may happen, the tumbler should be held over some large dish.) In this experiment it should be noticed that the paper does not press close against the rim of the tumbler after the inversion. It hangs rather loose, having dropped down or sagged a little, thus allowing the air above the water to expand a trifle, decreasing in pressure. 5. Fig. 16 (App. No. VIlIj shows a bottle closed with a rubber stopper through which two glass tubes, a and b, open at both ends, ex- tend. To one of the tubes, a, is attached a rub- ber tube, r. The bottle and the two glass tubes are full of water. By applying the lips to the outer end of the tube r water can b • " drawn " into the mouth, when the tube b is closed by a finger at the top? a \ Fig. 16. ('an this be done 38 PHYSICS. 6. Show some form of the Cartesian diver (No. XIV), explaining why it sinks when greater pressure is put upon the water in which it is placed. 43. Pumps. — Many contrivances for making fluids run from one place to another are called pumps. A flow may be caused by decreasing the pressure at the place where the fluid is to be delivered or by increasing the pressure at the place from which it is to be removed. EXPERIMENT. Show in operation glass models of the "lifting-pump" (App. No. IX, Fig. 17) and "force-pump" (App. No. X, Fig. 18), dis- cussing their action. L u flflfi ¥ © [ D \ or Fig. 17 Fig. 18. FLUID PRESSURE. 39 ^4 44. The Siphon. — The apparatus illustrated in the fol- lowing experiment is called the siphon. It is found in a great variety of forms and is of much use. EXPERIMENT. Take two glass tubes, each about 6 in. long, connected by a rubber tube about 1 ft. long. Fill the whole with water, then close each end with a finger. Hold one end be- neath the surface of the water in the gallon jar (Fig. 19) ; remove the fin- ger from that end, and bring the other end, still closed, down outside the jar to a level lower than the water surface. Is the water-pressure against the finger that closes the tube now greater or less than the atmospheric pressure upon an equally large surface? If greater, the water will run out when the finger is removed. If less, the air will run in and drive the water up in the tube when the finger is removed. Try the experiment. Repeat the experiment, but now hold the outer end of the tube, before opening it, higher than the level of the water in the jar. 45. Balancing Columns. — The method of finding specific gravities that is suggested by the following experiment is called the method of balancing columns. In the form here shoAvn it cannot well be used with liquids that naturally mix with each other, as alcohol and water do. Later this general method will be used in a form that does not bring the two liquids into contact with each other. Fig. 10. EXPERIMENT. Take a bent glass tube (App. No. XIII, Fig. 20) each arm of which is about one foot long, and pour water into it till both arms are 40 PHYSICS. Fig. 20. about half full ; then pour kerosene into one arm till it is nearly full. Does the water now stand as high in the other arm as the kerosene does in the first arm ? Can you from this experiment see a third method for finding the specific gravity of a liquid ? QUESTIONS. (1) Does water stand at the same level in the spout as in the main part of a watering-pot? (2) If one branch of a U tube (see Fig. 20) were larger than the other, would water stand at the same level in both? (3) Is it necessary in finding the specific gravity of a liquid by the method indicated in Art. 45 to have the two branches of the tube equally large ? (4) Does the height of mercury in the tube of a barometer depend upon the size of the tube? (We neglect at present w T hat is called the capillary effect. See Second Part.) (5) If the height of the barometer mercury-column, of specific gravity 13.6, is 76 cm., how tall a column of water could be sustained by the atmospheric pressure if there were a vacuum above the water ? (Give the answer in ft. as well as in cm.) (6) A water-tank 10 ft. deep is to be emptied by means of a tube used as a siphon. What is the least length the tube can have? (7) With ordinary atmospheric pressure what is the greatest height to which water may be raised by means of a pump working above it? (8) Do you understand the operation of the " trap" which allows water to flow from a sink to a sewer, but does not allow gas to come from the sewer to the sink ? CHAPTER IV. THE LEVER. 46. Definition and Illustration. — Civilized men do most of their work with tools or machines. Many tools and many parts of machines consist of a piece of iron or wood or other material movable to a certain extent upon a sup- port called a pivot, or axis, or fulcrum, by means of which a force applied in one direction at a certain spot may pro- dace another force different in direction or in magnitude, or in both, at another spot. Such a tool or part of a machine is called a lever. One of the most familiar examples of the lever is a crow- bar. A hammer, as used to draw out a nail from a board, is another example. Each half of a pair of scissors is a lever. We shall study some very simple forms of the lever to find out what relations hold between the forces exerted at different points. EXERCISE 8. THE STRAIGHT LEVER: FIRST CLASS. Apparatus : The lever and supporting bar (Xo. 17) fastened to the; long horizontal bar that reaches above the table from end to end. Two scale-pans (Xos. 18a and 18b). A set of weights (No. 19). Hang one scale-pan carrying a load of 8 oz. on the right-hand end of the lever at a distance of 14 cm. from the middle, as in Fig. 21. Hang the other pan, with an equal load, on the left-hand end of the lever, at such a distance from the middle that the lever will bal- ance, that is, stay horizontal when once placed so, even when the ap- 41 42 PHYSICS. paratus is jarred somewhat by tapping the short bar to which the lever is attached. Then make a record like this : Right dist. fr. centre. Left wt. fr! centre Ri ^ ht wt - (l+8) = 9oz. (1 + 8)= 9 oz. 14.0cm. (The space here left blank (in the record) is to be filled by the left- hand distance which the student finds necessary to make the apparatus balance.) Change the right-hand weight to 7 oz. , keeping its place unchanged, and move the left-hand weight, still 9 oz., to some new position which will make the whole balance, in spite of jarring as before. Make a record, as before, of the weights and distances, putting it just beneath the record for the first arrangement. Change the right-hand weight to 5 oz. without changing its place, and find what position the left-hand weight, still remaining 9 oz., must have in order that the lever may balance. Record the distances Fig. 21. and weights for this case under the records already made for the first and second cases. One more case may be taken, in which the right-hand weight be- comes 4 oz., still at 14 cm., which will give a fourth line in the record table. More observations with different arrangements might be made, but it is better to make a moderate number of good observa- tions than a large number of hasty or careless ones, THE LEVER. 43 By studying the record table now made the student should find a rule by which, when the two weights and one distance are given, the other distance can be found by calculation ; or when the two distances and one weight are given, the other weight can be found by calculation. QUESTIONS. (1) If a mass of 6 oz. is suspended from a point 4 cm. to the left of the centre of the lever in Exercise 8, how great a load placed at a distance of 10 cm. to the right of the centre will make equilibrium ? (2) A mass of 8 oz. is suspended from a point 5 cm. from the centre of the lever and is balanced by a mass of 10 oz. How far from the centre is the latter placed? (3j Two masses, 4 oz. and 12 oz. respectively, are to be suspended from a lever. Describe three possible arrangements of the masses, any one of which will cause them to balance. (4) A boy pushing down at one end of a lever 6 ft. long pries up a stone weighing 100 lbs. at the other end. The fulcrum is 2 ft. from the stone. The weight of the lever itself is neglected. How great is the force exerted by the boy ? 47. More than Two Weights. — In the preceding Exer- cise the class found out how to make the two weights hung [ from the lever balance each other. Let us ask now what the rule for balancing would be if there were more than two weights in use, as in Fig. 22, for instance. EXPERIMENTS. We will make the apparatus balance with four weights, two on each side. We will call the weight nearest the centre on the left hand weight Xo. 1, which we will write TF,, for short. The other weight on the left-hand side we will call Xo. 3, or TT 3 . weights on the right hand we will call Wz and TF 4 . When the whole balances, we will call The distance of Wi from the middle, D x , " " W% " " " D* t " u M\ " " " D<. tt3 tt Fig. 22. m The two 44 PHYSICS. Now if we go back for a moment to the case of two weights, which the class has studied, and if we call these Pi and P 2 , and their dis- tances from the middle d x and d 2 , we can state the rule for balancing in this way : Pi X ^1 must equal P 2 X d 2 . In the new case, where we have four weights, we may guess * that the rale is (Wi X A) + (TT 8 X P 3 ) = (TF 2 x D 2 ) + (TF 4 X B A ) 9 and then test the truth of our guess by trial. Try other like cases. 48. Circular Lever. — In the experiments with which we have just been engaged the weights have been suspended from the top of the lever on a level with that part of the pivot upon which the lever rests. In other experiments which are to follow we shall not always be able to keep this arrangement, I and we have now to find out what ' would be the effect of hanging one or more of the weights from points higher or lower than the point of support of the lever. For this purpose we shall use No. FlG - ~ 3 - XV, the piece of apparatus shown in Fig. 23, in which the straight lever thus far used is replaced by a circle of wood about 8 inches in diameter, supported by a screw passing horizontally through the centre. Such a circle, or disk, of wood comes under the general definition of a lever. EXPERIMENTS. We will hang at b and / such weights as will balance each * Shrewd guessing, followed by a test, should be encouraged by the teacher as a means of extending knowledge. In fact, it is the con- stant resource of the investigator. THE LEVER. U other, leaving the disk in equilibrium, and will then move one of the weights to a point vertically above or vertically below its pres- ent place ; that is, from /to e or g, or from b to a or c. Shall we still have equilibrium ? Try. We will now turn the disk a little, so that the lines ab c and efg will be no longer quite vertical, and will see whether now a weight at e or at g has just the same effect as if at/. Try. A careful note should be made of conclusions for future use. 49. Centre of Gravity. — In the experiments upon the lever thus far, the lever itself, whether a bar or a disk, has balanced, when left to itself without load. We have, therefore, not had to consider the weight of the lever itself. But many levers are used in such a way that their own weight helps or hinders the operation to be performed with them. To understand such cases we must learn something about what is called the centre of gravity of a body. EXPERIMENT. Take a board, cut in any irregular shape, like Fig. 24, for instance. Fig. 24. "Bore several small holes straight through the board, 46 PHYSICS. and put into each hole a wire nail that will fit close, long enough to project about half an inch on each side of the b ard. Tie a bullet at one end of a thread and make a loop in the other end. Put this loop over one hook of a piece of wire bent into the shape shown in Fig. 25, and then rest the nail a, Fig. 24, in the hooks of the same wire, so that the board and the string carrying the bullet will both hang free, the string Fig. 25. near the face of the board.* Mark with a pencil the course of the string downward across the board. Then suspend the board by the nail b and mark the new course of the string. Proceed in this way with all the nails and note the point where the various pencil-marks cross each other. Finally, place the board horizontal and balance it upon the flat head of a lead-pencil, noting how near the head of the pencil comes to the crossing of the lines marked on the board. Definition. — By such experiments as this we come to see that there is within the board a certain point which always hangs just beneath the support when the board comes to rest suspended from any one of the nails. We see that the same point has to be just above the support when the board rests upon the pencil-top. In short, the board acts in these experiments as it would if all its weight were concen- trated at this particular point. This point might be called the centre of weight or centre of heaviness of the board, but it is commonly called the centre of gravity. 50. Weight of the Lever. — The following Exercise is in- tended to make the pupil more familiar with the idea of centre of gravity, and to show how it may be taken account of in the use of the leyer. EXERCISE 9. CENTRE OF GRAVITY AND WEIGHT OF A LEVER. Apparatus : The lever of No. 17, detached from its supporting bar, and a small block (No 21), the two being fastened together, as in * The whole apparatus as shown in Fig. 24 will be called No. XVI.) THE LEVEB. 47 Fig. 26, so as to make one body, the whole of which will be called Fig. 26. the lever in this Exercise. A slender wooden cylinder (No. 13). A 1-oz. scale-pan (18 A or 18 b). A 1-oz. wt. from No. 19. To find the centre of gravity of the lever, balance it as nearly as you can, bar and block fastened together, in a horizontal position on the cylinder laid on the table (see Fig. 26), the cylinder being kept at right angles with the lever. Find in this way at what particular mark of the bar the centre of gravity is, and record this mark — for example, 9.1 cm. Then suspend the 1-oz. scale-pan carrying a 1-oz. wt. , 2 oz. in all, from any convenient point near the free end of the bar, and letting this end project beyond the edge of the table-top, balance the whole, as now arranged, as nearly as you can, on the cylinder laid on the table as before (see Fig. 27). Now record the mark from which the scale-pan hangs, 33.4 cm., we may suppose, and the mark which is just over the middle of the cyl- inder when the whole balances, 21.6 cm., let us say. ZDZ This case is like that of the lever studied in Exercise 8. The cylin- p IG# 27. der now taking the place of the screw as a support, we see that the left-hand weight is 2 oz., " " " distance is 33.4 -21.6 = 11.8 cm., " right-hand weight is the weight of the lever, " " " distance is 21.6 -9.1 = 12.5 cm., that is, the distance from the support in Fig. 27 to the centre of grav- ity of the bar and block. We do not as yet know the weight of the lever, but we will call it Wi> and see whether we can find its amount by calculation. If we apply the same rule that was found to hold true in Exercise 8, we shall have 2 X 11.8 = W, X 12.5, 48 PHYSICS. which gives for the weight of the bar and block 2 X 11.8 Wi = ,. - ' =1.89 oz., nearly. The value of TT 2 obtained in this way by the pupil should be com- pared with the weight of the bar and block as found by the teacher with some balance, e.g. No. XVII, much more sensitive than the spring-balance used by the class; for if the method of this Exercise is carefully followed it will give the weight of the lever more accurately than the spring-balance is likely to do. QUESTIONS. (1) An oar, the centre of gravity of which is 3 ft. from the end of the handle, weighs 4 lbs. It rests in the rowlock at a point 2 ft. from the end of the handle. How great a force applied at the end of the handle will keep it balanced? (2) A boy weighing 100 lbs. is see-sawing alone on a plank 20 ft. long weighing 50 lbs. The boy's centre of gravity is 1 ft. from one end of the plank. How far from the same end of the plank is the fulcrum ? (The centre of gravity of the plank is at its middle). (3) If the plank mentioned in the preceding problem is to balance on a fulcrum 8 ft. from one end, with a 100 lb. boy 1 ft. from this end and another boy 1 ft. from the other end, how much must the second boy weigh ? (See Art. 47). Ans. 54 T 6 T lbs. 51. Remarks. — We have now found out how to take account of the weight of the lever itself, when we need to do so. We know that all its weight may be regarded as concentrated at a certain point, which we call the centre of gravity, and we have tried one case in which the weight of the lever itself, acting at the centre of gravity, balanced a certain weight suspended from the bar. In common levers, like the crowbar, the weight of the bar itself is sometimes very important, when the f alcrum is a long distance from the centre of gravity of the bar. Centre of gravity will be taken up again, and the differ- ent kinds of eqtiilibrium, stable, unstable, and neutral, will be considered in the Second Part. We will now return for the present to cases of the lever THE LEVER. 49 where the centre of gravity lies, as in Exercise 8, just under the point of support of the bar. In such cases the weight of the lever itself does not tend to make the bar tip in either direction from its horizontal position. Classes of Levers. 52. Lever of the First Class. — In the levers which we have studied thus far the support, or fulcrum as it is often called, lies between the lines of suspension of two weights. This kind of lever, whether it is a simple bar or a disk or an object of irregular shape, whether its centre of gravity lies at the point of support or not, is called a lever of the First Class. 53. The Power, Power-arm, etc.- — To take a simple and convenient case, we will consider in Fig. 28 a circle sup- ported at its centre, F. We will suppose that this lever is used for the purpose of support- ing a weight TF, and the force used for this purpose, whether it is applied by means of another ! weight, as in the figure, or by means of the hand, or in any other way, we will call the power. We have seen in the experi- ments of § 48 that, as the lever now stands, it makes no difference whether W is sus- pended from the point which now carries it or from some point higher or lower in the same vertical line, which is called the line of action of W. A like statement can be made for P. We shall call the shortest distance from P 's line of action to the fulcrum the power-arm, and the short- Fig. 28. 50 physics. est distance from JF's line of action to the fulcrum the weight-arm. 54, Law for First Class. — In order that P and W may- just balance each other we must have, as can be seen from Exercise 8, power X power-arm = weight X weight-arm. Tins is the law for a lever of the first class. 55. Levers of the Second and Third Classes. — But we may have a case, like that shown in Fig. 29, in which the line of action of the weight lies between the fulcrum and ihe line of action of the power. This arrangement gives us what is called a lever of the second class. Fig. 29. Fig. 30. There is still a different case, shown in Fig. 30, where the line of action of the power lies between the fulcrum and the line of action of the weight. This is called a lever of the third class. In the second and third classes of levers, as in the first class, the shortest distance from the fulcrum to the line of THE LEVER. 51 action of the weight is called the weight-arm, and the shortest distance from the fulcrum to the line of action of the power is called the potver-arm. The pupil is to find out by means of the following Exer- cise whether the laws of the second and third classes of levers are as simple as the law of the first class. EXERCISE 10. LEVERS OF THE SECOXD AND THIRD CLASSES. Apparatus: The lever (No. 17) supported as in Exercise 8. A scale-pan (No. 18). A set of weights (No. 19). A spring-balance (No. 7). Suspend the pan with a load of 8 oz. at a point 5 cm. from the middle of the lever, and, on the same arm of the lever, at a distance of 10 cm. from the middle, pull upward with, a spring-balance, con- nected with the lever by means of a loop of thread, until the weight is balanced and the lever becomes horizontal. You have here a lever of the second class. Read the spring-balance and record as follows : Lever of Second Class. Weight. Weight-arm. Power. Power-arm. 9 oz. 5 cm. .... 10 cm. Try other similar cases, and study them all until you are able to write down the law for this class of levers. Then with the same apparatus place the spring balance between the fulcrum and the line of the weight. You will now have a lever of the third class. Try various cases and record as before Lever of Third Class. Weight. Weight-arm. Power. Power-arm. Law. QUESTIONS. (1) A lever supported at its centre of gravity is used to lift a weight of 100 lbs. applied at a distance of 1 ft. from the fulcrum, The 52 PHYSICS. power is applied 5 ft. from the fulcrum and on the opposite side from tbe weight. How great must the power be ? Must the power be applied upward or downward ? (2) If the power were placed on the same side of the fulcrum as the weight, everything else being as described in the preceding problem, how great would the power have to be ? Would it be applied upward or downward ? (3) If the power were 50 lbs. applied at a point 2 ft. toward the right from the fulcrum, and if the weight were applied 8 ft. toward the right from the fulcrum, how great could the weight be ? (4) If a weight of 5 lbs. were placed 4 ft. toward the right from the fulcrum, and a weight of 7 lbs. 6 ft. toward the right from the ful- crum, how far from the fulcrum toward the left must a force of 10 lbs. be applied in order tD make the whole balance ? Arts. 6.2 ft. In the four preceding problems the weight of the lever has not been considered, because the centre of gravity has been supposed to be at the point of support. Suppose now that the lever weighs 4 lbs. and that its centre of gravity is 3 ft. to the right from the fulcrum, and with this new condition go over each of the four problems again. 56. Force at the Fulcrum. — If we take a case like that shown in Fig. 31, it is plain that 4 oz. applied 7 cm. from the centre will balance 2 oz. applied 14 cm. from the cen- tre, but it may not be per- fectly plain how great the pull on the fulcrum itself is. We will, therefore, in the next FlG - si. Exercise try the experiment in one or two simple cases and see what the result will be. EXERCISE II. FORCE EXERTED AT THE FULCRUM OF A LEVER. Apparatus: The lever of No. 17 freed from its support. Two scale-pans (Nos. 18a and 18b). Two 1-oz. wts. and one 2-oz. wt. from No. 19. The spring-balance (No. 7). A piece of copper wire about 1 mm. in diameter bent into the form of a hook {h in Fig. 32). A piece of thread about 15 cm. long. ID o THE LEVEB. 53 Suspend the bar from the balance in the manner indicated by Fig. 32. Xote and record the weight of the bar alone. Then suspend one scale-pan with a 1-oz. weight from one arm of the bar, and the other scale-pan with a 2-oz. weight from the other arm in such a way as to balance, taking care not to let the pans and weights spill. Xote r and record the reading of the bal- ance. Then make the loads (pan FlG 32 and weight) 2 oz. on one side and 4 oz. on the other, and read and record. Try any other experiments that you can with the weights furnished, until you feel reasonably sure that you know the relation between the weights applied and the pull on the balance. Then state what this relation is. 57. Laws of the Lever. — In each of the cases in Exercise 11 we have applied two downward forces to the bar in sus- pending the two scale-pans with their loads, and have found these two forces to be balanced by another force exerted upward by the spring-balance. It will be well for us to study such cases very carefully, for similar ones are often found. Suppose we are to make three parallel forces, A y B, and (7, just balance each other when all are applied to the same body. Can we from what we have now learned tell any- thing about the relative magnitude and the arrangement of these forces ? We know that — 1st. All the forces cannot point in the same direction. Let us suppose that G is opposite in direction to A and B. 2d. Tlie force C must he equal to the sum of the two forces A and B. B\/ 3d. The line along which C is applied fig. 33. must lie between the lines along ichich A and B are applied. I 51 PHYSICS. 4zth. A X shortest distance from line of A to line of C = B x shortest distance from line of B to line of C. (See Fig. 33.) These rules apply as well to horizontal forces as to ver- tical forces. Try three spring-balances laid parallel to each other on a table and pulling at some light horizontal bar — a lead-pencil, for instance. (Or try the " checker-board " with larger spring-balances, if the apparatus of the Second Part is available.) 58. Pulleys. — We have already learned to consider a disk pivoted at the centre as a kind of lever. When such a lever is worked by means of a cord or band lying upon its cir- cumference, it is called a pulley. We shall now see that the pulley form of lever has some great advantages. EXPERIMENTS. (1) Take the p alley shown in Fig. 34 (No. XV), and let ns first use the largest circle only. Fig. 34. If we fasten two equal weights, W x and TT 2 , to the ends of a string and pass the string across the top of the pulley, we shall of course find that they balance each other. THE LEVER. 55 But suppose we used two strings, one for TFi and the other for TT 2 , fastening each string to a pin or tack at point A, but letting each string rest in the groove of the pulley, so that the final position of the two strings will be represented by Fig. 34. Will two equal weights balance each other under these conditions ? The question is quickly answered by trial, and by turning the pulley a little one way or the other we can try the experiment with A in a variety of positions. (2) Next try the effect of a horizontal pull, P, applied by a spring- balance at the top of the pulley to balance a weight W, as in Fig. 35. (Remember that in this position the reading of the ordinary 8-oz. balance is about J oz. less than the real force exerted by it, be- Fig. 35. cause the spring of the balance does not now support the weight of the hook and bar, which is about \ oz.). Find by experiment whether the force P must be greater or less than, or equal to, the direct pull of the weight W. (3) Balance a weight on one circle of the pulley by a weight on another circle, and find the simple relation which holds between the balancing weights and the radii of the circles. 59. Advantages of a Pulley. — We see that the advantage of a pulley, as compared with a simple bar-lever, is that the pulley enables us to vary the direction of our power at will 56 PHYSICS. and to lift a weight a much greater distance than we could with a bar-lever no longer than the diameter of the pulley. In fact, the distance through which we can lift the weight by means of the pulley depends merely upon the length of the string that supports the weight. 60. Windlass, Capstan, etc. — The windlass (see Fig. 36) is a familiar apparatus consist- ing of an elongated pulley, d, called the drum or cylinder, turned by a power applied at the handle, A, and acting through the lever, or crank, c. The crank and handle are sometimes called a tvinch. The fig. 36. same name is sometimes applied to the whole windlass. If the power is applied at right angles with both h and £, and if the cord sustaining the weight is small, we have, very nearly, P X length ofc — W X radius of d. A capstan is the same in principle as a windlass, but has a vertical drum, so that the lever travels in a horizontal circle. On ship-board capstans are frequently worked by means of a number of men walking about the drum and pushing against the levers, or capstan-bars, of which there may be several applied to one cylinder. 61. Movable Pulleys. — In the pulleys thus far studied by us the pivot has been fixed in position; but pulleys with movable pivots are frequently used. EXPERIMENTS. (1) Take the small metal pulley (No. XVIII) and arrange it ac- cording to the indications of Fig. 37, P being the pulley, M a weight THE LEVEE, Oi suspended from an axis through the centre of the pulley, B a spring balance, and h sl hook to which one end of the string passing beneath the pulley is attached. To what class of levers does the pulley in this position belong? What, then, should be the rela- tion between the weight, which is M plus the weight of the pulley itself, and the pull exerted by the spring-balance? Find by experiment whether the conclusion reached is correct.* (2) Let us now try an arrangement, like that shown in Fig. 38, in which we have one pulley, A, hooked to a bar overhead, and a double pulley (No, XIX), B, which moves up and down with the load M. Let us consider what should be the relation between the pull P and the weight W, which is M plus the weight of B, in this case. In the case tried in Experiment 1 we had two strings holding up the pulley P. We have now four strings holding up the pulley B. After thinking upon the matter for a little time, trying to study out what is the relation between P and TFwith this arrangement, let us try the experiment as we have already tried it in the simpler case, noting the force shown by the spring-balance when M is moving steadily up, and again when it is moving steadily down, and taking the mean between these two forces as the one that would be required to bal- ance the weight, W, if there were no friction. * In making this trial one must remember that friction is often large in pulleys, even when they are well oiled, as this one should be. Now when the load is being steadily raised the hand carrying the spring-balance must lift harder than it would if there were no friction, but when the load is being steadily lowered, the hand, pull- ing just hard enough to prevent the load from, hurrying, is assisted by the friction. The mean between the reading of the balance going up and the reading of the balance coming down will show, very nearly, what the pull required to sustain the load would be if there were no friction. M Fig. 38. 58 PHYSICS. 62. Another Law for Relation of Power to Weight. — The law, "4th" in § 57, for the relation between potver and lueight is extremely useful, and, properly applied, is suffi- cient for very complicated cases of the lever and pulley ; but in some cases it is more convenient to make use of a differ- ent form or statement of this law, a form which makes use of the relative distances moved over by the power and the weight in any operation of the lever, or pulley, or combina- tion of these, that may be in action. 63. Search for the Law. — If we study the various cases of lever, pulley, and combination of pulleys that have been described in the preceding pages, we shall see that when- ever the iveight is greater than the poiver the weight moves a less distance than the power does in any given operation of the apparatus; but whenever the weight is less than the power, friction being supposed zero, the weight moves a greater distance than the power does in the operation of the apparatus. 64. Statement of the Law. — If we study the matter more closely, we shall find the following rule or law sug- gested, though we cannot say that it is proved by our pre- ceding experiments: PX D p = WxD w9 where P stands for the power applied ; W " " " weight lifted; D p " " " distance the power moves ; D w " " " " " weight moves. APPLICATIONS OF THIS LAW. (1) It is evident that this law can be readily applied to a case like that of Fig. 38. We can see at once that if the pulley B were lifted one inch while P remained stationary there would be four inches of loose string under B. To make the string taut again, P would have to rise four inches. In actual use P and B rise at the same time, P moving four times as fast as B, TEE LEVER 59 (2) In Fig. 39, representing the rear wheel and gear of a bicycle, let the diameter of the tire be 28 in. ; " " " " small sprocket-wheel be 2 \ in. ; " " " " large sprocket-wheel be 5 in. ; " length of pedal radius be 6 in. Fig. 39. If the weight of the rider, 150 lbs., rests entirely upon the pedal shown, in its present position, how great a weight acting in opposi- tion, as in the figure, will just neutralize the driving-power ? The circumference of the circle described by the pedal is (see Ex. B) 2ft X 6 inches, or 12;r inches. One revolution of the pedal-crank makes the tire revolve 2 times, which drives the bicycle forward 2 X 27T X 14 inches = 567T inches. The weight W will be lifted as fast as the bicycle moves forward. If Dp stands for the distance the power P moves downward from its present position during any very short time, and if D w stands for the distance W moves upward during the same time, we have Dp \D lc \ :12tt :56tt. Hence the law W X D w = P X D p (see § 64) gives WX 56/r = PX 12k, or W=j\P= T 3 T X 150 = 32. 1 + (lbs.) QUESTIONS. (1) A large pair of shears is used to cut a wire. One handle of the shears being held fixed, a power of 25 lbs. is applied to the other handle at a distance of 2 ft. from the pivot. The wire cut is placed 2 in. from the pivot. How great is the resistance offered by the 60 PHYSICS. wiie ; that is, bow great a force applied just over tlie wire would drive the blade through it ? (2) A man lifts 10 lbs. of coal on a shovel. His left hand is at the end of the handle ; his right hand is 18 in. distant from his left hand ; the centre of gravity of the coal is 36 in. distant from the left hand. The shovel itself weighs 6 lbs. and its centre of gravity is 21 in. from the left hand. (a) What is the direction and magnitude of the force exerted by the left hand ? (Consider the right hand as the fulcrum.) Ans. 11 lbs. (&) What is the direction and magnitude of the force exerted by the right hand ? (Answer this question as if the left hand were replaced by a weight.) (3) A body weighing 160 lbs. is suspended from a pole resting on the shoulders of two men, A and B, of equal height. The point of suspension is 3 ft. from A's shoulder and 5 ft. from B's. How much of the weight does each man bear ? (4) Six men are working at a capstan, each exerting a force of 20 lbs., each at a distance of 6 ft. from the centre. The diameter of the coils in which the rope is being wound on the drum is 1 ft. How great is the strain on the rope, all friction being neglected ? (5) A man finds that by moving one point of a machine, consisting of levers and pulleys, forward 10 in. he can move another point of the machine 1 in. If a force of 5 lbs. is applied at the first point, how great a resistance applied at the second point will be required to neutralize it, if there is no friction in the machinery ? (6) Can the class name any tools or machines, not already men- tioned in this book, in which levers or pulleys are used ? CHAPTER V. THREE FORCES DIRECTED THROUGH ONE POINT: THE PARALLELOGRAM OF FORCES. 65. Introductory. — In studying the lever we have usually, though not always, had parallel forces to deal with, forces acting straight up or straight down. But very often we have to do with bodies that are acted upon by forces not parallel to each other. Thus when a ladder standing upon the ground leans against a house we have at least three forces acting upon the ladder: 1st, the earth's attraction, or, as we call it often, the tv eight of the body, which acts as if the whole substance were at the centre of gravity; 2d, the push of the ground against the foot of the ladder, which push is not straight upward; 3d, the push of the wall against the top of the ladder. Again, a flying kite is acted upon by the earth's pull, straight downward ; by the force exerted by the air, which force, because of the wind, is not straight upward ; by the pull of the string, which pull is not straight downward. The way to begin the study of such cases is to study the case of three forces all acting straight from or straight toward a single point. We shall take such a case in Exer- cise 13, measuring the forces by means of spring-balances. 66. Errors of Spring-balances. — It is quite possible in specific-gravity work to get accurate results with an in- accurate balance ; for the specific gravity of a body is found by taking the ratio of two quantities, both found by weigh- ing with the same balance, and if the balance should give 61 62 PHYSICS. the weight of each as n times its true value the ratio of the two false weights would be the same as the ratio of the true weights, whatever the value of n. But though a balance may give the weight of everything as n times its true weight, n being the same for all parts of the scale, it is very unlikely that several balances will all be wrong in just the same way; and as in Exercise 13 we shall need to use three balances in combination, it is neces- sary for us to give more careful attention to their errors than we have given heretofore. The following Exercise is intended to show how the errors of a spring-balance may be found and may be recorded in a form convenient for future use. EXERCISE 12. ERRORS OF A SPRING-BALANCE. Apparatus : A spring-balance (No. 7). A set of weights (No. 19). Thread for suspending the weights. A measuring-stick (No. 3). (Although weights reckoned in ounces are referred to in this Exer- cise, it may be performed equally well with suitable weights reckoned in grams.) Balance in the Vertical Position. — Suspend the balance by its ring from some convenient support so that the index will be not higher than the eye of the observer. Make five careful readings with the loads indicated in the first column below, and record these readings in a second column ; thus, for example: True Load. Reading. Error.* oz. - 0.2 oz. - 0.2 2 " 2.0 " 0.0 4 " 4.2 " +0 2 6 " 6.3 " +0.3 8 " 8.1 " +0.1 * That is, the quantity which must be subtracted from the reading in order to find the true weight. THE PARALLELOGRAM OF FORGES. 68 Now draw in the notebook a straight line a little more than twice the length of the scale of the balance, and mark off on it points corre- sponding to the loads and readings given above. Then represent the errors by vertical distances, measured down from the points indicating the readings when the errors are negative, and up when they are positive, showing these errors on a rather large scale, 0.5 cm. per 0.1 oz., for instance. Draw a curve through the extremities of these vertical distances, as in Fig. 40. This curve will enable us to tell with sufficient accu- racy for our present purposes the errors of any other readings made with the balance in the vertical position ; for instance, if the balance to which Fig. 40 relates reads 1 oz., we may assume that the error is very nearly 0.1 oz. and that the true weight is very nearly 1.1 oz.; if the reading is 3 oz , we may assume that the true weight is about 2.9 oz., and so on, the error for each case being found by measuring from the point indicating the reading up to or down to the curve. The Same Balance in the Horizontal Position. — We will suppose at first that the same index is used for the horizontal as for the vertical readings. Lay the balance flat on its back and tap it gently several times.* Then take its reading, which we will suppose to be — 0.5 oz. In this case it would require a force of 0.3 oz. to pull the index down to the position it occupied with no load in the vertical test. To bring the index to any reading in the horizontal use of this balance will, there- fore, require a force 0.3 oz. greater than the weight which, applied to the hook in the vertical use of the balance, would bring the index to the same point. It would in the case here supposed be sufficiently accurate for our purposes to add 0.3 oz. to any horizontal reading, and then correct this increased reading by means of the curve given in Fig. 40. When a different index is used for horizontal readings, the princi- ple is the same. For example, let us suppose the second index reads — 0.1 without pull in the horizontal use of the balance. We say, this reading exceeds the vertical reading without load by 0.1 oz. We must, therefore, subtract 0.1 from all horizontal readings made with this second index, and then correct these reduced readings by means of the curve in Fig. 40. * This method assumes that the movable parts of the balance are restrained by friction only, which the tapping overcomes. Sometimes the slot in the balance-face is not long enough to allow the index to find its unrestrained position. 64 PHYSICS. The curve so often referred to should be marked with the number of the balance and kept for future use. EXERCISE 13. THREE FORCES IN ONE PLANE AND ALL APPLIED AT ONE POINT: PARALLELOGRAM OF FORCES. Apparatus: Three 8-oz. spring-balances, each provided with two small blocks (No. 22) to go under its sides and hold it flat on its back when it is lying upon the table. The rectangular block (No. 9). The measuring-stick (No. 3). A sheet of paper. Thread. Take two pieces of strong thread, one about 12 inches, the other about 6 inches, long, and tie one end of the short thread to the middle of the long one. Fasten the three loose ends to the hooks of the spring-balances; then lay the latter upon the table, putting the blocks under their sides, as in Fig. 41, and let one student pull at each bal- Fig. 41. ance, taking care that the slit of each balance- face is in a straight line with the thread, until no one of these reads less than 3 oz. THE PARALLELOGRAM OF FORGES. 65 It will be found that any variation in the angles which the strings make with each other will require a change in the forces. Evidently there is some connection between the directions of the strings and the forces necessary to balance each other. The object of this Exercise is to make out what this connection is. Put under the threads a sheet of paper, and draw on this paper, just under each thread, apencil-mark parallel to the thread, and then write down alongside each pencil- mark the force in the direction of that line, as shown by the spring-balance. The balances must be held very still while these lines are being drawn, and must be read before any change occurs in the direction of the lines.* To draw a line place one side of the block (No. 9) close alongside one branch of the thread, taking care not to push the thread out of place, and then run the point of a well-sharpened pencil along the edge of the block under the thread. Draw the other lines in the same way, doing all very carefully. Each student in turn should make a set of lines, and record along- Fig. 42. side them the proper forces. The directions of the pulls should be varied somewhat by each experimenter, in order that his lines and forces may not be exactly like those of others. Take now the wooden ruler (Xo 3), and extend the three lines toward each other till they meet at one point. This they will do if * It is well to fasten the ring of each balance to some object heavy enough to hold the balance in place, thus relieving the experimenters, who might grow tired and unsteady in holding the balances long enough to permit of drawing the pencil- marks properly. 66 PHYSICS. they have been drawn originally just under the threads. If they do not all meet at one point, a new line should be drawn parallel to one of them in such a position as to pass through the crossing of the other two lines, and this new line, the dotted line in Fig. 42, is then to be used in place of the original line. The three lines as now drawn will represent accurately the directions of the three forces. Now measure off from the common point along the line A a dis- tance of 1 cm. for each ounce (or each 30 gm., if the forces are meas- ured in grams) of the force which was exerted along that line, and put a small arrow-head (see Fig. 42) at the end of this measured dis- tance. Erase that part of line A which lies beyond the arrow-head. Do the same with lines B and C that has been done with A. The three arrows thus obtained, all reaching from the same point, repre- sent the magnitude and the direction of the three forces exerted by the spring-balances. Now with A and B of Fig. 42 as two of the sides draw a parallelo- gram, taking pains to make it accurate.* Then make a parallelogram with B and C as sides, then one with A and G as sides. Compare the * One line may be drawn very nearly parallel to another by means of a device illustrated by Fig. 43. LI is a line already drawn. The block (No. 9) is so placed that for an eye placed at E the edge mn E Fig. 43. appears to be close to LI and parallel to it. Then a pencil-mark is made along the edge op. A better method is to set the edge op on the line LI and then guide the block to a new position by sliding it along the straight edge of a ruler at right angles with LI. THE PARALLELOGRAM OF FORCES. 67 length and direction of the line C with the length and direction of the diagonal of the parallelogram AB\ the line A with the diagonal of the parallelogram BC; the line B with the diagonal of AG. From a study of the Exercise make a rule showing how to find the direct. on and magnitude of a force G which put with two forces represented by the lines A and B (Fig. 44, below) will just balance them. Applications of the Parallelogram of Forces. The rale found in the preceding Exercise, and which is called the 'parallelogram of forces, is peculiarly easy to apply in cases where two of the forces are at right angles with each other. A number of such cases will now be discussed. (1) A force of 7 lbs. pulls north from a certain point, and a force of 4 lbs. pulls east from the same point. How large must a third force be to hold them in check, and what will be its general direc- tion? Solution. — Fig. 45 indicates the method of working the problem. Fig. 44. Fig. 45. Fig. 46. The directionoi the third force is evidently southwest, in a line with the diagonal of the rectangle, of which the base is 4 and the height 7. The magnitude of the third force is evidently equal to the length of the diagonal, which a simple rule of geometry shows to be v 4 -2 + r - = 8 - 06 +• (2) A boy weighing 50 lbs., represented by the point b in Fig. 46, is seated in a swing 10 it. long, represented by the line 8b. A horizon- 68 PHYSICS. tal pull holds him 4 ft. to one side from the natural position of the swing. (a) How great is the pull of the swing- rope ? (b) How great is the horizontal pull ? Solution. — Represent the weight of the boy by the line b W, made 1 cm. long, we will suppose. The other two forces acting on b must be such as to make a balance with this force. Draw the line bW exactly equal and opposite to bW, and complete the parallelogram as in the figure. A little geometry shows that the triangle MW is similar to the tri- angle Sbn. The line bi represents the pull of the swing-rope, triangles just mentioned are similar, we have U-.bW ::Sb : Sn, and, as the bi=(Sb-i-Sn) XbW = (10-*- VW^~4: 2 ) xbW'=lMxbW But b W = b W, which represents a force of 50 lbs. Hence bi rep- resents 54.5 lbs. Ans. to {a) = 54.5 lbs. The pull upon the rope is therefore greater than the boy's weight. The line bh y which is = i W, r epresents the horizontal pull. We have bh'.bW : :nb:nS, Ans. to (b) = 21.8 lbs. bh = (4 -r- VW -4?)XbW = 0.436 X b"W\ The force represented by bh is therefore 0.436 X 50 lbs. =21,8 lbs. (3) A mass of 20 lbs. is suspended from the point p (Fig. 47), where a string is bent at a right angle. The ends of the string are fastened to two nails , i^i and i\T 2t which are at the same height. The part ATi^ is 4 ft. long, the part N^p is 8 ft. long. (a) How great is the pull upon iV a ? Fig. 47. (b) How great is the pull upon JV 2 ? Solution.— Represent the weight by pW. Draw p W equal and opposite to pW. Complete the parallelogram. THE PARALLELOGRAM OF FORCES. 69 The triangle pqW is equal to the triangle W'rp and similar to the triangle N*pNi. The side N^N* = V& _j_ 42 = 8.94 ft. nearly. The pull on i\^ is represented by pq, and we have pq : p W : : N& : JMfi, or pg = (8 ■*- 8.94) X p IT'. Therefore pq represents very nearly 0.895 X 20 lbs. = 17.90 lbs., which is the answer to question (a). Similarly we find the pull on Jjf % to be very nearly (4 ■+■ 8.94) X 20 lbs. = 8.95 lbs., which is the answer to question (5). The pull along the 8-ft. part of the string is just one-half as great as that along the 4-ft. part. The LtfCLEN'ED Plane. 67. Introductory. — The parallelogram of forces will enable us to understand a contrivance very often used for raising heavy weights. It is a common thing to see barrels of flour or other heavy objects loaded upon wagons by roll- ing them up a plank or a pair of rails, placed with one end on the ground and the other upon the wagon, so as to make the ascent gradual instead of straight up. The flat slanting surface up which the body is rolled is called an inclined plane. Sometimes a body is lifted by forcing an inclined plane, the slanting face of a wedge , under it, as in Fig. 48. Fig. 48. Sometimes the force used by an experimenter or a work- man with the inclined plane is parallel to the inclined sur- face; sometimes it is parallel to the base-line of the plane, the horizontal surface of a wedge, for example. We will consider each case in order, seeking for the con- 70 PHYSICS. nection between the weight, steepness of incline, and force to be applied. 68. Force Applied Parallel to Incline. — This case is illus- trated by Fig. 40, where L reDresents the length of the incline; B " " base " " H " " height" " W " weight of the body on the incline, applied straight downward from the centre of gravity of the body; W is the equal and opposite of IF; N represents the force exerted upon the body by the plane X, a force which is straight outward from the surface of the incline if there is no friction (see Chap. VI) between the body and the incline; P represents the pull, parallel to the plane L, which with the force N will just balance W. By comparing the dotted triangle with the triangle whose sides are L, B, and H we see that P : W (or IF) : : H: L, THE PARALLELOGRAM OF FORGES. 71 or PXL=WX H, P= Wx (H+L). EXPERIMENTS. Take apparatus No. XX and adjust it as indicated by Fig. 50, put- ting 7 oz. upon the pan, so that P= 7+ 1 = 8 oz. Then raise or Fig. 50. lower the incline till the weight W will barely roll up the incline when the apparatus is purposely jarred slightly. (The incline cannot be quite so steep when this takes place as it might be if there were no friction. If a knot is made in the thread near where it passes over the pulley at the top of the incline, a very slight movement up or down the incline can be detected by watching the position of this knot. A slight movement is enough.) As soon as this adjustment is made read H, the length of the ver- tical scale from the top of the base-board to the under side of the in- cline, and record in the way indicated in the table below (upper row of numbers). Then without changing P rase the incline somewhat more, until W will, when the apparatus is jarred, barely roll down the incline. (The incline must be somewhat steeper for this than it would have to be if there were no friction.) When the proper adjustment is made, read the new value of //and record it in the second line of the table below. To find the //that would make P just balance W if there were no friction, take the mean between the two values now recorded. Then 72 PHYSICS. find the L that would correspond to this value of H, L being the dis- tance along the inclined scale from the hinge to the point of crossing the vertical scale. P W H L Going up. ..8 oz. 16 oz. .... .... " down.. 8 " 16 " To balance ... 8 oz. 16 oz. If time permits, make P = 6 oz., then 4 oz. ; and in each case repeat the operations just described. 69. Force Applied Parallel to Base. — This case is illus- trated by Fig. 51. Fig. 51. The line W is not here, as it is in Fig. 49, the hypothe- neuse of the dotted triangle; but it is evident that the dotted triangle is similar to the triangle made up of X, B, and H. P is the force applied parallel to the base, and just sufficient, with N> to balance W. We have, from a com- parison of the triangles, P : W (or W) ::I1:B, or P X B= W X H 9 P= WX (H+B). THE PARALLELOGRAM OF FORCES. 73 EXPERIMENT. For experiments in which the power is applied parallel to the base- line we cannot well make use of a string running over a pulley. We must apply the power by means of the spring-balance, as shown in Fig. 52, the long slot cut through the incline lengthwise allowing us to do so. Fig. 52. Find by trial a steepness of incline that will make P about 7 oz. , and, keeping this steepness unchanged for the time, find how large P is when it is pulling W slowly and steadily up the incline, and how large when it is letting TFrun with equal slowness and steadiness down the incline. Take the mean* of these two values as the one that would be needed to balance W if there were no friction. We record, then, for this case : Going up... . 11 down. w 16 16 To balance. . . 16 where B is the length of the base-line from the hinge to the foot of the vertical line, along which IL is measured. If time permits, lower the incline and try various degrees of steep- *The mean of the two values of P is not, in this case, exactly the quantity wanted, because the greater pull of P when IF is going up the incline makes W press harder against the incline when going up than when going down, thus increasing friction. The mean value of P. as now found, is a little greater than the value wanted, but so little that the error is not important, 74 PHYSICS. ness, so that P will be in one case about 5 oz. and in another case about 3 oz. 70. The Wedge. — The ivedge, as commonly used (see Fig. 48), is a case of the inclined plane with the applied force parallel to the base. It differs from the case shown in Fig. 51 in one respect. In Fig. 51 the body raised has a motion parallel to the base of the plane, while the plane itself has not. A wedge commonly has a motion parallel to its own base, while the body raised or otherwise moved by it does not have a motion in this direction. The principle involved in the two cases is quite the same, and for a wedge used to lift a weight we have, leaving friction out of ac- count as before, P=WX (H+B), where H stands for the thickness of the wedge, and B for its length. 71. The Screw. — The screw is an ingenious form of the inclined plane, as the following experiment will show. EXPERIMENT. Cut out a long narrow triangle of paper, (see Fig. 53), and then wind it upon a lead-pencil, beginning at the end £Tand keeping the line B all the time at right angles with the length of the pencil. The line L will make a regular spiral around the pencil, corresponding to the thread of a screw. 72. Pitch of a Screw. — The distance from one turn of the thread to the next turn, measured parallel to the length of the screw, is called the pitch of the screw. THE PARALLELOGRAM OF FORGES. 75 It is evident that in the ordinary use of a screw one revolution moves it toward or backward the length of its pitch. 73. Use of the Screw in Lifting. — A very large iron screw, called a jack-screw, is frequently used for lifting very heavy bodies. The power is applied to such a screw by means of a long handle or lever, which projects from the head of the screw at right angles with its axis, its central lengthwise line. Leaving friction out of account, we can find the relation between power applied and weight of the body lifted thus : Let P = the power applied to the handle at right angles with the handle and with the axis of the screw ; A = the distance from the point of application of P to the axis of the screw ; r == the radius of the screw itself; p = the pitch of the screw ; W= the weight of the body lifted. The force P produces at the thread of the screw a force P' = P x (A -f- r). (See Exercise 10.) This force at the thread is like the power used to drive a wedge. The circumference of the screw at the thread, which = 27rr, corresponds to the length of the base of the screw, while the pitch corresponds to the thickness of the wedge. We have, then P' X 27rr= Wxp, P'=z Wx (p + 2*r), or Px(iv r) = W X (p + 27tr) ; whence P X 2?rA = WXp. Observe that we have here, as we have had so often 76 PHYSICS. before, the rule, Power X distance the power moves = weight X distance the weight is lifted. Definitions. 74. Equilibrant. — A single force that will just balance, or make equilibrium with, two or more others is called their equilibrant. In Fig. 46 b IF is the equilibrant of hi and bh; U " " " " bW" bh; bh " " " " bW" bi. 75. Resultant. — A single force that can exactly replace two or more others, so as to produce the same effect upon the body acted on, is called their resultant. In Fig. 46 bW is the resultant of bi and bh. The resultant and equilibrant in any given case are equal in magnitude, but opposite in direction, so that the two would exactly balance each other. QUESTIONS. (1) What is tlie resultant of pq and pr in Fig. 47 ? What is their equilibrant ? (2) Draw three lines leading from one point, giving to them such magnitudes and directions that they will represent three forces in equilibrium with each other. (3) Replace two of the lines in the preceding problem by two others that will also represent equilibrium with the third line. (4) A telegraph-wire pulls north from a post with a force of 12 lbs. ; another pulls west from the same post with a force of 16 lbs. (a) How great is the resultant pull on the post? (b) If a third wire is put in to neutralize the pull of the other two, should it pull more nearly south than east, or more nearly east than south ? (5) Two sticks of equal length, OA and OB in Fig. 54, each rest ing one end upon the ground, meet at a right angle in a frictionless joint 0. From this joint is suspended a mass of 5 lbs., the weight of which is represented by the line OW. THE PARALLELOGRAM OF FORCES. 11 (a) How great is tlie total force exerted by each stick against the ground, the weight of the stick being left out of account ? Ans. 3.54 — lbs. (6) How great is the vertical push ex- erted by each stick against the ground? Ans. 2.5 lbs. (c) How great is the horizontal push exerted by each stick against the FlG - 54 - ground? Ans. 2.5 lbs. (We see from this problem that the total vertical force exerted at A and B is just equal to the weight of the suspended mass, the weight of the sticks not being considered. If we think of AOB as one end of the roof of a house, the answer to (c) shows the tendency of the roof to push the walls apart. This tendency is met in actual roofs by beams or rods connecting A and B.) (6) A wedge 1 ft. long ou its base and 2 in. thick is used to lift a weight of 300 lbs. in a case where friction may be left out of ac- count. How great is the force, parallel to the base, required to drive the wedge ? (Friction not being considered, the force required to keep the wedge moving after it is started is no greater than the force required to hold it in place so as to make equilibrium, as in the discussion of §70.) (7) A safe weighing 2000 lbs. is resting on an inclined plane 12 ft. long, one end of which is 2 ft. higher than the other. How great is the force, parallel to the incline, required to keep it from sliding down ? (8) A jack-screw having a pitch of £ in. and a handle 2 ft. 1 in. long is used to lift a mass of 5000 lbs. How great must be the power applied to the end of the handle ? CHAPTER VI. FRICTION. 76. Introductory. — When we push a heavy block along on the top of a table we feel a certain resistance. We know from experience that by making the surface of the table and the surface of the block yery smooth we can lessen the resistance. This resistance, the amount of which depends upon the condition of the rubbing surfaces, is ccdled Friction. Friction always opposes motion, whatever may be the direction of the motion, that is, it merely tends to stop the motion. It never helps to push the block back to the position where it started. We shall in Exercise 14 measure in a number of cases the force required to keep a block moving steadily along on a sheet of paper laid upon a level board, and shall study these cases with the purpose of finding out some useful facts, or laws, concerning friction between solid bodies. EXERCISE 14. FRICTION BETWEEN SOLID BODIES. Apparatus : A spring-balance (No. 7). A rectangular block (No. 9). Set of weights (No. 19). A smooth sheet of paper about 1 ft. wide and 1^ ft. long. Thread. We shall first consider the velocity of the motion, that is, ice shall ask whether the force required to keep up a slow steady motion is greater or less than that required to keep up a more rapid steady motion. 78 FRICTION-. 79 Lay the block on one of its broad sides, and attach it to the spring- balance by a thread passing around but not under the block. Load the block with weights until the force required to maintain a slow steady motion is about 3 oz. Draw the block parallel to its grain along the sheet of paper several times with a very slow steady motion, and then several times with an equally steady motion two or three times as fast. (As the paper is likely to grow somewhat smoother under the repeated rubbing, the experimenter should not make all his slow trials first, but should change from slow to fast and fast to slow a number of times.) Record your conclusion as to whether the slow or the more rapid motion requires the greater force. We shall next try to find out whether, the total weight being the same as before, it is easier or harder to draw the block on a narrow side than on a broad side. Use the same block and the same load of weights, pulling it now, as before, parallel to its grain. (The side upon which the block slides should in all cases be clean, and the broad and narrow sides which are compared should be, as nearly as practicable, equally smooth. The thread must not be between the rubbing surfaces in any case.) Record your conclusion as to whether the broad side or the narrow side offers the greater resistance to the motion. Finally y we shall ask what connection there is between the total mass drawn and the force required to draw it. For this purpose vary the weights placed upon the block, using not less than 6 oz. for the least and as much as 16 oz. for the greatest load. Add to the load in each case the weight of the block itself, and make the record in the following form, W being the load and b the weight of the block : W -f b* F (Force Required). Look for any simple relation between (1F+ b) and F. * It is well to begin with the lightest load, proceed in regular order to the heaviest, then go back in exactly the reverse order to the lightest, recording both trials made with each load and taking the mean of the two for final study. 80 PHYSIOS. The experiments just described will teach a number or useful facts about friction between two solid substances, but one must be careful not to apply the conclusions here arrived at to extreme cases, ex- tremely slow or very fast motion, for id stance ; or to cases where the pressure is great enough and the edge of the sliding body narrow enough to cause an actual cutting of the body into the surface over which it should slide. 77. "Laws of Friction." — The so-called laws of friction between solids are: 1st. Friction is independent of the velocity of one surface across the other, other things being equal. 2d. Friction is independent of the area of the rubbing surfaces i other things leing equal. 3d. Friction is proportional to the total pressure of one surface against the other, other things ieing equal. The experimenter in Exercise 14 need not be surprised or disappointed if his observations do not agree exactly with these statements. In fact, the " laws " are not strictly true; but they are near enough to the truth to be of very great use. 78. Coefficient of Friction. — If the pressure between two surfaces, at right angles ivith each of 'them, is called P, and if the friction between the two surfaces is called F, the ratio F-+- P is called the coefficient of friction. In Exercise 14 the ratio F -^ (TF+ b) is the coefficient of friction. There is a method of finding this coefficient without measuring either P or F. It makes use of an inclined plane and the parallelogram of forces. In Fig. 55 is supposed to be a body resting upon the incline AB, which is just steep enough to keep moving with uniform velocity, in spite of friction, if it is once started down the incline. The line W represents the weight of the body. This FRICTION, 81 is equivalent to a force OP at right angles with the incline and a force OM down the incline. It is the force OP that Fig. 55. causes the friction. It is the force OM that maintains motion in spite of the friction. If the body moves with uniform velocity down the incline, as we have supposed, the force OM must be exactly equal and opposite to the resistance of friction. For if OM were greater than the friction, the body would move faster and faster down the incline; while if OM were less than the friction, the body would move more and more slowly down the incline.* While the body is moving downward friction is represented by the arrow pointing from toward A, equal and opposite to OM. Therefore, in accordance with the definition given at the beginning of this article, we have coefficient of friction = OM -f- OP. * It require- force to set any body in motion and it requires force to stop any body that is in motion. If a body is moving along in a straight line with uniform velocity we know that the various forces acting on it balance each other. This matter is discussed further in the Second Part. 82 PHYSICS. A comparison of the triangle OWM, in which WM = OP, with the triangle ABC shows that they are similar, and hence OM: OP : : AC: PC. That is, the coefficient of friction — AC -^ PC. EXERCISE 15. COEFFICIENT OF FRICTION. Apparatus : The same block tliat was used in Exercise 14. A flat board (No. 20) about 15 cm. wide and 50 cm. long for the block to slide on. A sheet of paper to cover one side of this board. Some means of raising one end of the board and adjusting it so that the block will just slide down it ; another block similar to the one which slides, or any similar object, will do for this purpose. A 30-cm. measur i ng-stick . Place one end of the board on the table and the other on the sup- port. Vary the steepness of the board by varying the position of the support, until such an inclination is found that the block, once started slowly, will barely continue in motion down the board. Then lay off on the table beneath the board a distance of 30 cm., measured from the edge where the board rests upon the table, and from the end of this line measure H, the vertical distance up to the under side of the board. The coefficient of friction will be H^- 30. If the same block, the same side of the bloc'v, and the same kind of paper are used in this Exercise as in Exercise 14, the value of the coefficients obtained in the two Exercises should be compared. 79. Friction in Applied Mechanics. — Friction is one of the most important conditions in the construction and operation of very many mechanical appliances. It enters largely into the list of resistances to be overcome, as in the rolling friction of the car-wheels upon the track or of wagon-wheels upon common roads. Every axle revolves in its bearings with a measurable amount of friction, which can be diminished but not overcome by oiling the surfaces FRICTION. 83 in contact. On the other hand, many machines and mechanical appliances would be valueless without friction. Upon this the efficiency of belting, of brakes, of nails and screws of every description, is dependent. The driving- wheels of engines or of electric street-cars, the feet of men or of horses, would be unable to produce or maintain loco- motion without the aid of friction. If its operation were suspended, every river would become a cataract, soon run- ning itself out. Rolling Friction. 80. Introductory. — The friction encountered by a mov- ing body is usually much less when it is on wheels or rollers than when it slides, though it is true that on snow runners are better than wheels. The wheels of ordinary carriages do not get rid of sliding friction altogether, for the surfaces of the axle and the hub slide over each other. The " ball bearings " of bicycles do away with this sliding friction almost completely. 81. Coefficient. — The coefficient of rolling friction of iron wheels on iron rails may be as small as .002,* so that a pull of 4 lbs. may keep in motion a carriage weighing 2000 lbs. The coefficient of sliding friction of smooth dry iron upon iron is perhaps .15 or .20. 82. Slipping of Wheels. — When people were first con- sidering the use of steam for dragging railroad trains, they thought it would be necessary to provide the driving-wheels of the locomotive with cogs fitted to a cogged rail along the track. This device was found to be unnecessary for ordi- nary work, but it is used on very steep inclines running up the sides of mountains. * Rankine, Civil Engineering. 84 PHYSICS. Even upon ordinary railroads, when the rails are wet and there is a heavy train to be set in motion, the driving- wheels sometimes slip and revolve, while the train refuses to start. The frequency of the puffs from a locomotive depends upon the speed of revolution of the driving-wheels, and when an engine that has been p tiffing very slowly in starting a train suddenly gives three or four puffs in very quick succession, we may conclude that the driving-wheels are slipping on the rails. Engines are provided with sand- boxes, from which sand can be sprinkled upon the rails in front of the driving-wheels when slipping occurs. Friction between Solids and Fluids. 83. Unlike Friction between Solids. — The laws of fric- tion between solids and fluids are very different from those which hold between solids. Friction between solids and fluids changes comparatively little with change of pressure, but it changes a good deal with change of velocity. The resistance of the air is an important obstacle to rapid motion, as in the case of a railroad train, and the frictional resistance of the water to the hull and propeller of a steamer demands most of the steam-power required to propel the vessel. 84. Friction in Tubes. — The friction of liquids or gases flowing rapidly through long tubes is very considerable, as the following experiments will show. EXPERIMENTS. (1) Take a rubber tube 2 or 3 in. long and about 0.6 cm. in diame- ter of bore. Cue off a piece about 20 cm. long. Fill a large glass ■Jar with water. Using the short piece as a siphon, keeping the lower end about 10 cm. beneath the surface of the water in the jar, find the number of seconds required to fill a small tumbler with the water delivered, FRICTION, 85 Try the same experiment with the long tube, keeping its outlet also 10 cm. below the surface of the water in the jar. Compare the rates of delivery in these two cases. (2) Take a rubber tube 2 or 3 m. long and about 0.15 cm. in diame- ter of bore. Cut off a piece 10 cm. long. Light a candle. Put out the candle-flame by Wowing through the short tube. See how far from the outlet of the tube the flame must be placed in order to survive the blowing. Repeat the trial, using now the long tube. Compare the distances in the two cases. Something more concerning friction of water in tubes is given in the Second Part. QUESTIONS. (1) A body weighing 20 lbs. rests upon a horizontal surface upon which its coefficient of friction is 0.2. How great is the force re- quired to keep the body moving along the surface ? (2) It requires a force of 20 lbs. to keep a certain body moving along a horizontal plane, the coefficient of friction being 0.3. What is the weight of the body ? (3) A sledge weighing 10 lbs. can be drawn along a certain level surface by a force of 0.25 lb. How great may we expect the force to be which will just maintain motion when a load of 50 lbs. is placed on the sledge ? (4) A sledge weighing 50 lbs., having runners 1 in. wide, is dragged along a floor by a force of 15 lbs. How great a force would be required if the runners were twice as wide ? (5) According toRankine's Civil Engineering, the coefficient of slid- ing friction of loose earth on earth may be as much as 1, although it is generally less. Suppose a bank of earth, with 1 for the coefficient of friction, to be made of such steepness that the outer surface, if started, will continue to slide downward. (a) If a pole reaches 10 ft. straight downward into such a bank, how far along a horizontal line is the lower end of the pole from the surface ? (b) How great is the angle which the surface of such a bank makes with a horizontal plane ? CHAPTEE VII. THE PENDULUM. 85. Use in Clocks. — Before leaving the subject of Mechanics and going to that of Light it is well to learn something about pendulums, which are used to control the motion of clocks. If you were to examine the works of an old-fashioned clock, you would find the power which drives it in a heavy weight working upon a kind of pulley by means of a long cord, but the device which governs the speed of the works and allows the motion to be neither too fast nor too slow is the pendulum. As a crowd of men at a turnstile, however they may try to force their way, can pass no faster than the swinging turnstile permits, so the clock-weight, which if the control were removed would run down at once with a furious buzzing of the wheels, is allowed by the pendulum to descend only very slowly, a very little distance at every swing of the pendulum, and not at all when the pendulum does not move. The rate at which the clock-wheels can move, then, depends upon the length of time required for each swing of the pendulum. We will try a few simple experiments to find out something about the laws of pendulum motion. EXPERIMENTS. Description of Apparatus. — A convenient method of suspending a simple pendulum is shown in Fig. 57, where B is one end of a wooden bar, which is bevelled off ou the side from which the pendulum 86 THE PENDULUM. 87 hangs. G is a cork fastened to the top of the bar and having in it a slit made by a sharp knife, through which slit the silk thread, S, o 3 o 2 Fig. 56. I FI&.57. passes. If this part of the thread is waxed, the fastening thus ob- tained holds the pendulum securely, although it is very easy to in- crease or decrease the length of the pendulum at will. The length of the pendulum is to be measured from the under side of the bar to the centre of the ball. It is intended that the length of No. 2 and No. 5 in Fig. 56 shall be the same as that of No. 1, that the length of No. 3 shall be one-fourth that of No. 1, and the length of No. 4 one-ninth that of No. 1. It is therefore convenient to make the length of No. 1 just 36 inches, which will require 9 inches for the length of No. 3, and 4 inches for that of No. 4. The suspended body is a bullet in the case of each pendulum except No. 5, where it is some lighter object — a marble, for example. The whole apparatus is called No. XXI. (1) How does the time required for a single swing depend upon the lengthy or width, of the swing ? Set No. 1 and No. 2 swinging at the same instant and with the same width, or length, of swing, and watch them both for a little while until it is plain that under these circumstances they keep to- 88 PHYSICS. gether, No. J. taking just as long a time for one swing, or for any number of swings, as No. 2 does. Then draw the ball of No. 1 about one inch aside from its position of rest, and the ball of No. 2 about fifteen inches aside from its position of rest, and release both balls at the same instant. Watch the two for some little time, a quarter of a minute or longer, and see whether at the end of that time they begin each swing together, as they did at first. If they do not, observe which one has gained upon the other, and, after one or two repetitions of the experiment, write down an answer to the question which the experiment was intended to meet. This answer should state which swing, the long or the short, if either, takes the longer time, and whether the difference in time is large or small compared with the time of either swing. (2) How does the time required for a single swing depend upon the length of the pendulum from the support down to the centre of the ball? Let one person, holding a watch in his hand, draw ball No. 2 several inches aside from its position of rest and, releasing it at a convenient moment, give a signal to the class, and let the class count the number of single swings till, at the end of 20 seconds from the start, a signal is given to stop counting. In a similar manner the number of swings of No. 3 in 20 seconds and the number of swings of No. 4 in an equal time are found, and the observations for the three pendulums are recorded in a table, as follows : Number Time of Length Square Root of Swings. One Swing, of Pend. of Length. 36 6 9 3 4 2 The numbers to fill the fourth column must be found from those in the second and third columns. A comparison of the fourth col- umn with the sixth column will probably show that there is a close relation between the time of swing and the length of a pendulum.* (3) Weight of Pendulum-ball. — Finally, a comparison of No. 1 and No. 5, set in motion at the same time and with the same w T idth of * It is interesting and even amusing to watch pendulums 1 and 3 or 3 and 4 swinging at the same time, both being started at the end of a swing at the same instant. Pendu- Whole lum. Time. No. 2 20 sec. " 3 (( " 4 (C THE PENDULUM. 89 swing, will show whether the time of swing depends much upon the nature of the suspended body. It will doubtless be noticed that the width of swing of the lighter body diminishes more rapidly than that of the heavier one. This gradual loss of motion is due to the resistance of the air. The re- sistance is about the same for both bodies if they have the same size, shape, and velocity, but a light body is more quickly stopped by a given resistance than a heavier body. This is the reason why one cannot throw an acorn or a piece of cork so far as one can a stone of the same size. 86. Springs in Place of Pendulums. — It has been said above that pendulums are used to control clocks, but many clocks and all watches are controlled by means of vibrating springs; for these, like pendulums, are very regular in their swings and so are good time-keepers. The controlling springs (see the " balance " of a watch, Second Part) must not be confused with the much larger driving-brings^ or " matw-springs, " which are used in watches and in most clocks of the present day. More will be said about pendulums in the Second Part. QUESTIONS. 1. If one pendulum is 9 inches long and another is 64 inches long, how will the time of vibration of the first compare with that of the second ? 2. If pendulum A, 39 in. long, vibrates once in a second and pendulum B vibrates once in 5 seconds, what is the length of B ? CHAPTEE VIII. NATURE OP LIGHT : VISIBILITY OF OBJECTS. 87. Light is Something that Travels. — We say that a lamp gives, or gives out, light. This is true. Light is something that comes to our eyes from any object and enables us to see the object. A substance through which light can travel is called a medium for light. We have ways of measuring the time required by light to travel a given distance in air and in many other media. 88. Measurement of the Velocity of Light. — One of the simplest methods for measuring the velocity of light is that devised by the French physicist Fizeau. It consists essentially of a source of light, from which a bright beam may be obtained, a toothed wheel which may be made to revolve in a plane at right angles to the course of the beam of light, and a plane mirror. Apparatus is provided by means of which the rate at which the wheel revolves can be exactly measured. The beam of light passes through the space between two adjacent teeth of the wheel, travels a distance of several kilometers, is then reflected by the mirror, and returned over the same path by which it passed out. If the wheel is at rest, the beam as it returns will repass the aperture between the teeth through which it passed out. But it is easy to see that if the wheel could be revolved fast enough a tooth might be brought into the path of the returning 90 NATURE OF LIGHT: VISIBILITY OF OBJECTS. 91 rays in time to intercept them. Still more rapid revolu- tions would bring a new gap between teeth into the path of the returning rays, and so on. In fact alternate eclipses and appearances of the returning rays are produced when the wheel is revolved at a high and continually increasing velocity. From the rate of motion of the wheel and the distance traversed by the beam it is not difficult to calculate the velocity of light. As a result of measurements made by somewhat different means from those just described, the velocity of light has been ascertained to be about 300,000 kilometers, or 186,000 miles, per second in a vacuum. The velocity in air is a little less. 89. Light is of Various Kinds. — Light as it comes from the sun, or from most lamps, is of many different kinds, all blended together so that the eye does not distinguish one kind from another; but when this mixture of light falls upon certain objects, pieces of glass called prisms, for in- stance, the mixture is broken up and we see the different colors, EXPERIMENT. Hold a glass prism (No. XXXI) in the direct sunlight in such a position that light after passing through the prism will fall upon a white surface not in the direct sunlight. This breaking up of light is considered further in § 134. 90. Light a Wave-motion. — Before the nineteenth cen- tury many people believed light to consist of particles of matter, actually shot out in some way from the luminous body. These supposed particles were called corpuscles (that is, little bodies), and this theory as to the nature of light was called the corpuscular theory. We now believe that light is not a substance, but a kind of wave-motion, a shiver, which is sent along through bodies with great velocity and to very great distances, 92 PHYSICS. although the particles of the body, or medium, transmitting this wave-motion travel very small distances on either side of their positions of rest. More will be said about this in the Second Part of this book. 91. Color and Wave-length. — The different kinds of light, which produce in us the sensations of different colors, are distinguished from each other by differences of wave- length. Waves which produce the sensation of red, and which we often call red waves, are longer than the so-called blue waves, which produce the sensation of blue. One tint of red has a wave-length of one thirty-thousandth part of an inch. One tint of blue has a wave-length of one fifty- five-thousandth of an inch. 92. Light Travels in Straight Lines.* — When direct sunlight enters a darkened room through a small hole, one can usually trace its course and boundary in the room by means of the air-borne dust particles which are lighted up by it. It is easy to see that the boundary, the side, of the learn of light is straight. This is one of the familiar facts which show that light travels in straight lines. Practical applications of this property of light are found in the. prac- tice of sighting rifles, cannon, and other firearms; in the method of glancing along the edge of a board, which the carpenter adopts to see whether it is straight; and in the various surveying operations, in which points are located by sighting with the unassisted eye, or by means of fine slits in metal plates, or by the aid of small telescopes. 93. Light " Pencils and Rays." — If a beam of light is * This statement holds good only in cases in which the light travels in a medium or substance of uniform composition through- out. Even under such circumstances there are certain exceptions to the general rule of rectilinear propagation. These occur where light passes close by the edges of objects, but the effects produced, although very interesting and beautiful, are not sufficiently promi- nent to make their study in this book necessary or desirable. NATURE OF LIGHT: VISIBILITY OF OBJECTS. 93 slender, it is a pencil of light. If the pencil is very slender indeed, it is called a ray of light, and is represented in drawings by a single line. 94. Camera Obscura^ — This name means dark chamber. EXPERIMENT. Push the small tube of No. XXIV, closed end foremost, into the larger, and then, pointing the apparatus toward a window, look into the smaller tube and move it back and forth in the other till the best image of the window or of objects outside is obtained. It is evident at once that the image is upside down, that is, that the bottom of the image represents the top of the object. This is due to the fact that the light-rays, coming from the object and traversing the very small aperture in the end of the tube, cross each other in their passage, as in Fig. 58, where the object is represented by the arrow AB. Fig. 58. For instance, the cone of rays A A! from the tip of the arrow and the cone of rays BB' from the other end, cross at mn, and appear in the image at the spots A' and B' respectively. If the aperture mn were gradually made larger, the spots A! and B\ illuminated from A and B respectively, would grow larger and larger. The same would be true of the spots illuminated from other points of the arrow; and at last the growing spots would so overlap each other that the image would be lost in a mere blur of light on the screen. 94 PHYSICS. 95. Shadows. — From the fact that light travels in straight lines, it is easy to see that it will be cat off from a portion of space behind any illuminated opaque object, just as waves of water are cut off by a breakwater, leaving a region of calm water behind it. The simplest case is that in which the light-giving object is as small as possible. EXPERIMENT. Light a bat- wing gas-jet or a kerosene lamp with a broad, thin flame, and cast the shadow of a lead-pencil, held vertical, on a sheet of white paper, having first the edge and then the broad side of the flame toward the pencil. Note the great difference in the sharp- ness of outline in the two cases. 96. Umbra. — A shadow with a perfectly sharp outline could only be obtained by using as the source of light a mere point. To illustrate what would be the result if this Fig. 59. could be done, the student should examine Fig. 59. Of the light-rays proceeding from the point £, the cir- cular opaque object OF intercepts all which strike its sur- face, thus forming a shadow whose shape is in this case the frustum of a cone, OS TV.* The black space *STin the screen is not the whole shadow, but a section of the entire shadow OS TV. A perfect shadow like this, equally dark at all points, is called an umbra. 97. Penumbra. — Suppose now that the source of light is of appreciable size, a candle-flame, for example: then the * That is, a cone with its top sliced off by a section parallel to the base. NATURE OF LIGHT: VISIBILITY OF OBJECTS. 95 opaque object eats off all illumination from some por- tions of the screen, and from other portions cuts off only a part of the light, as in Fig. 60. That part of the screen which receives light from part of the flame AB, but not from all of it, will appear a partially shaded ring, P'S'SP, around the central area of total shadow. This ring forms what is known as the penumbra (from two Latin words meaning almost and shadow). On account of the comparatively large size of most sources of light most shadows are surrounded by a wide margin of penumbra. The student will find the best examples of clear-cut shadows in those cast upon near sur- faces by opaque bodies exposed to electric arc-lights, and he Fig 60. may compare the dim and indistinct shadows of the leaves of shade-trees exposed to the sun, with those cast by the same objects exposed to the electric light at night, in which even the serrated margins of the leaves are sometimes clearly outlined. 98. How Light Weakens with Distance . Law of Inverse Square- — If two equally large surfaces are turned toward a very small flame, distant 1 ft. from one and 2 ft. from the other, the nearer surface will receive very nearly four times as much light from the flame as the more distant but- 96 PBTS1CS. face. It is easy to prove that this is true if light travels in straight lines diverging from a point (Second Part). If one surface is three times as far away as the other it will receive only one ninth as much light as the nearer one, and so on. The law, which holds when the diameter of the light- giving spot is very small compared with the distance from it to the receiving surface, may be stated thus : The amount of light received on a surface of given area from a given source of light is inversely proportional to the square of the distance from the source to the surface. This is called the law of inverse square. It follows from this law that if a lamp L sends ,to a given surface at a distance D 311st as much light as another lamp U sends to the same surface at a distance D\ the light- giving powers of the two lamps, which powers we will call P and P\ must be such that P: P':: D 2 : D'\ Illustration. A candle-flame 30 cm. from a white card and an incandescent elec- tric lamp 120 cm. from the same card light it up equally. What is the relative power of the two sources ? Pi (for the lamp) : P c (for the candle) : : 120 2 . 30 2 . Hence P = P c X 16. 99. Photometry ; Rumford's Photometer. — It is a matter of great practical importance to compare the illuminating power of different lamps. This operation is called photom- etry, or light-measurement. It cannot be done by merely observing the lamps directly ; for the eye is unable in this way to distinguish slight differences of power, and if the lights are of somewhat different colors the unaided eye gives only the vaguest indications in regard to their comparative efficiency. NATURE OF LIGHT: VISIBILITY OF OBJECTS. 91 One of the simplest devices for measuring the relative power of two sources of light is RuniforcTs photometer, which compares the shadows cast by a rod placed in front of them. EXERCISE 16. USE OF RUMFORD PHOTOMETER. Apparatus : Two small kerosene lamps like No. 33. A cardboard screen and its support (No. 32 and No. 21). Any opaque rod about 1 cm. in diameter and 10 or 15 cm. tall, supported upright ; e.g., No. 13 standing in a hole bored m a small block, or a Bunsen burner, A meter rod. The ooject of the experiments will be to find whether a flame sends more or less light from its broad side than from its edge, and, if so, how much, The flame should be made as large as they can well be without smoking. The apparatus should be arranged as in Fig. 61, A C Fig. 61. Jl L is one of the lamps to be compared, and L the other ; R the rod AB the screen, and Sl and 8& the shadows. The lamps should be so arranged that lines drawn from their centres to the centre of H will make nearly equal angles at i? with the line CD, drawn at right angles to the screen through the centre of i?, and on this line the observer should be placed. The shadows should be near each other, but must not overlap. It is plain that the shadow corresponding to L is illuminated by light from L' and that the one corresponding to U is illuminated by light from L. Place the lamps equidistant from the rod. and, shielding the eyes 98 PHYSICS. from the direct light of the flames, adjust the flames, turned edge- wise to the rod, until the shadows are of equal darkness. Then turn one of the lamps about in place until its flame is flat- wise to the rod, and compare the shadows again, fixing the attention upon the middle of the more blurred one. If the shadows still appear to be of equal darkness, record the fact. If they do not, move one of the lamps toward or from the rod until the shadows appear equally dark, and then record the distance of each flame from the corre- sponding shadow. Try each lamp in turn flatwise, the other being edgewise. Be- tween the trials test the flames again in their original position, to make sure that they are still equal. If, on the whole, it appears that one aspect of the flame, broad side or edge, is more effective than the other, estimate the relative light- giving power of the two aspects from the measured distances, mak- ing use of the law of inverse square. 100. Bunsen's Photometer. — The form of photometer devised by the German chemist and physicist Bunsen yields, under suitable conditions, more accurate results than the apparatus just described, and is equally simple, but more difficult to use in an undarkened room. EXPERIMENT. Drop a little paraffin on a sheet of heavy, unsized, white paper, — thin drawing paper, for instance. Heat the paper by placing on it a moderately hot iron weight or a can of hot water, until the paraffin is entirely melted and soaked evenly into the paper, so as to make a roughly circular spot about 3 cm. in diameter. Cut out of the paper a circle about 12 cm. in diameter with the spot just prepared in its centre. It will be noticed that the spot is translucent ; that is, it allows some light to pass through it, although objects cannot be clearly seen through it. If one looks from a darker portion of the room toward a brigther portion with this screen interposed, the translucent spot will appear brighter than the ring of opaque paper around it, while, under the reverse conditions of illumination, the opaque ring will appear brighter than the spot. Mount the screen in any convenient way ; for example, in a block (No. 21). In making photometric observations with this screen the NATURE OF LIGHT: VISIBILITY OF OBJECTS. 99 illumination from one source of light is to be allowed to fall at right angles on one side of the screen, and that from the other source is to fall at right angles on the other side. The screen is then to be moved back and forth between the two lights until a position is found in which the appearance of the screen, as tested by the con- trast between the central spot and the rest of the surface, is exactly the same on both sides when viewed from the same angle. The illumination on the two sides is then equal, and the distances from the lights to the screen will afford a means of comparing the power of the lights, as already indicated in § 98. It is hardly worth while to attempt this experiment in an undarkened room. 101. Effect on Light of the Body on which it Falls. — When light-rays meet the surface of a body they may be : a. Regularly reflected : that is, sent off from the surface in a direction which can be calculated or foretold, if we know the direction in which they are to strike the surface, as sunlight is reflected by a mirror. b. Irregularly reflected or scattered: that is, sent back or off from the surface in many different directions, as sun- light is sent back from the surface of white cloth or white paper. c. Transmitted : that is, allowed to pass through as sun- light through clear window-glass. d. Absorbed : that is, neither reflected nor transmitted, but swallowed up, as sunlight by a lamp-black surface upon which it falls. It usually happens that more than one of these effects is produced by the same body at the same time. 102. Visibility of Objects.* — Very few of the objects we see shine by their own light, as we can tell by testing them in the dark. They merely give off the light, or some part * For much interesting and valuable matter upon this subject see Rood's Text-book of Colo?', Appleton & Co. 100 PHYSICS. of the light, which has fallen upon them from the sun, or from some other light-giving body. Of course we see many things every day upon which neither the sun nor any lamp is directly shining. We see them by what is called "daylight. 53 This, however, is sunlight, although it may not have come straight from the sun to the objects that we see lighted up by it. It may have gone from the sun to a mass of clouds, from the clouds to the surface of fields or streets or walls of houses, and from these surfaces into corners where the sun itself is never seen. It is extremely fortunate for us that all external objects do not treat the light which falls upon them in exactly the same way. If they did, all things would be of one color, and we could distinguish only light and shade. We have something like this condition after a fresh fall of snow which has covered roofs and trees as well as the ground. There are always, however, parts of trees and houses not completely covered by the snow, and this fact enables us to keep our bearings fairly well. If everything were covered by the snow, our eyes would not be of much more help to us in broad daylight than they are in the dead of night. 103. Colors of Transparent Bodies. — Colored pieces of glass, colored liquids, and other transparent bodies, gen- erally owe their color to the fact that they are not trans- parent to all kinds of light. The light which enters them, sunlight, for example, usually consists of many different colors blended together; and they rob this light of those colors which suit their own constitution, transmitting the rest. It is the transmitted, the rejected, light which we get from them that gives them their apparent color. The light which they absorb is turned to something else in the absorption, and is no longer light. It is usually turned into heat. NATURE OF LIGHT: VISIBILITY OF OBJECTS. 101 104. Colors of Opaque Bodies. — Most bodies with which we art? familiar do not appear to transmit light. We cannot see through them, and we call them opaque bodies. In fact, most so-called opaque bodies are not perfectly so. If they are made into very thin sheets, the sun can shine faintly through them. Even when they are in thick masses, the light penetrates a very little distance beneath the surface, where some of it is absorbed, and some, being reflected by interior particles, returns to the outside. This returning light is usually different in color from the mix- ture of lights that entered, certain parts having been absorbed more than others. 105. Light from Surface of Colored Bodies. — The light reflected from the real external surface of non-metallic colored bodies receiving white light is usually not colored. The following experiment shows an illustration of this fact: EXPERIMENT. Let a beam of direct sunlight, entering a window, fall very ob- liquely upon a sheet of colored glass in such a way that the reflected beam will fall upon a white surface. Observe the color of the reflected light. Certain materials, silks, for example, may reflect white light, from the outer surface, together with considerable colored light that has penetrated this surface and has been sent back from the interior. The white light gives the sheen, but in the spots where this is strong the color is not at the same time very evident, being made to look pale by the large amount of white light mixed with it. The following experiment will show how the color coming from an object maybe deepened by diminishing the amount of white light reflected from the external surfaces of its numerous particles. EXPERIMENT. Grind a lump of sulphate of copper to a fine powder and observe how faint the blue color becomes ; then wet the powder with water, 102 PHYSICS. which adds nothing but prevents some of the external reflection, and note the decided deepening of the blue. In velvet * the ends of the fibres, which reflect but little white light, are turned outward, and the light which pene- trates the surface and then returns to the outside is deeply- colored. * See Rood, p. 79. CHAPTEE IX. REGULAR REFLECTION OF LIGHT. 106. Reflectors. — Smooth, even surfaces, like the surface of still water, polished glass, or polished metal, reflect light regularly (§ 101). Transparent reflectors are not convenient for ordinary use: partly because light which we do not want may come through them from behind; partly because they reflect really well only such light as falls upon them very obliquely. EXPERIMENT. Let M t Fig. 62. be a piece of clear window-glass, L a lamp, and E the position of the observer's eye. The rays LM and J/^rnake a W *T large angle with the line 3IX, which is the normal to the surface of the glass. Observe the comparative brightness of the flame itself, and its picture or image, seen by reflection from 31. Xotice with what degree of clearness objects back of 31, as, for instance, points on the wall WW, can be seen through 31 in the direction E3I. Gradually move the lamp and the eye toward the point X, until at last both lamp and eye are as nearly as possible on the line X3L While making these changes of position observe any resulting changes in the brilliancy of the image of Z, and in the clearness with which objects on the line WW are seen through the glass. 103 104 PHYSICS. The reflecting surface which we make use of in a common mirror is not the front surface of the glass, but the metallic surface at the back. The glass is merely a convenient transparent support for the metallic layer, keeping it in shape and protecting it from being tarnished, as it soon would be if exposed to the air. Reflection from a Plane Mirror. 107. Where the Image Is. — A plane mirror is a flat mirror. We shall study curved mirrors later. When we place an object in front of a plane mirror and stand in a proper position we see an image, or "reflection," .^____ °^ the object, and we say that we see the object, or its image, in ~ the mirror. If M, Fig. 63, is •P t »P A the mirror, a point of the •P 2 .p 3 object, and P 1? P 2 , P 3 , and P 4 are the positions of four eyes, all fig. 63. may see at the same time an image of the point in the mirror. Our first Exercise in light is intended to answer the question whether all these eyes see the same image, that is, whether all are looking toward the same point, and if so, where this point is — in front of the mirror, or behind it, or at its surface. EXERCISE 1 7. IMAGES IN A PLANE MIRROR. Apparatus : A mirror (No. 23). A rectangular block (No. 9). A rubber band to hold the mirror to the block. Two straight- edged wooden rulers (Nos. 24a and 24b). A meas- uring-stick (No. 3). A sheet of thin white paper about 12 inches by 20 inches. A small block (No. 25). Attach the mirror to the large block by means of the rubber band in the manner shown by Fig. 64. -^ .i ., n Fig. 64. Draw a straight pencil -mark across the sheet of paper at its middle, and set the back surface of the mirror BEGULAR REFLECTION OF LIGHT. 105 \7* 3 directly over and parallel to this line, the middle of the mirror being very near the centre of the sheet. See Fig. 65. Draw on the sheet of paper in front of the mirror a triangle, each side of which shall be several inches long, and no corner of which shall be less than three inches from the mirror. It is well to have one angle of the triangle not directly in front of the mirror, but somewhat to one side, like point No. 1 in the figure. Place the small block in such a position that the vertical pencil-mark which it bears shall be directly over point No. 1 of the triangle. Then lay a straight- edged ruler (Fig. 66), upon the paper in such a position that one of its long horizontal edges, PQ, shall point directly to- ward the image of the vertical pencil-mark, as' seen in the mirror.* The ruler should be so placed that the line of sight will strike near one end of the face of the mirror. Then with a well-sharpened pencil draw upon the paper a fine clear mark alongside that edge of the ruler which lies just beneath the line PQ (Fig. 66) along which the sight has been taken. Mark this line 1, because it points toward the image of the vertical pencil-mark when this mark is over point No. 1. Next, without disturbing anything else, Fig. 66. p i ace the ruler in a new position, far removed sidewise from the position just occupied, sight as before, draw another line alongside the ruler, and mark this line also 1. Then with the ruler in a new position, about half-way between the first two, if this is convenient, draw a third line in the same way, and mark this also 1. All this time the small block has remained unmoved, and the pencil-mark upon it has pointed straight down at point No. 1. * Many persons cannot do this at first unless they are especially instructed. A person who is not near-sighted should hold his eye eight or ten inches distant from P, and should then direct the ruler in such a way that the point P, the point Q, and the image of the vertical pencil-mark seen in the mirror shall all lie in one straight line. Do not try to look along the vertical side of the ruler, but hold the eye high enough to see all the time the top of the ruler Fig. 65. 106 PHYSICS. Now place the small block so that the pencil mark shall point straight down at point No. 2. While it is in this position draw three straight lines toward the image and mark each one of these 2. Finally, put the pencil mark over point No. 3, draw three straight lines toward its image, and mark each of them 3.* When the three sets of lines Lave been drawn, the two blocks and the mirror are removed from the paper, and each line is then length- ened f until it cros es both the others of the same set ; that is, each No. 1 line is continued toward or beyond the mirror till it crosses the two other No. 1 lines. Then the No. 2 set and the No. 3 set are treated in the same way. After each set of lines has been extended in this way, it will be in order to answer the qu* stion whether all the lines of any one set lead to the "same point or nearly so, and, if so, where is this point situated with respect to the mirror and to the point whose image it is. If the image of each point, No. 1, No. 2, and No. 3, can be thus found, connect the image-points with each other by straight lines^ and thus make an image-triangle. Then fold the sheet of paper carefully along the pencil-mark by which the mirror was placed, and holding the folded sheet against a window, so that the light from without will shine through it, compare the size and shape of the two triangles and their relative positions with respect to the line along which the paper is folded. * While drawing all these lines the experimenter should look fre- quently to see whether the back of the mirror remains in place. It may be thrown out of place by a little blow or by rubbing the paper hard to remove pencil-marks. f If a line has to be extended far it is well to use two rulers, A and B, as shown in Fig. 67. First A is put into position and a line ~7T Fig. 67. is drawn alongside it. Then, while A remains unmoved, B is care- fully brought close to it, as the figure shows ; then B is held firmly in place whil^ A is pushed forward to tLe position indicated by the dotted lints. B is then removed without disturbing A, and again a line is drawn alongside A. In this way a line may be continued nearly straight for a considerable distance, REGULAR REFLECTION OF LIGHT. 107 'I he general rule for placing the image of any point should be recorded when it is found. The final result aimed at in this A B Exercise should be to enable the student to tell, without farther ex- periment, in any new case given him (Fig. 68, for instance, in which A B is the line upon which the mirror stands), the position of the image of points No. 1, No. 2, No. 3, and No. 4, and so the shape and position of the image of the figure at the corners of which these points lie. 108. The Law of Reflection.— In Fig. 69, MM is a I\ mirroi surface, CD a normal to this surface, OC a ray in- cident at the point C, and CP the same ray after reflection. The angle i is called the angle of incidence. The angle r is called the angle of reflection. The "law of reflection" is, that the angle of reflection is equal to the angle of incidence. This law is easily proved on the basis of what we have 108 PHYSICS. learned in the preceding exercise. The line of proof is this : The image of is at I. The angles at E are right angles ; EI — EO ; EC is common to the two triangles ; hence the triangle CEI is similar to the triangle CEO. Then angle % = angle EOC = angle EIC = angle r. 109. Real and Unreal Images. — If the rays of light pro- ceeding from a point are by any means really brought together again at a different point, as in Fig. 79, then the second point is called a real image of the first. A real image has an actual existence in space, and will show as a picture upon a properly placed white screen. If the rays of light proceeding from a point are by any means made falsely to appear to diverge from a different point, as in Fig. 70, then the second point is called an / / v V Fig. 70. unreal, or virtual, image of the first. A virtual image has no real existence in space, and would not show upon a screen placed where it appears to be. Evidently the image formed by a plane mirror is an un- real image. 110. Images of Images. — If any of the rays from (Fig. 71) after reflection from the mirror A fall upon a second plane mirror i?, they will be treated by this second mirror REGULAR REFLECTION OF LIGHT. 109 just as if they really came from I x \ that is, we shall, look- ing into the mirror B in the right direction, see an image of the image /„ and this second image, 7 2 , will appear just Fig. 71. as if it were the image of an actual object, sending rays from I r The rays reflected first from A and next from B might then fall upon a third mirror, and give an image of the image 7 2 , and so on; but at each reflection there is some loss of light, and an image formed after many reflections might be dim. 111. Positions of the Various Images. — Let A and B in Fig. 72 represent the positions of two plane mirrors meeting at right angles with each other at the point C. Let be a small object placed between the mirror faces. We shall have one image, 7 l3 formed by mirror A with- r 3*£_ Ji Fig. 72. 110 PHYSICS. out any help from mirror L\ and another, 7 2 , formed by B without help from A. There is also 7 3 , the image of / l5 seen in B\ and there is 7 4 , the image of 7 2 , seen in A, I b and J 4 fail at the same spot. We cannot with this arrangement of the mirrors get images of I 2 and / 4 ; because rays leaving mirror A as if diverging from I A would not strike the face of i?, and rays leaving mirror B as if diverging from 7 3 would not strike the face of A. Observe that and its images fill the corners of a rec- tangle. If were midway becween the mirrors, the rectangle would be a square, with at its centre. If the angle between the mirrors were made a bit less than 90°., I z and 7 4 would fall apart. If the angle were made 60°, one sixth of a circle, lying half-way between them, and its images would fill the corners of a regular hexagon having Cat the centre, If the angle were 30°, one tivelfth of a circle, and its images would fill the corners of a tweive-sided figure. EXPERIMENT. Place the hinged mirrors of No. XXV upon the board, with the reflecting surfaces making an angle of 90°, the point 1, Fig. 73, being midway be- tween them. Place a lighted candle, of such length that its flame will not be above the upper edge of the mirrors, ex- actly over the spot 1. Note the positions of the images of the candle seen in the mirrors. Put pegs into the holes behind the mirrors in such posi- tions that to an observer placed in front of the mirrors, so as to see the images in the mirrors and the pegs over the mirrors, the pegs will appear to coincide with the images. Then make the angle between the mirrors 60° and place pegs to coincide with the images. Finally, try an angle of 30°. REGULAR REFLECTION OF LIGHT, 111 112. The Kaleidoscope. — The preceding passages give an explanation of the kaleidoscope, No. XXVI, with which most oeautiful effects of endless variety can be obtained. The kaleidoscope uses bits of colored glass instead of a candle flame, and sometimes has three mirrors put together at angles of 60°. QUESTIONS AND PROBLEMS. (1) In a lighted room at night the glass of a window will serve as a mirror. In daylight unsilvered glass with a black cloth behind it may be used in the same way. Can you explain this ? (2) Soon after the moons of Mars were discovered in 1877 some one announced in a newspaper that one of these moons could be seen near Mars by looking at the reflection of that planet in a common mirror. It is true that a faint bright speck appeared near the image of Mars as thus seen, which did not appear when the planet was looked at di- rectly, but the true moons could be seen only by the aid of powerful telescopes. Can you, after trying the experiment with any bright star, explain the appearance seen in the mirror ? (3) Write some short word as it would appear in a mirror if the printed page containing it were reflected in the mirror. (4) A person standing in the middle of a room 20 ft. wide looks with one eye into a mirror 2 ft. square set in the wall of one side of the room. How many square feet of the wall behind himself could he see reflected in the mirror if his own image did not obstruct the view? {Suggestion : Draw a diagram representing the position of the ob- server, the mirror, the reflected wall and its image, all on a horizon- tal plane.) (5) A candle-flame is placed half-way between two plane mirrors which meet at an angle of 40°. How many images appear, and how are they arranged ? Reflection from Curved Mirrors. 113. Spherical and Cylindrical. — Most curved mirrors are parts of spherical surfaces. We shall, however, study mirrors which are parts of cylinders. They are more con- venient for our use than spherical mirrors, and they are less expensive. 112 PHYSICS. We shall use both the convex, or bulging, and the concave, or hollowed, face of the mirror. 114. Centre of Curvature, etc. — MM, Fig. 74, represents a cut through a cylindrical mirror at right angles with the straight lines of its ^3i surface. This cut is of course a part of a circle. C, the centre of the circle, is called the centre of curvature of the mirror. The point is called the centre of the mirror. The line CO, extended to any distance fig. 74. in either direction, is called the principal axis of the mirror. Any straight line extending, like CR, through C and across the line MM, but not through the point 0, is called a secondary axis of the mirror. EXERCISE 18. IMAGES FORMED BY A CONVEX CYLINDRICAL MIRROR. Apparatus : The mirror (No. 27). A measuring-stick (No 3). Small block (No. 25). Rulers (No. 24 A and b). Sheet of white paper. The plane mirror (No. 23) and its supporting block (No. 9), Hold the mirror with its straight edges vertical, and look at the image of your own face in the convex surface. You will see that the image is distorted, appearing too narrow for its length. Hold the mirror with its straight edges horizontal, and the image will be dis torted in the opposite way, appearing too wide for its iength. The object of the following experiments is to give a better understanding of these curious effects. Set the mirror on the table and bring one end of the plane mirror close to the surface of the curved mirror, as in Fig 75 Then place the small block in front of both mirrors, as in Fig. 75, in such a position that you can see the block reflected in both mirrors at the same time. Do the two images thus seen appear of the same height ? Do they appear of the same width ? REGULAR REFLECTION OF LIGHT. 113 Fill out, if you can, the following statement : Lines of the object which are parallel to the straight lines of the cylindrical mirror appear in the cylindrical mirror plane mirror. Lines of the object which are at right angles icith the straight lines of the cylindrical mirror appear in the cylindrical mirror in the plane mirror. Fig. 75. Remove the plane mirror. Holding the base of the curved mirror firmly in place, make a fine, clear, pencil-mark on the paper along the outer edge of the mirror. Then mark on the paper the point (7, which is the centre of curvature of the mirror. About 5 cm from the front of the mirror draw an arrow 6 cm. long, marking the ends and the middle as in Fig. 76. Then place the small block so that the vertical pencil-mark which it carries will point straight down at point No. 1. With the straight-edged ruler draw two lines, well apart, toward the image of this vertical line as seen in the mirror, avoiding parts of the mirror, if there are such, that do not give a good image of the line. Mark each of these lines 1. Then draw two lines for point No. 2 and two for point No. 3, in the same way. 2 > »>^> - Fig. 76. 114 PHYSICS. Then clear the paper and prolong each pair of lines till it conies to a crossing-point. The three points thus found will locate the images of object-points No. 1, Noi 2, and No. 3, respectively, and a line con- necting these three image-points will give an idea of the shape of the image-arrow, whether it is straight or not, and whether its curvature, if it has any, is in the same general direction as the curvature of the mirror or in the opposite direction. Draw a straight line from each marked object-point to the corre- sponding image-point, and prolong these three lines until they cross each other. Note where the crossing occurs. Is the image longer or shorter than the object ? Is it nearer to, or farther from, the mirror than the object is ? (It must be understood that the pupil is asked these questions only in regard to the particular case that he has tried. He cannot tell with- out further experiments or further instruction whether the answers he gives in this case would be true for all cases of objects reflected in mirrors such as he is using, for he does not know that the distance of the object from the mirror may not decide all these questions. The fact is, however, that, if he has found correct answers to the questions asked for his one case, the same answers will be true for the same questions in all cases with convex cylindrical mirrors. The effects seen with concave mirrors are much more complicated.) Fig. 77. 115. "Law of Reflection " Still Holds.— With curved mirrors, as with plane mirrors, the law (§ 108) angle of in- REGULAR REFLECTION OF LIGHT. 115 cidence = angle of reflection holds. With the help of this law we can see why the image of a point is nearer the mirror, when this is convex, than the point itself is. Let (Fig. 77) be the object-point in front of the convex mirror JO/, the centre of curvature being at C. A line drawn from C to any point of the mirror is at right angles with the mirror at the point of crossing. Two rays going from to the mirror-front appear after reflection to come from /, which is nearer the mirror than is. 116. Principal Focus and Focal Length. — If rays come from some very distant point on the principal axis (§ 114), they are practically parallel to each other when they reach the mirror. Two such rays are represented by r 1 and r 2 (Fig. 77). Applying the law of reflection to them, we find that after reflection they appear, as r/ and r 2 ', to diverge from a point P, which is very nearly midway between the reflecting surface and the centre of curvature- The point P is called the principal focus of the convex mirror. It may be defined as the point which marks the image of an object-point situated a long distance away from the mirror on the principal axis, or as the point from which rays coming to the mirror parallel to the principal axis appear to diverge after reflection. The distance, measured along the principal axis, from the principal focus to the reflecting face is called the focal length of the mirror. The principal focus and the focal length play a very im- portant part in the science of curved mirrors and lenses (§ 136). More will be said of this later. See § 124. 117. Concave Mirrors. — If the concave side of the mirror were used, it is easy to see from Fig. 78 that rays from a point near the mirror-front would after reflection appear to come from a point /, which is farther from the mirror 116 PHYSICS. than is. It is evident that the rays from are more nearly parallel to each other after reflection than before. Rays from a point 0', somewhat farther from the mirror than 0, appear after reflection to come from a still more distant point, I\ and these rays are nearly parallel after reflection. It is easy to see that if the object-point were put somewhat farther still from the mirror, the rays pro- ceeding from it might, after reflection, be parallel to each other. They would appear to come from a point as far as possible behind the mirror. If the object-point is placed still farther away from the mirror, as at in Fig. 79, the rays may after reflection be actually converging, and cross at a point I in front of the mirror. This image I is a real image (§ 109), and if is bright enough, the image /may be seen, like a picture, on a piece of white paper or cloth placed in the right position. REGULAR REFLECTION OF LIGHT. 117 We see that the centre of curvature lies between the object-point and the image-point / in the case shown by Fig. 79. This is always so in the case of real images formed by concave mirrors, unless the object-point is at (7, in which case the image-point also falls at C. If the object-point were placed where / now is, in Fig. 79, the image-point would fall where now is. EXERCISE 19. IMAGES FORMED BY A CONCAVE CYLINDRICAL MIRROR. Apparatus: The same as in the preceding Exercise, and in addition a common pin. Preliminary . Remove the mirror from the base- board ; place the latter upon the paper* and mark on the paper the point and the curved outline of the board. Make the distance OA 4.2 cm., and draw the arrow A 4 cm. long. Draw radii from G through the ends of A. Make the distance GB 3.5 cm., and draw B from radius to radius. Make the distance GD 1.5 cm., and draw D from radius to radius. (All this should be done before the regular Exercise begins.) Place the mirror in position as in Fig. 80, and, keeping the eye about 20 cm. from it, look at the images of A, B, and D. Do the images of A and B point in the same general direction, from left to right in the figure, as the arrows themselves ? Is the same answer true of D and its image ? Are the images of A and B longer or shorter than the arrows them- selves ? At the centre of A stand the pin upright, and laying the two rulers on the paper, point one edge of each toward the image of the pin, contriving to have these edges make a considerable angle with each other. In this way the position of the image is located. Is it behind the mirror or in front ? Is it, then, a real image or an unreal one ? 118 PHYSICS. By the same method locate the image of the pin when erected at the centre of B and when at the centre of D, asking and answering in each case the same questions that were asked when the pin was at the centre of A. If time permits continue the Exercise as follows : Extend the two radii r r by the lines r' r' drawn on the paper, as in Fig. 80. Draw the arrow E, 6 or 8 cm. distant from G, marking points 1, 2, and 3, upon it. Locate the image of each of these points by the method used in the preceding Exercise with the convex side of the mirror, drawing upon the paper the lines of sight and the image of the arrow. 118. Principal Focus of Concave Mirror. — Tlie principal focus of a concave mirror is the point to ivhich rays, coming to the mirror parallel to the principal axis, converge after reflection. In other words, it is the point which marks the image (real) of a very distant point on the principal axis. As in the case of a convex mirror, the principal focus lies very nearly midway between the reflecting surface and the centre of curvature. 119. Rule for Placing Images. — Fig. 81 illustrates an Fig. 81. Fig. 82. easy rule for finding the position of an image in a convex mirror. Let AB be the object. Draw one ray from A straight toward the centre of curvature. This ray will return on REGULAR REFLECTION OF LIGHT. 119 itself after reflection, coming as if from C. Draw another ray from A parallel to the principal axis. This will after reflection appear to come from P, the principal focus (§ 116). Both reflected rays appear to come from A\ which is therefore the image of A. The image of B is found in the same way. If the object is a straight line, as in this figure it is cus- tomary to represent the image by drawing a straight line from A' to B '. This is inaccurate, as Exercise 18 should show. Fig. 82 shows the same method applied to a concave mirror. 120. Distorted Images. — In Exercises 18 and 19, and in all the figures that have been given representing cylindrical mirrors, we have been dealing with rays which are, both before and after reflection, parallel to the plane on which the mirror rests. If we make use of other rays, as we do when looking obliquely down at the mirror face, we see things sadly twisted, the effects thus obtained being too difficult for oar profitable study. 121. Relation of Cylindrical to Spherical Mirrors. — If we were to use a spherical mirror, placed with its principal axis horizontal, and employ only horizontal rays striking the mirror on a narrow horizontal strip through its middle, we should get effects quite like those we have already studied. All the figures from 71 to 82 would apply as well to a spherical mirror so used as to a cylindrical mirror. Indeed, these figures are like those commonly given to show the effects obtained with spherical mirrors. For general use spherical mirrors are better than cylindri- cal mirrors, because they can be used from more points of view without giving badly distorted images. 120 PHYSICS. EXPERIMENTS. With a concave spherical mirror 5 or 6 inches wide (No. XXVII) 3f-~~" M interesting lecture-table experiments may be made in a slightly darkened room, the image of a candle-name or, better, gas-flame being thrown upon a screen so as to be visible to all in the room. The screen should be uf tracing Lq cloth or oiled paper, so that the image upon it S may be seen from both sides. An opaque screen q 9 should hide the flame itself from the eyes of the class. Fig. 83 suggests a good arrange- ment, MM beng the mirror, G its centre of cur- vature, L the flame, S the opaque screen, and S' the tracing- cloth screen. The positions of L and S may be greatly varied and may be interchanged, but the least g' distance of either from mirror the should be Fig. 83. rather more than one half the radius of curva- ture of the mirror, if real images are desired. 122. Principal Use of Spherical Mirrors. — Although spherical mirrors are sometimes used to form images, as in certain telescopes, probably their most important use is to concentrate light upon some object that cannot otherwise be well seen. Thus, the small objects which are to be looked at with a microscope need to be brightly illuminated, and a concave mirror is commonly used to throw light upon them. 123. The Ophthalmoscope. — Often a physician wishes to see what is wrong in the depths of a patient's eye. To do this tfie interior of the eye must be especially lighted up. If this is done by holding a flame in front of it, the flame dazzles the eye of the observer and therefore is of little use. The difficulty is overcome by means of the REGULAE REFLECTION OF LIGHT. 121 ophthalmoscope, Fig. 84, where M is a curved mirror with a hole in the centre; L is some source of light, placed so that the rays proceeding from it to the mirror pass by re- flection into the eye of the patient, represented by E\ and marks the position of the observer's eye. This simple application of the concave mirror was made by the great physicist Helmholtz, and it has probably won for him more popular fame and gratitude than all his other work. The most remarkable thing about many inventions is the fact that they were not made earlier. 124. Formulas Relating to Curved Mirrors. — In the fol- lowing formulas, which are here given without proof, D o = the distance of object-point from mirror, D t = the distance of image of object-point from mirror, F = focal length of the mirror. For a convex mirror we have A + A F For a concave mirror we have - + - = - A A F when the object-point is farther from the mirror than the principal focus is, and J_ JL_ _ 1_ A A ~ F when the object-point is between the principal focus and the mirror. It is doubtful whether work done with the cylindrical mirrors will be accurate enough to give results agreeing with these formulas. Similar formulas are used with respect to lenses. 122 PHYSICS. QUESTIONS. (1) A small object is placed close to a convex mirror. (a) Is the image real or virtual ? (b) If the object is moved farther and farther away from the mirror, will the image at any time become real ? (2) If one looks at the image of his own face in a convex mirror, will the nose appear too prominent and the forehead and chin re- treating, or will the opposite be true ? (3) If a small object is placed close to a concave mirror — (a) Is the image real or virtual ? (&) If the object is moved farther and farther away from the mirror, will it reach such a position that its image will be real ? If so, what is that position ? (4) (a) Have you in using any single mirror, plane or curved, seen a virtual image that was inverted, as compared with the object? (b) Have you seen any real image, formed by a single mirror, that was right side up, as compared with the object ? (5) Do you see anything wrong with the physics of the following statement, copied from a prominent newspaper ? — There are times when the public sees things in a convex mirror, in which they appear broad, robust, and expanded. There are times when the public sees things in a concave mirror, in which they appear cramped, narrow, and contracted. CHAPTER X. REFRACTION OF LIGHT. 125. Introductory. — In the experiment made with a prism the class may have noticed that the light did not go in the same direction after leading the prism as before entering it. Some members of the class in looking into pools or vessels of water may have noticed that objects beneath the surface are not exactly where they seem to be. EXPERIMENTS. (1) Place a straight stick in an oblique po-ition, partly in and partly out of water. Notice the apparent bendiDg or disconnection of the stick at the surface of the water. (2) Place on a table a pan (No. XXVIII), 15 cm. or more in diam- eter and with nearly vertical sides 4 or 5 cm. high. Place a small coin on the bottom of the pan, and adjust the head in such a position that the side of the pan will just hide the more distant por- tion of the coin from the eye at E (Fig. 85). Maintain the head in this position by resting it against any convenient sup- port ; k:ep one eye closed, and look with the other into the pan, just beyond the farther edge of the coin, while another person slowly pours in water. Have the pouring stopped as soon as the whole of the coin becomes visible. (3) Repeat experiment 2 with the eye held vertically above one edge of the coin, with a slender stick or a stout wire laid across the top of the pan, nearly in the line of vision, to serve as a point of departure from which to measure the apparent displacement of the coin, if any should be observed. 123 Fig. 85. 124 PHYSIC 8. 126. Interpretation of the Preceding Experiments. — Objects always appear to the eye to be in the direction from which the rays are travelling at the moment of entering the eye. Evidently, then, since the coin appeared to rise when the water was poured into the jar, in Experiment 2, the light-rays which proceeded from the coin must have been bent aside in some way by the water. In Eig. 85 the straight line CE, which passes from the left-hand edge of the coin C to the pupil of the eye at E, represents the course of a light-ray from that point before the water was poured into the pan. Any ray that passed farther to the right than CE would be intercepted by the side of the pan ; any ray that passed farther to the left, or more nearly vertical than CE, would miss the eye : hence it is evident that, so long as the pan is filled with air only, and the eye kept in the position shown, the coin cannot be seen. But as soon as water is poured into the pan, the rays no longer travel in straight lines from the object C to the eye. £ach ray suffers an abrupt change of direction at the sur- face of the water, and from this it follows that such rays as those which take the general course CS in the figure are finally brought to meet the eye at E. As a result of the bending C becomes visible, and its farther edge is seen apparently at 6 y ', in a position somewhat raised above the bottom of the pan. Experiment shows that the course CSE might be retraced by a ray. That is, a ray leaving E in the direction E8 would reach C by the line SC. QUESTIONS. (1) If normals were drawn to the surface of the water, at the points about S where the rays emerge, would the bending of each ray be towards or from the normal (in the air) ? REFRACTION OF LIGMT. 125 (2) If the rays were passing from E to C, would the bending at the surface of the water be toward or from the normal (in the water) ? When we look straight clown into water at any small object it appears to be in its true direction from the eye, but nearer than it really is. Fig. 86 indicates w T hy this is so. represents the object, E the eye, much magnified, and 0' the apparent position of the object. 127. Angles of Incidence and Refrac- tion. — The change of direction which a ray of light under- goes when it passes obliquely from one medium into another is called refraction. The amount of the bending, or refraction, which a ray of light suffers at any surface depends partly upon the two substances which meet at this ^ surface, and partly upon the angle, i (Fig. 87), which the ray makes with a line JYjV, which is at right angles with the surface at the point where the ray strikes the sur- face. If the space above the line AB represents the air-space, and that below this line the water, or glass, or whatever substance it may be that lies there, solid or liquid, the course of the ray is changed at the surface in such a way that the angle r which it makes with NN inside the solid or liquid is smaller than the angle ». C -B N Fig. 87. 126 PHYSICS. The angle i in Fig. 87 is called the angle of incidence. The angle r is called the angle of refraction. If the ray were represented as coming in the opposite direction, that is, first along R and then along /, r would be the angle of incidence and i would be the angle of re- fraction. The ray would be bent just as much at the sur- face as it is when going first along / and then along R. 128. Index of Refraction, — When the direction of I is changed the direction of R is changed. The way in which the change of one depends upon the change of the other is easily shown by means of Fig. 88. /, /', and I" show three rays all of which come to the point and then sep- arate, the first going along i2, the second along i?', the third along R" . The circle whose centre is at C is drawn with any convenient length of radius. The dotted lines, w, n\ n'\ and m, m\ m" 9 are drawn from the points where the rays cut the circumference to the line NN 9 at right angles. BEFR ACTION OF LIGHT. 127 If this figure has been drawn so as to acord with the results of experiments on light-rays, we shall have n __ n' _ n" m m! m'" and any one of these equal ratios is called the Index of Re- fraction of the second medium, heloiv AB, with reference to the first medium, above AB. n If now we can measure — in any given case, we shall have a quantity which is very useful in physics, for by means of it we can calculate at once the value of a new m to go with any new n\ that is, we can, if we know the index of refraction and the angle which any ray makes with NNvcl one medium, find without further experiment the angle which the same ray makes with NN in the second 71 medium. Exercise 20 shows how to find the ratio — for m the case of air and glass. EXERCISE 20. INDEX OF REFRACTION OF GLASS* Apparatus : A piece of plate glass (Xo. 28). Articles 3, 24a and 24b. A sheet of paper and three pins. Place the glass, G (Fig. 89), on the paper P. Stick one pin up- right at the point 1 close to one of the polished edges of the glass ; stick the other pin at 2 close to the other polished edge. Look with one eye from the position S through the whole width of the glass at pin No. 1. Move the eye toward 3, looking all the time through the glass at the pin. It will presently be noticed that the pin seen through, the glass is not in the same direction from the eye as the same pin seen over the glass. That which is * I owe tbe plan of this admirable Exercise to Mr. F. M. Gilley of the Chelsea High School. It is described in Gilley's Principles of Physics, Allyn & Bacon, Boston. — E. H. H. 128 PHYSIOS. seen through the glass is an image of the real pin, and it is upon this image that the attention should be fixed. Continue moving the eye in the general direction of 3, keeping it, however, about 30 cm. from the glass, until the image of pin No. 1 is just hidden behind pin No. 2. Then place a pin at 3, in the same straight line with the eye, pin No. 2, and the image of No. 1. Draw a fine pencil-line upon the paper close to the glass edge touched by pin No. 2. Then remove the glass. The line now drawn marks the position of the refracting surface. The line 1-2, Fig. 89, shows the direction, within the glass, of a 1 >*L G S A. 2 p \3 Fig. 89. Fig. 90. certain ray from 1. The line 2-3 shows the course of the same ray after it leaves the glass at 2. The line NN' , Fig. 90, is drawn nor- mal to the refracting surface at the point of emergence. From this point equal distances are laid off, to B and to C. Lines are drawn from B and from G to the line i\W at right angles. CE-z- BD = the index of refraction from air to glass. EXERCISE 21. INDEX OF REFRACTION OF WATER. Apparatus : Articles 3, 14, 15, 24a, 24b, 29, 30, and a sheet of paper about 6 inches square. Put the partition i^in place, as shown in Fig. 91, and pour water into the jar until its surface comes within 1 or 2 mm. of the middle tooth of the partition. Then by means of the plunger (No. 14), at- tached to the side of the jar by means of its clasp, raise the level of BEFBACTIOjST OF LIGHT. 129 the water till the apparent distance between the middle tooth of the partition and its reflection in the water surface is less than 1 mm. (To see this reflection well, one should look through the wall of the empty part of the jar.) Then the brass index b is attached to the jar, as shown in Fig. 91, and is raised or lowered, with the tip p touching the glass, until an eye on the line Cg, 20 or 30 cm. from the jar, can barely see p> the very tip of b, apparently in a straight line with Cg. This setting should be made with care, and after it is made the experimenter must look to see whether the tooth at G is clear of the water. If its lowest edge touches the water the setting is useless, and all of the adjustments must be made anew before a reading is made. P d/-m-} Fig. 91. Fig. 92. When all the adjustments have been successfully made, measure carefully the distance from the top of the jar down to the tip, p, of the index, the measuring-stick being kept outside the jar. Measure now the inside diameter of the jar. Measure also, unless it is already known, the distance * of G below the top of the jar. * It is well to have this distance, which is somewhat troublesome to measure accurately, given by the teacher. Partitions of different depths might be used in order to vary the angles of incidence and refraction. If the jar used in this Exercise is not pretty level at the top, or if the partition is not just at the middle of the jar, it is well, after making one setting of the index and one measurement of its position, to turn the jar about, transferring the index to the other side, and make a new setting and a new measurement. The mean of the two measurements thus made should be nearly free from any error caused by irregularity of the jar or of the partition's position. 130 PHYSICS. Now make a drawing, of full natural size, of the sides of the jar (inner lines), the water surface and the partition, as in Fig. 92, con- tinuing the partition line, by means of dots, well down into the jar. Put p in its proper place, and then draw the lines pC and Cg. Lay off Cd = Cg, and then draw the lines n and m. The index of refraction from air to water is — . m 129. Index Different for Different Colors. — In Exercise 20 the observer may have noticed a tint of bine or of red at the edge of the image of the pin. The fact is that light of various colors comes from the pin, and that the rays are not all refracted alike, the blue being refracted more than the red. The index of refraction is therefore different for light of different colors, but for our present purpose we need not dwell upon that fact. We get a sort of average index by the method of Exercises 20 and 21. 130. Relation between Index of Refraction and Velocity of Light. — The velocity of light in any transparent sub- stance depends on the nature of the substance. It is greatest in a so-called vacuum. It is least in the most highly refractive substances, and, indeed, the index of re- fraction for any given substance depends upon the rate at which light travels through it. This is sometimes illustrated by an analogy suggested by REFRACTION OF LIGHT. 131 the march of troops over ground of various kinds. Suppose a column of troops to be marching over smooth ground, represented by the space to the left of the line SS ' in Fig. 93. The front of the column being at AB, let the line SS' represent the border of a marsh or other difficult ground. Upon entering, the right of the column, B, first encounters the marsh, and the soldiers at B will fail behind those of the rest of the front. In consequence of this the column will, one part after another, wheel to the right until, when the whole front has entered the marsh, it will have the new direction shown by the line A'B'. Substitute for the column of troops a beam of light, and for the marsh a highly refractive transparent substance, and one may get some notion as to how refraction depends upon the retard- ing effect of refractive substances upon light-rays. 131. Total Internal Reflection: Critical Angle. —In Fig. 94 we have air above the horizontal line and w^ater, Fig. 94. glass, or some such transparent medium below the line. A ray of light R x may come from beneath to the surface at such an angle with the normal that it will after refraction at be parallel to the refracting surface. The ray R t com- ing up to at a larger angle with the normal will not pass out to the air, nor will it skim alons; the surface. It will 132 PHYSICS. be reflected at the point 0, the surface acting as a perfect mirror, and will follow the course R'^ the angle of reflec- tion being equal to the angle of incidence. The angle a, ivhich must not be exceeded if the ray is to 2mss ont into the air, is called the Critical Angle. The reflection which takes place when this angle is ex- ceeded is so good that it bears the especial name total reflection. EXPERIMENTS WITH TOTAL REFLECTION. (1) With the eye at E, Fig, 95. look at right angles into a glass prism shaped like ABC, at the same time holding an object at 0. Note the position of the image 0' and its re- markable distinctness. (2) In Fig. 96 SS' is a disk or square of thin wood about 10 cm. wide, LO is a piece of knitting-needle about 8 cm. long. The wood floats in water which fills a vessel to the brim AB Push the needle down until its upper end is nearly level with the upper surface of the board, and look down obliquely through the water, close past the margin of the board, at the lower extremity of JE± Fig. 96. the needle. Now draw the needle up, little by little, through the floating board until the point is reached at which the needle just vanishes from view, the line of sight being made at last as nearly horizontal as possible. Lift the board from the water, and note how much of the needle still projects below the board. REFRACTION OF LIGHT, 133 When the point is at 0', the light-ray going from it to S passes out into the air. When the point is at 0, a light-ray OS suffers total reflection along SR. The angle OSN 1 , or its equal SOL, is nearly equal to the critical angle. Of course no great accuracy can be expected here. Effect of Transparent Plates and Prisms. 132, Transparent Plates. — A plate of glass, or other transparent material, with plane parallel sides, as in Fig. 97, refracts light which enters it obliquely, but refracts it equally and in the opposite direction when it comes out at the opposite side of the glass, so that the entering and emerging rays are parallel to each other, although, as Fig. 97 shows, they do not lie in one straight line. Evidently a thick plate of glass will, other things being equal, set the emergent ray farther to one side, from the line of the original ray, than a thin plate will. 133. Prisms. — A. prism, in the study of light, is usually a piece of glass, or other transparent material, bounded by three rectangular and two triangular faces. DEF in Fig. 98 represents one end of such a prism. J) Fig. 97. It is evident that light entering the face DE from air will be refracted toward the normal NM. Going through the prism to the face DF 'it passes out into the air, being re- 134 PHYSICS. fracted again, this time from the normal M'N\ so that the two refractions have bent the ray far from its original direction. The total bending or deviation suffered by a ray in pass- ing completely through a prism depends on a number of things. 1st. On the angle which the two faces passed through make with each other. This angle is called the refracting angle : see D in Fig. 98. The greater this angle is, other things being equal, the greater the total deflection will be. We have seen in § 132 that if the two faces are parallel the total deviation is zero. 2d. On the color of the ray. This fact has already been noticed. Eed light is deviated less than blue light. 3d. On the angle tvhich the ray makes with the first sur- face. The total deviation is least when the ray strikes in such a way as to follow, within the prism, a course parallel to ZE7, ."Fig. 99, which makes the distance AI equal the dis- tance AE, and makes the refraction equally great at both surfaces. EXPERIMENT. Repeat the experiment of § 89, varying the angle at which the sunlight strikes the first face, in order to show that there is one in- A clination which gives a less total deflection of the light than any other position. REFRA CTION OF LIGHT. 135 4th. On the material of the prism. All kinds of glass do not refract equally. 134. Dispersion: The Spectrum. — The separation of rays of different colors by a prism is called dispersion. The spot or band of colored light produced by the dis- persion of a sunbeam is called the solar spectrum. It is customary to divide the spectrum into seven regions, called red, orange, yelloiu, green, Hue, indigo, violet, and to call the general colors of these the primary colors, to dis- tinguish them from those formed by compounding two or more of them. This division of the spectrum is a mere matter of convenience. We might name a hundred colors of the spectrum if we chose to do so. So long as we keep to any one refracting material the dispersion is, in general, greater when the average deviation of all the rays is greater. Thus with a given prism the dis- persion is least when all the rays go through the prism as the ray IE goes in Fig. 99. When prisms of different material are used, two kinds of glass for example, one may disperse the rays more than the other, while producing no greater average deviation of all the rays ; or one may disperse the rays about as much as the other while deviating them, as a whole, much less. 135. " Achromatic " Prisms. — Two prisms of nearly equal dispersive power but of unequal deviating power may be combined, as in Fig. 100, making a com- pound prism which produces considerable deviation with very little final dispersion. Such a combina- tion is called achromatic, that is, colorless. Achromatic combinations of lenses (§ 149) are used in many optical instruments. Fig. ioo. 136 PHYSICS. Lenses. 136. Shapes of Lenses. — A lens is, usually, a piece of glass whose two faces are parts of spherical surfaces. Sometimes there is a cylindrical surface between the two spherical faces. Fig. 101 shows various lenses as they would look if cut through the middle. Fig. 101. Lenses are classed as convex, or converging, and concave, or diverging. Convex lenses are all thicker in the middle than at the margin, and cause parallel light-rays to con- verge, as in Fig. 102. Concave lenses are thinner in the Fig. 102. Fig. 103. middle than at the margin, and cause parallel light-rays to diverge, as in Fig. 103. Some of the lenses used in the most accurate optical in- struments have convex or concave surfaces, which are not strictly parts of spherical surfaces. Such lenses possess certain advantages over spherical-surface lenses (see § 147). 137. Definitions Relating to Lenses. — The lenses we shall use will be much like No. 1 in Fig. 101. The two sides are supposed to be just alike. REFRACTION OF LIGHT. 137 To understand such a lens better we will make use of Fig. 104. (7 is the centre of the spherical surface of which A SB is a part. It is called the centre of curvature of the face ASB. C is the centre of curvature of the face ABB. The straight line HCOO'K, continued to any distance in each direction, is called the principal axis of the lens. Any straight line going, like LM, obliquely through the centre of the lens is called a secondary axis of the lens. If the two faces of a lens are exactly alike, as we suppose them to be here, any ray of light going through the centre of the lens, the point 0, will hare the same direction after leaving the lens as before entering it, because the two little- spots of surface at which it enters and leaves the lens are parallel to each other, so that the ray is affected just as if it were going through a plate with parallel faces.* is called the optical centre of the lens. Eays entering a convex lens "parallel to its principal axis, as in Fig. 102, are refracted in such a way that after leav- ing the lens they will cross this axis. They do not all cross at one point, but if the faces are near together, and are very small parts of spherical surfaces, as in our lenses, such rays will cross at or near a certain point, F, on the principal * The direction of the ray within the lens is, of course, not quite the same as its direction before entering. This fact is not shown in Fig. 104. 138 PHYSICS. axis, and this point is called the principal focus of the lens. There are two principal foci, one on each side of the lens. See points i^and F' in Fig. 104. The distance from the principal focus to the nearer face * of the lens is called the focal length of the lens. Focal length is a quantity of very great importance in dealing with lenses, and the next Exercise will show how to find it by experiment. For this purpose we need to have the light come to the lens in rays nearly parallel to each other and to the principal axis. This we can do by taking the light from any small spot of any distant but distinct object; for instance, a chimney or a church-spire outlined against the sky. EXERCISE 22. FOCAL LENGTH OF A CONVERGING LENS. Apparatus : The lens (No. 31) mounted on a block. A meter-rod (No. 2). A small block (No. 21) bearing a white cardboard screen (No. 32). A common pin. Fig. 105. First Method. — Place the lens and the screen upon the rod, as in Fig. 105, and point the rod at some distant object, seen against the sky, in such a way that the light from this object will pass from the lens and then fall upon the screen. Move the screen back and forth * Bee Appendix I, REFRACTION OF LIGHT. 139 until that part of the image * which lies on or near the principal axis of the lens is made as distinct as possible. Then by means of the grad- uations of the meter-rod, or by an independent measuring-stick if this is preferred, note the distance from this part of the image to the nearer face of the lens. This is the focal length. Second Method. — Remove the screen from its block and put the pin upright in its place. Let the pin, thus mounted, be placed on the meter-rod, about as far from the end of the rod as the pupil usually holds a book from his eyes when reading. Place the lens somewhat farther from the same end of the rod. Place the eye at this end of the rod and, looking sharply at thepm, direct the rod and adjust the lens in such a way that the light from some distant object will pass through the lens and form an image in the air close to the pin. To decide whether the image is nearer the eye than the pin is, move the eye to and fro, to the right and the left, watching the pin and the image. f If the pin is more distant than the image, it will, when the eye is moved toward the right, appear to move across the image toward the right. If the pin is nearer than the image, it will, when the eye is moved toward the right, appear to move across the image toward the left. The rod should not be held in the hands during this test, but should be placed on some steady support. Continue the adjustments until the test described fails to show which of the two, the pin or the image, is nearer the eye. Then meas- ure the distance from the pin to the lens. It should be the focal length of the lens. Compare the values of the focal length given by the two methods. The second method is more difficult, but it is instructive, and it *The image is formed because light coming from anyone small spot of the object is brought to a small spot again by the lens. The image is made up of such small spots each in its own place. For the purposes of this Exercise the distant object need not be more than 30 or 40 feet from the experimenter. The images on the screen will be much more distinct if the apparatus is used in the back part of the room, well away from the windows. f To see the reason of the test just described, close one eye and hold the two forefingers, some inches apart, in line with the other eye, so that one finger hides the other. Then move the eye to the right and left, and notice the apparent movement of the fingers with respect to each other. 140 PHYSICS. can be used in cases where the image is too faint to show clearly upon the screen. 138. Discussion of Exercise 22. — It is common to speak of the rays coming to a lens from a distant object as parallel rays. . This does not mean that rays coming from different parts of the object to the lens are parallel to each other. It means merely that rays coming from any one spot of the object to the lens are parallel, or very nearly parallel, to each other. In fact, if rays from the different parts of a luminous body could be converged to the same point, the result would not be an image repeating the features of the original objects. It would be a mere point, or very small patch of light. The image seen in the Second Method is, like that of the First Method i a real image (§ 109), but it is in the air. As there is an image in the air, we may well inquire why this image cannot be seen by a whole class at once without the use of a screen. It is because the light forming the image in the air goes straight on through this image, and can be received only by placing one's self behind the image. The light which forms an image upon a screen is by the threads of the screen reflected back in all directions, and therefore some part of it reaches every eye. QUESTION. If a bright point were placed at the principal focus of a lens, what direction would the rays going from this point to the lens have after passing through the lens ? 139. Object-distance and Image-distance: Conjugate Foci. — Two points so placed tvith respect to a lens that an object placed at either of them will have an linage at the other are called Conjugate Foci of the lens. REFRACTION OF LIGHT. 141 EXERCISE 23.* RELATION OF IMAGE-DISTANCE TO OBJECT-DISTANCE : CONJUGATE FOCI OF A LENS. Apparatus : The same lens that was used in Exercise 22. A meter-rod. Block (No. 9). Small block (No. 21), with a cardboard screen (No. 32). Small kerosene lamp with an asbestos band around the chimney (No. 33). Arrange the apparatus according to Fig. 106. The hole in the as- Fig. 106. bestos band, lighted up by the name behind, is the object the image of which is to be received upon the screen. One end of the meter- rod is placed vertically beneath this illuminated hole. Place the screen at first at a distance from the object about equal to three times the focal length of the lens. Then move the lens back and forth on the rod between the object and the screen, and see whether in any position it gives upon the screen a clear image of the object. If it does, measure the distance from the lens in this position to the object, and write this distance as the first number in a record- column headed D (object-distance). Measure also the distance from the lens to the screen, and put this distance as the first number in a record-column headed Di (image-distance). If, with the present position of the screen and object, there is no position of the lens that will cause a distinct image of the object to fall upon the screen, move the screen one or two centimeters far- ther from the object, and then try again to get a good image. If still none is found, move the screen still farther away, continuing the trial till a distinct image is obtained. Then measure and record the * To economize space upon the laboratory- tables it will probably be necessary to have pupils work in pairs in this Exercise. Each pair should know the focal length of its lens at the outset, so as to lose no time in beginning the Exercise. 142 PHYSICS. D and Di as already described. (Very little time need be spent upon these first successive trials.) Then at one move place the screen about 10 cm. farther still from the object, find a position of the lens that will give a distinct image, measure and record D Q and Di as before. Without moving the screen, see whether there is any other position of the lens that will give a distinct image ; if there is, measure and record the D Q and the Di for this position of the lens. Move the screen 10 cm. farther away, and then do exactly as before. If there is time, move the screen two or three more times, adjust- ing the lens, measuring, and recording each time. It is better to make a moderate number of settings and readings well than a large number carelessly, but an error of one or two millimeters in these readings will be of no great consequence. 140. Discussion of Exercise 23. — The distance from object to image in any case of Exercise 23 is D Q + D { , and we may call this D oi . This distance was shortest in the first case recorded. Let each member of the class divide the D oi of this case by the focal length of his lens. Is there any general agreement between the quotients thus found ? When the screen was farther away, was there usually more than one position of the lens that would give a distinct image, the screen remaining unmoved ? If you were told that in a given case the D Q was 20 cm. and the A 60 cm., could you tell what the other possible D Q and Di would be for the same positions of object and screen ? Look at your record-columns for Exercise 23, and see whether they help you to answer this question. Let each member of the class call F the focal length of the lens which he used, and let him test the truth of the formula. 1 = 1 + 1 REFRACTION OF LIGHT. 143 or, what means the same, D xD =F(D +A), for all cases tried and recorded Ly himself in Exercise 23. PROBLEMS. (1) Do for a certain case is 50 cm. and Dx is 100 cm. How great is Ft (2) If Di is 80 cm. and Fis 20 cm., how great is D ? (3) If D Q =Di, we will call each D. (a) What in this case is the relation between F and D1 (b) How does this agree with your observations in Exercise 23? 141. Real Image Formed by a Lens. — In the preceding Exercises the object presented to the lens has been small, or has been at such a distance as to give a rather small image. It is now desirable to study larger images, and to study them with especial reference to their shape and size, rather than their distance from the lens. We shall in the next Exercise find the shape and size of an image of an arrow placed at right angles with the principal axis of the lens and not far from the lens. We shall not attempt to find the whole image at once, but shall find separately the images of several points of the arrow, and then make an approximate image of the arrow by connecting these points. EXERCISE 24. SHAPE AND SIZE OF A REAL IMAGE FORMED BY A LENS. Apparatus: The lens (No. 31). Measuring-stick (No. 3). Block (Xo. 21) carrying in the narrow slot on its top a piece of wire (No. 34) extending first horizontally and then downward (see Fig. 108). A ruler (No. 24). Block (No. 25). A sheet of paper about 30 cm. wide and 1 m. long, having near one end an arrow 8 cm long, drawn at right angles with a pencil-mark about 30 cm. long, and marked, or numbered, as shown by Fig. 107. Weights (No. 19) to hold the cor- ners of this sheet in place on the table. Arrange the apparatus as shown by Fig. 108, the centre of the lens over a point on the long pencil-mark, at a distance from the centre of 144 PHYSICS. the arrow about equal to one and a half times the focal length of the lens, and block Xo. 25 in such a position that the vertical mark upon its face points straight down to point Xo. 3 of the arrow. This ver- tical mark will now cross the principal axis of the lens, if the lens is accurately placed. Place the other block near the other end of the paper in such po- sition that the vertical part of the wire it carries shall be near the A1 ■m Fig. 107. Fig. 108. principal axis of the lens. Keep the eye 20 or 30 cm. distant from this part of the wire, on a level with the centre of the lens and in line with the centre of the lens and the vertical part of the wire. Look at this part of the wire so as to see it distinctly, and note whether you can see at the same time, near the wire, the image of the pencil- mark on the farther block. If so, find out by moving the eye to the right or left, as in Exercise 23, whether this image is more or less distant from the eye than the vertical wire is. Then move the block carrying the wire into such a position that the image and the wire seem to keep close together when the eye is moved a considerable distance to the right or left. When this adjustment is made, put a dot on the paper just beneath the vertical wire and mark this dot 3. It represents the image of object-point Xo. 3. Find in a similar manner the image-points 1, 2, 4, 5, corresponding to the object -points 1,2, 4, 5. The experimenter must take care not to let any idea he may have as to the position where an image-point ought to be affect his judgment in deciding where it is. After all the five image-points are found, connect them, No. 1 to No. 2, No. 2 to Xo. 3, etc., by means of straight lines, thus getting a rough representation of the whole image. REFRACTION OF LIGHT. 145 Draw from each object-point toward the corresponding image-point a straight line as long as the ruler (No. 23), and note the point where ^hese lines cross each other. 142. Formation of the Image in Exercise 24. — The for- mation of the image-points in Exercise 24 is illustrated by Fig. 109. One ray from the object-point A follows a secondary axis (§ 137) passing through the centre of the lens, and its direction after leaving the lens is the same as before entering it. (Its direction inside the lens is not quite the same, but the figure does not show this.) Another ray from A runs parallel to the principal axis (§ 137) before entering the lens, and will therefore pass Fig. 109. through the principal focus, /, on the farther side of the lens. The crossing of these two rays at A' shows the posi- tion of the image of A. In a similar way B\ the image of i?, is located. 143. Size and Shape of Image. — If a straight line is drawn from A' to B' in Fig. 109, we mav call this the length of the image, although the images of points between A and B will not lie on this line. It is evident from Fig. 109, and also from the figure obtained in Exercise 21, that the distance A'B' is to the distance AB as the distance of A'B' from the lens is to the distance of AB from the lens. The curved shape of the image obtained in Exercise 24, if the work has been correctly done, is due to the fact that the ends of the object-arrow are farther from the lens than the centre of the arrow, and to the further fact 146 PHYSIOS. that the focal length along a secondary axis is less than the focal length along the principal axis. This latter fact can easily be shown by direct experiment with either method of Exercise 22. 144. Virtual Image Formed by a Lens. — We see in Exercise 24 and in Fig. 109, where the object-point is farther from a lens than its principal focus is ; that the rays going from this object-point to the lens are bent by the lens in such a way that, after leaving it, they converge to a point again. We know, too, that if the object-point were placed at the principal focus the rays going from it to the lens would emerge from the lens parallel to each other. It is not difficult to see that, if the object-point were placed letiueen the lens and its principal focus, the rays going from it to the lens would be divergent still, after leaving the lens, though less divergent than before entering it. In the next Exercise we shall have a case of this kind. EXERCISE 25, VIRTUAL IMAGE FORMED BY A LENS. Apparatus: The same as for the preceding Exercise except that the sheet of paper need not be more than one half as long, and that the arrow upon it should be 4 cm. long and about 20 cm. distant from one end. Place the lens between the arrow and the nearer end of the sheet of paper, at a distance from the arrow equal to about two-thirds of its focal length, and in such a position that its principal axis extends over the middle point of the arrow. Place the small block (No. 25) with vertical pencil-mark pointing straight down at the middle point, No, 3. of the arrow. Turn the vertical part of the wire on the other block so that it will point up instead of down, and place this block some distance behind the other one. Holding the eye 20 or 30 cm. from the lens, look through the lens at the image of the vertical pencil-mark, and at the same time oxer the lens at the vertical part of the wire. Bring the wire into line with the image, and then by the usual test find which of them is the more distant. Move the wire back and forth until it coincides in REFRACTION OF LIGHT. 147 position with the image. Then mark with a figure 3 the point just under the vertica lpart of the wire. This represents the image of object-point No. 3. In a similar manner locate the images of points 1, 2, 4, and 5. Connect the image- points by straight lines, from 1 to 2, from 2 to 3, etc.. thus forming an image of the arrow. Draw a straight line from each image point to its corresponding object-point, and note where these lines will cross each other if con- tinued. 145, Formation of the Image in Exercise 25. — The images observed in Exercise 25 were virtual images. They could not be shown tipon a screen, and were not formed by the actual crossing of light rays. Fig, 110 will serve to illustrate the way in which virtual image-points are formed. Let AB be the object, placed between the lens LL' and the principal focus F\ To find the position of the virtual image of the point A, draw ^[/parallel tc the principal axis of the lens, This ray will, after leaving the lens, pass toward F, the principal focus * on the farther side, and so will appear to have come along the path 31 F. Draw another ray, AC\ passing through the centre of the lens This ray will, after leaving the lens, have the same direction as before entering it, and will be represented by the line CiV". If, then, we carry back the line CiVtill it crosses the line MF, also carried backward, the point A\ where the crossing occurs, is a point from which both of the rays appear to come. A' is, then, the virtual image of A. By a similar process B' is found to be the virtual image of B. P\ the image of the point P, is here represented as lying in the straight line between A' and B'. It is usually so * The dotted lines drawn from M and N to F 'in Fig 110 are not intended to show the actual course of the rays within the eye. 148 PHYSICS. represented in books. Exercise 25 shows that it does not lie there, The image A'B' is evidently larger than the object AB. Whenever a virtual image is forced by a convex lens, this image appears, to an eye placed in any ordinary position on the other side of the lens, larger than the'real object would ^ A' i -^^— "" Fig. 110. look if held at a comfortable seeing-distance from the eye. Hence the name magnifying-glass^ so commonly given to a lens used as in Fig. 110. 146. Application of Formula. — The formula used in § 140 to express the relation betweeen focal length, object- distance, and image-distance in the case of real images, can be adapted to use with virtual images by merely changing the sign of one term, so as to make i A A To illustrate the use of this formula it will be well to measure the distance from lens to object-point 3, and from lens to image-point 3, in the diagram made in Exercise 25, and try them in the formula, with the known value of F, REFRACTION OF LIGHT. 149 147. Spherical Aberration in Lenses. — All the rays going from a point to a lens A do not, after passing through the lens, converge to a single point 7. Those which go through the lens near its margin converge to a nearer point J\ This imperfection of a lens is called spherical aberra- tion. When a very clear-cut image is needed, it is customary to put a stop in front of the lens; that is, a thin metal plate with a hole which permits only those rays to pass which are near the principal axis of the lens. Lenses can be so constructed, with surfaces not quite spherical, as to do away with this defect in great part, for light of any one color, but such lenses are difficult to make and are uncommon except in large telescopes. 148. Chromatic Aberration in Lenses. — Ordinary lenses, made of a single piece of glass, give rise to colored fringes or borders about the images which they produce. The cause for this defect, which is called chromatic aberration, is this, that the objects looked at send more than one kind of light to the lens and that rays of different colors are not refracted equally by the lens, and so do not come to a focus equally near the lens. But little trouble from this source is experienced in the use of lenses of slight convexity, whose images are not to be further magnified; as, for instance, in spectacles and ordi- nary magnify ing-glasses. " Stopping out " the greater 150 PHYSICS. portion of the surface of a lens with a circular diaphragm, which allows light to pass only through a small portion of the lens near its centre, improves its performance greatly. How much help such diaphragms give by reducing spherical and chromatic aberration, may be learned by taking out some or all of the diaphragms of an ordinary cheap spy- glass, and then looking with it at distant objects in bright sunlight. 149. Achromatic Lenses.— Fortunately for the manufac- turers and users of optical instruments, it is possible to make an achromatic lens, or one, at any rate, which is practically achromatic. This is usually |b accomplished by uniting into one lens two sepa- rate lenses,* one, A, of flint-glass, and the other, B, of crown-glass, as shown in Fig. 112. A con- vex lens made in this way has, on the whole, a converging effect on parallel rays, while at the same time the superior dispersive power (§ 134) of the flint-glass enables the lens A, though of less fig. 112. refractive power than the lens i?, just to coun- teract the dispersive tendency of the latter. Many of the lenses used in optical instruments of the best quality are achromatic. Eye-pieces (§ 164), however, of the ordinary pattern do not require achromatic lenses. A large lens practically free from spherical and chromatic aberration is a marvel of skillful and patient work. Grlass suitable for making a large lens of the best quality is very difficult to procure, as a very slight flaw or unevenness of quality may spoil a large block. The shaping and polish- ing and testing of the largest lenses, after the proper kind of glass is obtained, is a work of years, and men who are * Sometimes more than two pieces are employed in making an achromatic lens. REFRACTION OF LIGHT. 151 skillful and patient enough to do it become known through- out the world. For many years the largest and best lenses for great telescopes have been made by Alvan Clark and his two sons of Cambridge, Massachusetts; but now all of these famous men are dead. The largest lenses ever made, 40 inches in width, were placed in the great telescope of the Observa- tory of Chicago University by the last of the Clarks a few weeks before his death in 1897. QUESTIONS AND PROBLEMS. (1) An object is placed at a great distance from a converging lens and on its principal axis. (a) What changes of position will the image of this object undergo while the object is moved along the principal axis up to the surface of the lens ? (b) In what part of this operation will the image be erect and in what part inverted ? (c) In what part will it be real and in what part virtual ? (2) In Exercise 25 the .virtual image of a straight line was found to be a curve. How should a line be curved with respect to the lens in order to make its virtual image a straight line ? (3) The focal length of a certain convex lens is 15 cm. (a) How far from the lens will the image be if the object is 30 cm. from the lens ? (b) How far if the object is 10 cm. from the lens ? (4) An object is 40 cm. from a convex lens and the image equally far from the lens. What is the focal length of the lens ? (5) If the object mentioned in problem 3 is 5 cm. long, how long will each of the images there mentioned be ? (In answering this question disregard the curvature of the images.) (6) A bright point, which is more distant from a converging lens than its principal focus is, sends white light to the lens. Which falls nearer the lens, the red image of the point or the blue image ? Why ? (7) What would be the answer to the questions in (6) if the point were between the lens and its principal focus? CHAPTEE XI. THE EYE: SIGHT AND COLOR. 150. Parts of the Eye. — The eye as an optical instru- ment consists of a liquid lens A (Fig. 113) called the aqueous humor, a solid lens, B, called the crystalline lens, a transparent jelly-like mass 0, called the vitreous humor, and a screen rr, called the retina, upon which the image of the object looked at falls. The aperture at the back of the FlG ' 118 ' eye is occupied by the optic nerve leading from the retina to the brain. 151. Accommodation. — Muscles attached to the lens B have power to change its form to some extent, thus adapt- ing the eye to see distinctly near or distant objects at will. This is called the power of accommodation. A normal eye, that is, an eye approved by physicians, has such shape as to give upon the retina distinct images of very distant objects without effort. In accommodating itself to see nearer objects such an eye has to make an effort, which grows greater as the distance lessens, but does not become painful until the object looked at is less than eight or ten inches from the eye. 152. Far-sight and Near-sight. — Some eyes lack the power of accommodation for near objects, and are called far-sighted, or long-sighted, although they cannot see dis- tant objects any better than normal eyes can. 152 THE EYE: SIGHT AND COLOR. 153 Some eyes are slightly egg-shaped, the retina being farther back than in normal eyes. These eyes are called near-sighted, or short-sighted, because they are well adapted for seeing near objects, while they cannot see distant objects distinctly. For some purposes near-sighted eyes have a certain advantage over normal eyes, for they enable their possessor to hold an object very near, when there is need, and so make it look larger than it would look to the normal eye. 153. Eye-glasses.' — Ear-sighted eyes must wear convex lenses to help them converge the rays from a near object to an image upon the retina. Near-sighted eyes must wear concave lenses to prevent the rays sent by a distant object from coming to an image in front of the retina. The Perception of Color. 154. The Color-sense. — In the retina are found the ends of the nerves through which we get the sensation of light and of color. Although the eye can distinguish scores of different tints, it is believed that the sets of nerves operating in the percep- tion of colors are very few, probably not more than three or four. Each set of nerves is supposed to give one peculiar color sensation and only one ; but the combination of these few primary color sensations in various proportions is sup- posed to give all the other color sensations. It is very generally believed that the primary color sensa- tions are three — red, green, and violet. 155. Mixing Color Impressions. — The most convenient way to find the effect of mixing color sensations is to place variously colored pieces of paper on some body which can be made to spin rapidly before the observer's eyes. Tops or other whirling apparatus No. XXXV, for example, can 154 PHYSICS. be used for this purpose, and indeed the whole outfit for this kind of experimentation is now readily obtained. EXPERIMENT. Place a red paper, a green paper, and a violet paper upon a whirl- ing apparatus, and so vary the proportions of the visible parts of these papers that when rapidly whirled before the eye they will produce the effect of gray. (In the study of color all shades of gray, from brilliant white to dead black, must be classed together as white, the difference between them being merely a difference of brightness.) 156. Complementary Colors. — It has already been shown that ordinary white light is composed of many different colors, ranging from red to violet, but it is not necessary to put together all of these colors in order to get the sensation of white. There are many pairs of colors, any one pair of which will give the sensation of white when its elements are mixed in the right proportions. The two colors making such a pair are called complementary to each other. Thus, according to Eood, red is complementary to green-blue, orange " " " cyan-blue (between blue-green and blue), yellow " " " ultramarine-blue, greenish-yellow " " violet, green te " purple. EXPERIMENT. Place blue and yellow disks upon the whirling apparatus, and so proportion the visible parts that when revolving rapidly they will produce the effect of gray. Try the same experiment with other pairs of complementary colors. 157. Fatigue of the Retina. — If one looks steadily for a short time at some strongly colored object held against a background of gray or white, that spot of the retina upon which the image of the colored object falls loses in part, for THE EYE: SIGHT AND CO LOB. 155 the time being, the power of giving the particular color sensation which it is furnishing, while its power of giving other color sensations may remain as great as ever. This temporary loss of power is called fatigue of the retina, and it may give rise to curious effects. EXPERIMENT. Hold a piece of bright green paper against a white background, and look very steadily at one spot on this paper for thirty seconds. Then look steadily at some one spot of the white surface for a few seconds and note any peculiar color effect that is observed. The color complementary to green will probably appear as a patch upon the white, the shape of this patch being exactly like that of the green paper. Try the same experiment with other colors. 158. After-images. — The effects observed in the follow- ing experiment are still more curious than those of § 157. EXPERIMENT. Look steadily for half a minute at some 'particular spot on a win- dow haying the sky as a background. Then close the eyes and wait a few seconds for the figure on the window to show out against the darkness. Watch the changes of color the figure undergoes. Ob- serye that details appear in this persisting image which were not noticed while the eyes were open. 11 After-images " like the one here mentioned, cannot be the work of memory. They must be due to some change of state in the retina, some real impression made there, which lasts for a considerable time but gradually passes away. CHAPTER XII. OPTICAL INSTRUMENTS. 159. Importance of Optical Instruments. — Much of the progress of science during the nineteenth century has been due to improvements in the construction of optical instru- ments and their more general use in scientific investigations. Improvements in telescopes and the invention and per- fection of the spectroscope have enabled the astronomer to discover, and even to measure, objects and motions whose existence was unsuspected by the observers of two genera- tions ago. The chemist is to-day able by means of the spectroscope to ascertain in a few minutes the presence, in a substance of unknown composition, of elements which it would have taken him days to detect by purely chemical means. To the physician, the food-analyst, the manufacturing druggist, and to those engaged in many other professional or technical occupations, the microscope is a necessary piece of apparatus, a tool of daily, almost hourly, use. Optical instruments comprise a great variety of combina- tions of mirrors, lenses, and prisms. Only some of the simpler ones can be referred to in an elementary book on physics. 160. The Photographer's Camera. — This instrument consists essentially of a box, in the front of which is fastened a convex lens or a combination of lenses, L (Fig. 114), the distance of which from a ground-glass screen, P, at the 156 OPTICAL INSTRUMENTS. 157 other end of the box. may be varied at will. An inverted and usually diminished real image of any o.itside object not too near L may be formed on P. When this adjustment lias been precisely made, the lenses are covered with an opaque cap; a plate of ordinary glass, coated with a film of gelatine made sensitive to light by the presence in it of cer- tain compounds, usually of silver, is substituted for P; the cap is then- removed, and the light is allowed to act for a A p L \/ \J Fig. 114. sufficient time upon the sensitive plate, after which the cap is replaced and the plate removed and " developed " into a photographic t; negative/* Those who are interested in practical photography will find in Exercise *24 some explanation of the difficulty ex- perienced in making all parts of the ground-glass screen show clear images at the same time; and in § 1-47 there is a suggestion as to the effect of ; * diaphragms " with larger or smaller holes. 161. The Magic-lantern. — This instrument, known also by various other names, stereapticon, for instance, requires a powerful source of light, such as a large kerosene-flame, or some form of calcium-light A (Fig. 115), in which a cylinder of quicklime is heated by a flame formed by burn- ing together oxygen ami coal-gas, or. bust of all. the electric arc-light. By means of a large lens B (Kg. 115), called 158 PHYSICS. the condenser, a powerful beam of light from this source is thrown upon the painted or photographed " slide," the Fig. 115. image of which is to be exhibited. This slide is pushed into the opening (7, a little outside the focus of a smaller convex lens or a pair of such lenses, D, and a greatly enlarged real image of the slide is thrown upon the screen. The throwing of large images upon a screen is called pro- jection of these images and apparatus used for this purpose is called projecting apparatus. 162. Projecting a Spectrum. — A kind of spectrum has been shown in the experiment of § 89, but a better disper- sion of the colors can be obtained by means of some device like that described in the following experiment. If sunlight is not available, the stereopticon, if provided with a calcium" light or an electric arc-light, can be successfully used, the prism being placed in the path of the rays after they have traversed the projecting lens. EXPERIMENT. By means of a porte-lumiere (No. XXX ) throw a beam of sunlight through a narrow slit at S, Fig. 116. Place a lens, L, in the path of the beam, and adjust it so as to throw a distinct image of the slit on a screen at I. Now introduce a prism, P (2s o. XXXII), in the OPTICAL INSTRUMENTS. 159 position shown in the figure, and then place the screen at RR', making the distance PR equal to PI The prism used may be of flint-glass, or, better, may be hollow and filled with the highly dis- persive liquid bisulphide of carbon. Examine the spot of colored light on the screen. (1) How many colors can be distinctly seen ? (2) Do they blend, or are they sharply Fig. 116. separated from each other? (3) Which color is most refracted? least refracted? Try the effect of passing the emergent pencil through a second prism similar to the first, and placed so as to re- fract the light in the same direction as the first. Try the effect with a second prism so placed as to refract in the opposite direction from the first. 163. The Simple Microscope. — In its least complicated form the simple microscope, or magnify] ng-glass, consists of a convex lens used, as explained in § 145, to form an upright magnified image of any small object. When much magnifying-power is required, two or even three convex lenses, mounted one over the other with their surfaces only a few millimeters apart, are often used. Such combinations are called doublets or triplets, according to the number of lenses composing them. They have certain advantages over single lenses of equal magnifying-power. The discussion in § 145 will helj) the student to see that the magnifying-power of a simple microscope is greater as its focal length is less. 164. The Compound Microscope, — For viewing objects under any but the lowest magnifying-powers, that is, in all 160 PHYSICS. cases wher^ the apparent diameter of the image is to be anywhere from 50 to 5000 times the actual diameter of the object, the compound microscope is employed. The essen- tial optical parts of this instrument, as usually constructed, are (see Fig. 117), an eye-piece, LL\ here represented as single, but generally consisting of two convex lenses, and an objective, 11, frequently consisting of from two to six pieces. These lenses are fixed in a brass tube so arranged that the distance between the eye-piece and the objective can be varied at will, within certain limits. A mirror, not here shown, which is adjustable to any desired angle, is asually employed for throwing light upon the object. The object to be viewed is placed on a platform beneath OPTICAL INSTRUMENTS. 161 the objective, and is strongly illuminated by light reflected from the mirror. A real, inverted, magnified image, A x B l9 of the object is formed within the tube of the instrument at a position somewhat nearer to the eye-piece than its principal focus. This real image is therefore magnified by the eye-piece, which forms an enlarged virtual image, A' B\ of it at a position not far from the object. The foci of the object-glass are at/ and/', those of the eye-piece at F' and F. The total magnifying -power of the instrument is that of the objective multiplied by that of the eye-piece. In general, the shorter the focal length (see Appendix I) of a microscope objective, the greater its magnifying-power. An objective of one inch focal length will, on a tube 10 inches long, give, with the lowest power eye-piece in com- mon use (the " A " eye-piece), a magnification of about 50 diameters; with an eye-piece of double the magnifying- power (" B " eye-piece) the total magnification will be about 100 diameters, and so on. EXPERIMENT. Fasten a page of fine print, P in Fig. 118, upright on a table in a good light. Set up in front of it a short-focus convex lens, L, at a distance from the page somewhat greater than the focal length. If Fig. 118. Hold another short-focus convex lens, L\ in various positions farther from the page until one position is found in which an eye close to L' sees through it an inverted, magnified image of the print, this being a virtual image of the real image formed by the lens L. This appa- ratus is a rude model of the compound microscope. 162 PHYSIOS. 165. The Astronomical Refracting Telescope. — This in- strument consists essentially of the long-focns object-glass, or objective, L (Fig. 119), mounted in one end of a tube, at Fig. 119. the other end of which is placed an eye-piece, L\ precisely similar to that of the compound microscope. The eye- piece can be moved toward or away from the object-glass in order to make the image appear most distinct. The real image of any distant object is, of course, always formed by the objective very near its principal focus. The foci of the eye-piece are at i^and F'. Astronomical telescopes are always furnished with achro- matic object-glasses (§ 149). EXPERIMENT. Mount upon blocks two convex lenses, one of 30 or 40 cm. focal length, the other of about o cm. focal length. Set them up on the table with their principal axes coincident— that is, with their centres on the same straight line at right angles to the centres of their faces. Mount a bit of tracing-paper or greased writing-paper, and place this screen in such a position between the lenses that the one of greatest focal length shall throw upon it a distinct image of some distant bright object. Look at this image on the translucent paper through the 5-cm. lens Choose such a position and distance as to give a clea- virtual image, as much magnified as possible, of the real image on the screen. Now remove the screen, and observe that the virtual image of the real image is still visible. 166. Efficiency of the Telescope. — The usefulness of the telescope as an aid to vision depends upon the following QUESTIONS AXD PROBLEMS. 163 points: (a) the clearness and sharpness of the image, or what is called the definition of the instrument; (b) the brilliancy of the image; (c) the amount of allowable mag- nification. Good definition depends upon the accuracy with which the leas is shaped and finished, and upon the quality of the glass, which should be free from flaws. Brightness depends upon the amount of light which can be concentrated in the different parts of the image. Hence a large objective will, other things being equal, give the best illumination. In some recent telescopes the objective has a diameter of 3 feet or more. The magnification, with a given eye-piece, is evidently very nearly proportional to the focal length of the objec- tive; but unless the objective is large, and furnishes much light, it is useless to give it great focal length, for the reason that the much-magnified image would be too faint to be seen to advantage. QUESTIONS AND PROBLEMS. (1) How could you find the weight of a body that will float, if you had no balance but had a vessel filled with water and a "graduated * glass flask — that is, a flask with marks upon it showing the number of cu. cm. required to fill it to certain depths ? (2) If a liter of hydrogen weighs .0896 gin. and if the sp. gr. of oxygen as compared with hydrogen is 16, what is the weight of 1 cu. m. of oxygen ? (3) A certain volume of mercury of density 13.6 weighs 216 gm., and the same volume of another liquid weighs 14.8 gin. Find the density of the second liquid. (4) A piece of iron weighs 200 lbs. in air and 172.5 lbs. in water. How great is its sp. gr. ? (5) A given body weighs 500 gm. in air and 400 gm. in water. (a) How great is its volume ? (b) How great is its sp. gr. ? (6) A board 12 X 6 X 1 in. weighs 1.5 lbs. What is its density in lbs. per cu. ft.? 164 PHYSICS. (7) A cubical block of wood 15 cm. along the edge weighs 1125 gm. What is its density ? (8) A 30 cu. cm. body weighs 10 gm. in water. How great is its sp. gr. ? (9) What is the volume of a body which weighs 25 gm. in air and 20 gm. in water ? (10) A body weighs 180 lbs. in water and 120 lbs. in a liquid that is 1.8 times as dense as water. Find the volume and the sp. gr. of the body ? (11) How much will a kgm. weight of sp. gr. 7 weigh in a liquid which is 0.8 as dense as water ? (12) A cubical box, 3 ft. square on a side, made of 2 in. plank of sp. gr. 0.5, has a bottom but no top. It contains a body weighing 100 lbs. To what depth will this box sink, upright, in water ? Ans. 6.7 in. nearly. (13) The sp. gr. of air, as compared with water, is about .00129 at 0° C. under ordinary atmospheric pressure. How many grams would equal the buoyant force exerted by air in this condition upon a cu. m. of any substance? (14) If the sp. gr. of a certain block is 0.3 and its volume 100 cu. cm. , how much of it would be submerged if it were floating in a liquid of sp. gr. 2. (15) A rod floats one-half submerged in a liquid of sp. gr. 0.9. How much of it would be submerged in a liquid of sp. gr. 3? (16) There is a uniform rod 6 ft. long and 4 in. square, of sp. gr. 0.5. What must be the sp. gr. of a cubical piece of metal 4 in. on the edge which, when attached to the rod, would just hold it sub- merged in water ? (17) If a diver with his suit weighs 200 lbs. and it takes ^ of a cu. ft. of lead, sp. gr. 11.4, to keep him submerged in fresh water, how many cu. ft. of water does he, in his suit, displace? (18) Two boys are pulling at a rope in opposite directions, each with a force of 25 lbs. (a) How great is the tension on the rope? (b) How great would you call the tension if the rope were tied to a beam and supported a weight of 25 lbs. ? (19) A uniform beam, 12 ft. long and weighing 300 lbs., rests, horizontal, on a fulcrum 2 ft. from one end. How much weight must be applied at this end to make the beam balance in its present position ? (20) (a) Find the direction, position, and magnitude of the equil- QUESTIONS AND PROBLEMS. 165 ibrant (§ 74) of two forces, parallel and in the same direction, one of which is 10 lbs. and the other 12 lbs., their lines of action being 3 ft. apart. (b) Find the direction, position, and magnitude of the resultant (§ 75) of the same two forces. (21) One end of a horizontal beam 20 ft. long and weighing 50 lbs. rests upon a wall, and the other end is supported by a rope that will bear only 85 lbs. A boy weighing 100 lbs. walks slowly along the beam from the wall toward the rope. How far from the rope will the boy be when it breaks ? (22) A hammer is use! to draw out a nail from a board. The head of the hammer rests against the board at a distance of 3 in. from the nail. A force of 50 lbs. is applied at right angles with the handle at a point 12 inches from the boar^l. How great is the force exerted by the hammer on the nail ? (This case is similar in principle to some of those discussed in connection with the pulley. See Ex- periments under § 58.) (23) If a force of 50 lbs. is applied at the end of the handle of a "jack-screw " 18 in. from the centre of the screw, and if one revolu- tion of this screw lifts a weight 0.5 in., how great is this weight, if there is no frict.on ? (24) A sled weighing with its load 50 lbs. rests on the side of a hill rising 1 ft. in a distance of 5 ft. along the incline. (a) How great a force acting parallel to the incline is needed to keep the sled from sliding downward if there is no friction ? (b) If the crust on the snow is just strong enough to bear the sled under these conditions, how much would the load on the sled have to be lightened in order that a similar crust might bear the sled on a level ? (25) An inclined plane rising at an angle of 45° has a load of 50 lbs. resting upon it. How large a horizontal force will be needed to keep this load moving up the incline if there is no friction ? (26) A horizontal force of 10 lbs. is required to keep a certain body moving along a horizontal surface with which its coefficient of fric- tion is 0.2. How great is the weight of the body ? (27) A mass of 100 lbs. rests upon an inclined plane 10 ft. long and 4 ft. high. (a) How great must be the resistance of friction to keep the body from sliding down the incline ? (b) How great must the coefficient of friction be ? (28) If a simple pendulum 1 m. long vibrates 58 times a minute, 166 PHYSICS. what is the length, of a simple pendulum that vibrates 116 times in a second ? (29) The length of a simple pendulum vibrating once a f econd in the latitude of New York is about 39.1 in. How many seconds a day would a clock lose if controlled by a simple pendu'um 40 in. long ? (30) Two lights, A and B, are placed 20 ft. apart. The power of A is to that of B as 4 to 9. At what point between them must a screen be placed in order to be equally lighted up on both sides ? (31) The distance of the planet Neptune from the sun being 2,800,000,000 miles, nearly, how long does it take a wave of light to go from the sun to Neptune ? (32) What is the height of a tree which casts a shadow 100 ft. long, when an upright rod 5 ft. tall casts a shadow 7 ft. long ? (33) The image of an upright stake 8 ft. tall, and 10 ft. from a window-shutter appears on a screen 4 ft. beyond the shutter. The aperture in the shutter through which the light passes from the stake to the screen is very small. How great is the length of the image ? (34) The clock on a wall indicates 9.30. What time will it appear to indicate if the observer sees the reflection of the clock in a mirror on the opposite wall but does not distinguish the numerals ? (35) A plane mirror lies up m a table and a pencil 6 in. tall stands upright on one edge of the mirror. How wide must the mirror be in order that a person whose eyes are 5 in. above its surface and 20 in. distant from the pencil may just see the whole length of the pencil reflected in the mirror ? (To be solved by drawing and meas- uring. The thickness of the glass is to be neglected. ) (36) Prove that if an object is placed in front of a plane mirror and the mirror is moved either toward or from the object, without turning, the image will move twice as far as the mirror. (37) Prove that if a candle is placed in front of a vertical plane mirror and the mirror is turned 45° about a vertical axis, the image of the candle will move through an arc of 90° around the axis of the mirror. (38) Two plane mirrors, A and B, are placed 12 cm. apart, facing each other and parallel. A small object is placed between them 4 cm. distant from A. Calculate the distance from A to the first and second images seen in it. Do the same for B. (39) Two plane mirrors, placed vertical, make with each other an angle of 60°. A candle is placed between them, but nearer one than the other. Draw a figure showing the positions of the various images of the candle. QUESTIONS AND PROBLEMS. 167 (40) If the radius of curvature of a concave spherical mirror is 50 cm., and if a candle is placed 40 cm. distant from the mirror, (a) How far from the mirror will the image of the candle be ? (b) Will this image be real or virtual ? (c) Will it be erect or inverted ? (d) If the candle-flame is 2 cm. long, what will be the length of its image ? (41) If the candle mentioned in the preceding problem was 10 cm. from the mirror, what would be the answers to the questions there stated ? (42) What would be the answers in problems 40 and 41 if the mirror were convex ? (43) Have you ever seen curved mirrors used except in a class-room or laboratory? If so, for what purposes were they used? (44) Define the term index of refraction. (45) The index of refraction of the earth's atmosphere is little greater than 1 with respect to the space outside this atmosphere. Does this fact delay, or does it hasten, the first glimpse of the rising sun ? (46) For which of the colors here named is the index of refraction of glass the greatest — red, green, yellow, blue ? For which of them is it least ? (47) How could you find by experiment the color complementary to any given tint ? (48) Show that the image formed by a convex lens may be either larger or smaller than the object. (49) Prove algebraically, and also graphically (after the manner of § 142), that when thy distance of an object from a convex lens is twice the focal length, the image is at the same distance on the other side. (50) A rod 5 cm. long held in front of a convex lens, at right angles with the principal axis, has an image 25 cm. long upon a screen dis- tant 100 cm. from the lens. How great is the focal length of the lens? (51) An object 4 cm. long, placed 20 cm. from a certain lens and at right angles with the principal axis, has a real image 10 cm. dis- tant from the lens. If the same object were placed 5 cm. distant from the same lens, (a) Would the image be real or virtual ? (b) How far from the lens would the image be ? (c) How great would the length of the image be ? APPENDIX I. FOCAL LENGTH, ETC., OF LENSES AND COMBINATIONS OF LENSES. It is customary to define the focal length, F y of a single lens as the distance from the focus to the nearest point of the surface of the lens, and in the formula — = — - -f- — to consider D and Di as measured Jo JJo JJ\ from the object and image, respectively, to the nearest point of the lens. With this interpretation of the letters, the formula is not ex- actly fulfilled by any actual lens. It holds strictly true only for the ideal case of a lens of zero thickness, but it is sufficiently near the truth for common purposes in the case of ordinary lenses. The for- mula is about equally accurate, for a4ouble convex lens, at least, when all the distances, F, D , and Di, are measured to the optical centre of the lens (§ 137). When a combination of lenses is used, as in a microscope-objective or a photographic camera, a formula similar to that just given can be applied, but the F, D , and Di occurring in it are not now measured either to the nearest point of the combination or to the optical centre. They are measured to certain other points determined by the radii of curvature, thickness, and refractive index of each lens, and the dis- tance between the two lenses. In the ordinary use of such a combi- nation, its magnifying power is substantially equivalent to that of a single ideal thin lens having a focal length equal to what is called the focal length of the combination. The calculation of the focal length of the combination is frequently very laborious. Dealers in photogiaphic objectives very frequently state as the focal length of a combination of lenses the distance from the principal focus to the near, r surface of the nearest lens. They sometimes call this the " back focal length,' ' or, rather, the * * back focus" of the coni- 168 APPENDIX I. 169 bination. It is a convenient quantity to use in the description of a lens, but it is not intended for use in the formula -= — — 4- -=r . F D G Z>i The term " equivalent focal length/' or " equivalent focus," is sometimes applied, in the case of a combination of two equal lenses, to the distance from the principal focus to a point midway be- tween the two lenses. APPENDIX II. INDICES OF REFRACTION OF VARIOUS SUBSTANCES COMPARED WITH A VACUUM. (See § 128.) Agate 1.540 Canada balsam 1.53 Diamond 2.5 Fluor spar 1.434 Glass (ordinary crown). 1.53 " ( " flint)... 1.61* Ice 1.31 Quartz 1.544 Rock salt 1.544 Selenium (crystals) 2.98 Alcohol .., 1.36 Petroleum (heavy) 1.45 (light) 1.4 + Water 1.333 Nitrogen 1.000298 Oxygen 1.000371 * The dispersive power (§ 134) of flint glass is nearly twice as great as that of crown glass. APPENDIX III. All the articles in the first list here given should be furnished to each member of the laboratory section. LIST OF ARTICLES REFERRED TO BY NUMBER IN THE "EXERCISES" OF THIS BOOK. No. 1. A 10-cni. section of a meter-rod. No. 2. A meter-rod, marked on one side in feet and inches. No. 3. A 30- cm. bevel- edged measuring-stick, marked on one side in inches. No. 4. A waterproofed wooden cylinder about 8 cm. long and 4.5 cm. in diameter, loaded internally with shot so that it will float nearly submerged in water. No. 5. A brass can about 14 cm. tall and 7 cm. in diameter, having a slightly declining, straight, overflow-tube, about 6 cm. long and 0.8 cm. in internal diameter, extending from a point about 1.5 cm., clear, below the top of the can (see Fig. 6). To prevent dribbling the junction of tube and can should be covered, internalJy, with a coat of paraffin melted on. No. 6. A braes catch-bucket with a wire handle, capable of holding about 175 gm. of water, and weighing not more than 50 gm. No. 7. An 8-oz. spring-balance graduated to 0.5 oz. (There is now in the market an improved balance, graduated on one side in 10-gm. [Tni^iiiliinfiinjiinf 1 Fig. 120. intervals and on the other side in 0.25-oz. intervals. It is, moreover, especially adapted for use in the horizontal position. This improved balance is desirable for this course.) 170 APPENDIX III 171 No 8. A rectangular waterproofed block of wood, about 7 cm. long and 4.5 cm. square on the end, so loaded internally with shot that it will sink in water, but not enough to make it weigh more than 225 gm. No. 9. A rectangular waterproofed cherry block about 7.5 cm. X 7.5 cm. X 3.8 cm. This block should be smooth, and therefore the water- pr ioflng should be done by soaking it in very hot paraffin. For the best results this soaking should be done in a vacuum. Excess of paraffin should be scraped off before the block is used. No. 10. A one-gallon glass jar of good quality. (It is poor economy to buy a poor jar and have it break with a liquid in it.) No. 11. A lump of roll sulphur weighing about 175 or 200 gm. It is not worth while to cast these lumps into regular cylindrical form. No. 12. A lead sinker with w r ire handle, weighing about 175 gm. No. 13. A waterproofed wooden cylinder about 1 cm. in diameter and 20 cm. long. Doweling-rod, furnished by hardware dealers, serves well when waterproofed. No. 14. A holder for keeping No. 13 upright in water. It consists of a waterproofed wooden rod about 12 cm. long and 1.3 cm. square on the end, provided with a clasp for attaching it to the side of a jar, and with two screw-eyes projecting from one side, the rings of which are large enough to let the cylinder No. 13 slip easily through them, but not large enough to allow the cylinder to tip far from the vertical position (see Fig. 10). No. 15. A cylindrical glass jar, about 14 cm. tall and 10 cm. in diameter, with level top. No. 16. A broad-mouthed bottle with ground-glass stopper, stand- ing not much more than 11 cm. tall with stopper, and weighing, when filled with water, about 175 or 200 gm. No. 17. A lever and supporting-bar. The lever is a 30-cm. section from a meter-rod, pivoted upon the smoothed cylindrical body of a brass screw which is driven horizontally into the end of a bar of hard wood about 25 cm. long, 5 cm. wide, and 3 cm. thick. A brass plate projecting from this bar and overhanging the middle of the lever prevents the lever from tipping far, while it allows suffi- cient freedom of motion. The lever itself, except for a distance of 2 cm. each side of the middle, is cut away so that its top is level with the upper part of the hole through the centre. There should be a 172 PHYSICS. screw-hole running downward through, the middle of the supporting- bar, to facilitate in attaching it, as shown in Fig. 21. No. 18 (A and B). Two brass scale-pans about 6.5 cm. square, each with its suspending threads weighing accurately 1 oz. (that is, not differing from this weight by more than .01 oz.). Each pan is suspended by four strong linen threads meeting in a knot about 20 cm. above the pan, two of them continuing in a loop about 4 cm. long above this knot. (Fig. 21.) No. 19. A set of iron weights, 8 oz , 4 oz., 2 oz., and two 1 oz. , making a total of 16 oz. No weight should be in error more than .01 oz. No. 20. A flat pine board about 50 cm. long and 15 cm. wide for use in the Exercises on Friction. No. 21. A cubical block of wood about 3.7 cm. on each edge. A groove about 1 cm. wide and 2 cm. deep extends through the lower part of the block with the grain of the wood. An ordinary short screw extends through one side of the block into this groove, and serves to fix the block in position upon a meter- rod. Across the grain at the top of the block is a slot about 0.1 cm. wide and 0.5 cm. deep. (Figs. 26 and 105.) No. 22. Two bits of wood, each about 8 cm. long and 1 cm. square on the end, for supporting the spring- balance in a horizontal position. (Fig. 41.) No. 23. A plate-glass mirror about 15 cm. long, 3.8 cm. wide, and 0.2 cm. thick, the coating on the back protected by paint or varnish. No. 24 (A and B). Two straight- edged rulers of some wood that will keep its shape well — white pine, for instance — each about 30 cm. long, 5 cm. wide, and 1 cm. thick. No. 25. A block like No. 21, but without the large slot and the screw. One side of this block is coated with white paper, and a vertical pencil-mark or ink-mark is made across the middle of this paper. (Fig. 108.) No. 26. A Walter Smith " school {rquare," or other equally good protractor. No. 27. A cylindrical mirror of nickel-plated brass, about 5 cm. tall and 8 cm. wide, cut from seamless tubing 4 inches in diameter and J inch thick, mounted upon a sem circular base-board of wood of the proper radius of curvature. The base-board should be about 1.5 cm. thick. No. 28. A piece of plate-glass about 7 cm. square and 0.6 cm. thick, APPENDIX III. 178 for Exercise 20 on Index of Refraction. Two opposite edges or narrow sides of the glass should be ground tolerably plane and polished sufficiently to allow seeing readily through the whole width of the plate (see Fig. 89). No. 29. A brass partition made to fit the small glass jar (No. 15) and to extend downward into the jar a distance equal to about one- third the diameter of the jar. It should be made of sheet brass Fig. 121. about .07 cm. thick, The method of shaping and adjusting the par- tition is suggested by Fig. 121, where A shows a side view and B an end view of the partition. The flanges shown in B are bent more or less in adjusting the partition to fit the jar closely, but without too much pressure. No. 30. An index of thin sheet brass made to clasp the side of the jar (No. 15). This index is a strip about 15 cm. long, before bending, and 1 cm. wide, tapered to a point at one end. To enable it to clasp the jar, about 3 cm. at the untapered end is bent over. (See pb in Fig. 91.) No. 31. A circular (not elliptical) double-convex spectacle-lens, having a focal length not less than 12 cm. and not more than 16 cm. The lens is mounted on a block similar to No. 21. (See Fig. 105.) No. 32. A white cardboard screen about 8 cm. square, of such thickness as to be held firmly in the narrow slot of the small block No. 21. (Fig. 105.) No. 33. A small kerosene lamp of such size and shape as to fit it for the use shown in Fig. 106. The lower partof the chimney is sur- rounded by a thin sheet of asbestos paper, having a hole 3 or 4 mm. in diameter at the height of the flame. No. 34. A wire, of the right size to fit into the narrow slot of No. 21, bent at a right angle, one arm about 6 cm. long, the other about 4 cm, (Fig. 108.) 174 PHYSICS. ARTICLES USED BY THE TEACHER, BUT NOT TO BE FURNISHED TO STUDENTS. (Most of them are referred to by number in the " Experiments " of this book.) No. I. A gauge for testing pressure at various points and in vari- ous directions in a jar of water. In Fig. 122, P is a pillar of wood or metal about 25 cm. tall ; G is a small glass thistle-tube about 1.7 cm. wide ; m is a tbin rubber membrane fastened water-tigbt across the mouth of C ; p and p are hard-rubber pulleys about 1.7 cm. in diam- eter, fitting closely on their axes ; r is a small rubber tube ; g is a glass tube ; i is a short column of water serving as an index. A band of strip-rubber, such as toy stores supply, connects the two pulleys p and p, so that by turning a, the axis of the upper pulley, between the thumb and finger, the gauge-face m may be turned upward, down- ward, or sidewise, without changing level. A student-lamp chimney, with stopper for one end, accompanies this gauge. APPENDIX III 175 No. II, Apparatus for bursting a bottle by an attempt to compress water within it. The essentials are a glass bottle, with a perforated rubber stopper which fits the bottle well when driven in its full length ; a strong frame for holding the bottle and keeping the stopper in place ; a rod, with convenient handle, to be driven water-tight down through the hole in the stopper, No. III. Glass tube about 1 m. long, closed at one end, connected by a strong rubber tube 25 cm. long with another glass tube 20 cm. long. .See Fig. 12.) No. IV. Strong thistle-tube (Fig. 13) about 2.5 cm. wide, covered at the mouth with strong sheet rubber and furnished with a thick- walled rubber tube about 20 cm. long. No. V. Small air-pump suitable for both exhaustion and compres- sion. Such a pump is frequently sold without base, but it is well to have a base, bell-jar plate, and one or two bell-jars. For many purposes a larger pump is desirable. No. VI. Bent glass tube for Boyle's law, the whole tube about 1.5 m. long (Fig. 14). No. VII. Common large rubber foot-ball, with a rubber tube about 30 cm. long attached to the key. (Fig. 15.) No. VIII. Small bottle provided with rubber stopper fitted with two glass tubes as in Fig. 16. No. IX. Glass model of lifting-pump (Fig. 17). No. X Glass model of force-pump (Fig. 18). No. XI. Hydrometer for liquids less dense than water. No. XII. Hydrometer for liquids more dense than water. No. XIII. Glass U tube (Fig. 20) about 60 cm. long before bending. No. XIV. Some form of the Cartesian Diver. No. XV. Eight-inch and four-inch wooden disks combined in one piece for use as a pulley. This piece is fitted with various pins (re- movable) for suspending weights. It is mounted much like the lever of No. 17. (See Fig. 34). No. XVI. Centre-of-gravity board, with suspension and plummet. (Fig. 24.) No. XVII. Platform balance weighing from 1 kgm. to 0.1 gm., provided with a set of brass weights. No. XVIII. Well-made small brass pulley with a hook or loop. (Fig. 37.) 1Y6 PHYSICS. No. XIX. Well-made small double brass pulley with hook or loop. (Fig. 38.) No. XX. An inclined plane,* shown about one-fourth natural size in Fig. 123. The roller should be of brass, accurately turned. It weighs with its frame jus<- 16 oz. The graduations of the scale may be in millimeters. The apparatus should be made with care. No. XXI. Pendulum-support and pendulum-balls (Figs. 56 and 57). Fig. 123. No. XXII. Three small packages of dyestuffs soluble in water, various colors. No. XXIII. Three glass plates, red, green, and blue, about 10 cm. square. No. XXIV. Camera obscura consisting of two pasteboard tubes each about 25 cm. long. The larger, about 5 cm. in diameter, is closed at one end save at the centre, where there is a hole about 0.1 cm. in diameter in a thin partition. The smaller tube, about 4 cm. in diameter, is closed at one end by thin tracing-paper. (See § 94.) * A number of excellent features in this apparatus are due to Mr. Sweet, formerly of the Rindge Manual-Training School in Cambridge. APPEXDIX II J 177 No. XXV. Make according to the following directions : On a board about 35 cm. square (Fig. 73) lay off a circle 30 cm. in diameter. Bore 12 holes, 1, 2, 3, etc., dividing the circumference in 30° parts. From the centre draw radii, making the angle a of 90°, /3 of 60°, and y of 30°. Provide pegs, about 15 cm. tall, to fit in all the holes. Mount two strips of thin " silvered '' glass, each about 20 cm. long and 10 cm. wide, on two boards hinged together in such a way that the angle between them may be varied from 30° or less to 90 D or more, the longer edges of the mirrors being horizontal. No. XXVI. An inexpensive kaleidoscope. No. XXVII A concave spherical mirror 12 or 15 cm. in diameter. No. XXVIII. A " granite-ware " ba^in 15 cm. or more in diameter. No. XXIX. Thin waterproofed board, pierced by knitting-needle for experiment on the critical angle. (Fig. 96.) No. XXX. A porte-lumiere. No. XXXI. Right-angled glass prism about 5 cm. long, for show- ing "total reflection." (Fig. 95.) No. XXXII. A pair of equilateral prism?, for experiment on pro- jecting the solar spectrum. (§ 162.) No. XXXIII. Set of about half a dozen lenses of various shapes 4 or 5 cm. in diameter. 'Fig. 101.) No. XXXIV. Set of about half a dozen convex lenses varying from 2 cm. to 50 cm. in focal length, the largest 6 or 8 cm. in diameter. No. XXXV. Rotating apparatus* suitable for carrying Maxwell's color-disks, etc. No. XXXVI. Set of color-disks, e.g. , those made by Milton Bradley. MISCELLANEOUS ARTICLES. Two pounds of clean mercury. Two pounds of assorted soft glass tubing, from 2 mm. to 8 mm. inside diameter. Six feet of rubber tubing, about 5 mm. inside, that will not col- lapse when connected with the air- p am p. An ounce or two of very small rubber tubing. Piece of thin sheet rubber about 6 in. square, for use with the gauge. (No. I.) Set of cork- borers. Three-cornered file for cutting glass tubing. * The well-known little tops with color-disks serve very well if larger forms of XXXV and XXXVI are not available. 178 PHYSICS. Screw-driver. Pair of wire-cutting pliers. One- half pound of naked copper- wire about 1 rnm. in diameter. LABORATORY TABLES. The laboratory tables used in the Cambridge grammar-schools are well suited to the work of this course. They are about 10 ft. long, 4 ft. wide, and 2 ft. 10 in. tall. They have white- pine tops about 1^ in. thick, and heavy white- wood legs. Extending from end to end over each table are two horizontal bars, about 2 in. by 3 in., ad- justable at various heights (which should range from 1| ft. to 3£ ft. by 3-iu. intervals) above the table-top, their ends, which are cut in tenons, sliding in grooves in the supporting posts. These posts are fastened to the frame of the table and rise through slots in the table-top, being flush with the ends of this top and about 10 in. distant from the sides. Pins of iron or wood placed in holes in these posts support the ends of the horizontal bars. To adapt these tables to Exercises 29 and 30 (Second Part of book), holes about 1-J in. in diameter should be bored through the top. For Exercises in heat (see Second Part) this table should have a gas-pipe running along the middle of the top from end to end, with three stop-cocks leading to the right and three leading to the left. This pipe should be readily detachable, as its pre c ence would be in- convenient in many experiments. Each table is intended to accommodate six independent experi- menters. APPENDIX IV. A FEW EQUIVALENTS IN THE ENGLISH AND METRIC SYSTEMS. 1 meter = 1.0936 yards. 1 " = 3.2809 feet. 1 " = 39.3705 inches. 1 kilometer = 0.6214 mile. 1 gram = 15.4323 grains = 0.0353 ounce- 1 kilogram = 2.2046 pounds avoirdupois. 1 yard = 0. 9144 meter. 1 foot = 0.3048 " 1 inch = 0.0254 " 1 mile = 1.6093 kilometers. 1 pound avoirdupois = 0.4536 kilogram. 1 ounce = 28.35 grams. The following are approximate equivale 1 decimeter = 4 inches. 1 meter = 1.1 yards. 1 kilometer = f of a mile. 1 kilogram = 24 pounds. 179 INDEX. Aberration, chromatic, spherical, 149 Achromatic, prism, 135 ; lens, 150 Angle, of incidence and of re- flection, 107; of refraction, 125; critical, 131 ; refracting, of prisms, 134 Archimedes, principle of, 19 Atmospheric pressure, 31 Axis of lens, 137 Balance, spring-, errors of, 61 Barometer, Aneroid, 34; Torri- celli's 31 Boyle's law, 35 Bunsen's photometer, 98 Camera, obscura, 93 ; photog- rapher's, 156 Centre, of curvature, 112; optical, 137 Chromatic aberration, 149 Coefficient of friction, 80 Colors, 92, 100 ; complementary, 154 ; mixing, 153 Composition of forces, 76 Conjugate foci, 140 Critical angle, 131 Curvature, centre of, in mirror, 112; in lens, 137 Density, definition of, 15 ; of water, 16 Dispersion, 135 Distance, measurement of, 3, 4 ; object- and image-, 140 English and metric units, App V Equilibrant, definition of, 70 Eye, 152 Eyepieces, 161 Floating bodies, specific gravity of, 23 Fluids, definition of, 28 ; liquid and gas, 34 Focal length, of mirrors, 115 ; of lenses, 138 ; formula for, 142, 148, 168 Foci, conjugate, 41 Focus, principal, of mirrors, 115, 118 ; of lens, 138 Foot, relation to meter, App. V Forces, parallelogram of, 61-69 Friction, 78 ; between solids, 80 ; coefficient of, 80 ; between solids and fluids, 84 ; rolling, 83 ; in tubes, 84 Fulcrum, definition of, 49 ; force at, 52 Gas, 28, 34 (rases, pressure of, 31, 35 Gravity, specific, 18 ; centre of, 46 Hydrostatic press, 36 Illumination, measure of, 96 Images, in plane mirrors, 104- 111 ; by convex mirror, 112 ; by concave, 117 ; distorted, 119 ; of lenses, 143-148 ; after, 155 Inch, relation to cm., App. V Inclined planes, 69-75 Incident angle 107 Index of refraction, 126 Inverse square, law of, 95 Kaleidoscope, 111 181 182 INDEX. Lantern, magic, 157 Length, focal, of mirrors, 115 ; of lenses, 138 ; formula for, 142, 148, 168 Lenses, definitions relating to, 136-138; shapes of, 136; achro- matic, 150 Lever, definition of, 41 ; circular, 44; weight of, 46; laws of, 50, 51, 53, 58 ; pulley, windlass, capstan, 56 Light, 90 ; velocity of, 90 ; theory of, 91 ; colors, 92, 100 ; pencils and rays, 92 ; weakens with distance; 95 Liquid, 28, 34 Liquids, pressure of, 28-30, 36 Machines, 41 Magic latern, 157 Mass, 17 Meter, relation of, to foot, App. V Metric system, App. V Microscope, 159 Mirrors, plane, 104; cylindrical and spherical, 111, 121 ; con- vex, 112; concave, 115; for- mulas for, 121 Objectives, 160 Optical instruments, 156 Parallel rays, 140 Parallelogram, measurement of, 9, 10; of forces, Chap. V Pendulum, 86 Penumbra, 94 Photometry, 96 Physics, definition of, 1 Plane mirrors, 104 Porte-lumiere, 160 Pressure, at different levels, liquids, 29, 30 ; gases, 34 ; in different directions, liquids, 29; gases, 33 Principal axis of lens, 137 Prisms, achromatic, 135 ; defini- tion of, 133 Projection, of images and of spec- truim. 158 Pulley, 54-57 Pumps, 38 Rays, parallel, 140 Reflection, of light, 103; from plane mirror, 104 — law of, 107, 114 ; total internal, 131 Refraction, 125 ; index of, 126; of glass, 127, 133; of water, 138; of air, 129; of different colors, 130; relation to velocity, 130 Resultant of forces, 76 Rumford photometer, 97 Screw, 74 Shadows, 94 Siphon, 39 Specific gravities, definition of, 18; formulas, 20; methods of obtaining, 18-27, 39 Spectrum, 35 Spring-balance, errors of, 61 Springs, in watches, 89 Stereopticon, 157 Telescope, 162 Torricelli's barometer, 31 Triangle, 6 Tubes, flow of fluids in, 84 Umbra, 94 Units of measurement, 5 Velocity of light, 90 Virtual image, of mirrors, 108 ; of lenses, 146 Volume, measurement of, 10-13 Water, density of, 16 Wave-lengths varying in differ- ent colors, 92 Wedge, 74 Weight, compared with mass, 17