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PHYSIOS
BY
CHARLES RIBORG MANN
THE UNIVERSITY OP CHICAGO
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GEORGE RANSOM TWISS
THE CENTRAL HIGH SCHOOL, CLEVELAND
• • •
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CHICAGO SCOTT, FORESMAN AND COMPANY
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Copyright, 1905, By SCOTT, FORESMAN AND COMPANY
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WITHDRAWN
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ROBERT O. LAW COMPANY. PRINTERS AND BINDERS. Ch(ICAOO
MARSH. AITKEN & CURTIS COMPANY, CHICAOO
PREFACE
Within the last twenty years the methods of teaching physics ' have been revolutionized. The reaction against the loose and desultory methods previously in vogue was started by the em- phasis given to laboratory work in the new books which then appeared. The movement gained impetus from the influence brought to bear on 'the schools by the Harvard Entrance Require- ments and the Report of the Committee of Ten. This pressure on the schools resulted in a demand for closer observation by the students, involving careful measurements.
As far as science is concerned, the most important result of this introduction of laboratory work into the schools has been the development in the public mind of a widespread recognition of the fundamental principle that knowledge is real and living to the individual, only when it is founded on personally observed facts and personal experience.
Nevertheless, during the past few years it has become clear that the present methods of applying this principle to instruction in physics have failed to arouse in the students that real enthusi- asm for the pursuit of the subject, which is indispensable for the mastery of its principles. This has been found true in spite of the fact that, in their own way, boys and giris have by nature and disposition the keenest interest in physical phenomena.
In recognition of this fact, many attempts have been made to develop a better system of instruction. Such a system ought to give the mental training that has been so much emphasized; but it ought also to inspire in the boys and giris a living enthusiasm for the subject, and to develop in them the scientific habit of mind, the ability to utilize their knowledge, and a just apprecia- tion of the significance of natural phenomena. If physics can be so taught as to develop in the student these elements of power, so vital to his future career, there can be no doubt that in due time its educative value will • be properiy appreciated and
its popularity restored.
iii
IV PHYSICS
This book is the contribution of two of the fraternity of teachers toward the attainment of this end. The method of instruction herein set forth has been developed after long experience and much experimenting with high school and college classes. The greater part of the book has been in manuscript for more than two years, and has been used in connection with class teaching. The results obtained have been so encouraging that the book is submitted to other teachers, in the hope that it may be of service to them.
In a recent work, President G. Stanley Hall comments at length on the decline of interest in physics in the high schools, and consequently also in the colleges, and he' suggests several remedies.^ He attributes this failure of physics to "the violence done to the nature and needs of the youthful soul by the present methods and matters." He points out that this violence consists in: 1. Neglect of the hero-ology of the science, of historical and biographical references, so that the learner is not made to "feel vividly a sense of growth." 2. "The rage to apply mathematics to the boy's brain processes," instead of appealing to his interest in concrete things. 3. The failure to realize that "very much thoroughness and perfection violates the laws of youthful nature and of groA\i;h." The young student "wants only answers that are vague, brief, but above all suggestive." 4. Neglect of the practical side of the subject, which is the side that appeals most strongly to the youth. **He is chiefly interested in the 'go' of things."
The methods of instruction which have proved helpful to us, and which are embodied in this book, are in harmony with many of President HalFs suggestions. We have endeavored to strengthen the presentation of the subject, and aid the teacher in three ways: ^ I. By arousing interest. II. By developing the scientific habit of ! thought. HI. By presenting some of the principles from the historical standpoint. Some of the ideas that have guided us in this endeavor are the following:
I. Interest. Interest is rarely stimulated in youth by elegant
* Adolescence, by G. Stanley Hall, Vol. II, pp. 154 seq., New York, AppletOQ, 1905.
PREFACE V
and abstract mathematical treatment; nor is it often aroused by rigorous logical demonstrations. It is aroused by beginning with some concrete thing that goes — ^something which is already familiar. Interest may be sustained by basing the discussion on these familiar and concrete things. Nothing helps more than to have the student feel that you are discussing with him some- thing concerning which he already knows a little, and of which he ; has long been desirous of knowing more. The authors believe that the adoption of an informal style and the use of arguments that i are physical, rather than mathematical, will also be helpful; for they have been mindful of the success of the great teachers, Fara- day and Tyndall, in imparting scientific ideas to untrained minds in this way. Mathematics is an excellent servant but a very bad master; so equations are used only where they are clearly a help to the student, and the development of each is carefully presented with the aid of physical, rather than mathematical concepts.
The aim has been to show the student that knowledge of physics : enables him to answer many of the questions over which he has ) puzzled long in vain. He is approached with the attitude: What do the forces of Nature do for us, and how do they do it? His self -activity is stimulated by this questioning attitude of text-book and teacher, and he is urged to investigate independently at home. His interest is not killed at the start by attempting to cram him with definitions of things to which he has no corresponding con- cept, such as indestructibility, impenetrability, and the like. Nor is he deceived by attempted definitions of undefinable con- cepts, such as mass, force, time, and space. On the contrary, the attempt is made to implant the concept and create the demand for its name, or definition, which is withheld Until the need for it ' is apparent.
II. The Scientific Method. Although interest may be ob- tained through the technical applicati^s of physics, the teaching must not consist in descriptions of these only, any more than of descriptions of laboratory apparatus only. The attainment of scientific principles is always the purpose or end of the argument; not inventions, nor yet laboratory experiments. Science must
VI PHYSICS
be shown to consist in that body of organized knowledge which makes invention possible. Beginning arguments with inven- tions, or general observations of phenomena, may not be the logical order, but it is more nearly the order in which Nature herself teaches, and the result of the argument does not lose in definiteness, clearness, or accuracy^ provided the laboratory is continually held up as the final court of appeal where all doubtful questions are settled.
Each chapter in this book is a continuous argument toward some principle or principles, and the entire book is an argument toward the conclusions stated in the last chapter. This treatment ; is intended to develop and foster the habit of scientific thinking. The attempt is made (1), to interest the student in observing care- fully and accurately first the familiar things about him, and then the things in the laboratory; (2), to interest him in detecting analo- gies and similarities among the things observed; (3), to train him in I keeping his mind free from bias and in drawing conclusions ( tentatively; (4), to make him see the value of verifying the con- i elusions and accepting the result, whether it confirms or denies. his inferences. The arguments in the various parts of the book are not all alike; there are many forms in which the scientific method may be used.
We have tried deliberately to give the student the impression that science leads to no absolute results — that, at best, it is merely a question of close approximation; of doing the best we can, and accepting the result tentatively, until we can do better. This attitude places the teacher also in the position of a learner and prohibits him from making use of didactic or dogmatic statements; for these are the bane of science as well as of other things. Science instruction, that does not develop mental integrity, freedom of the personal judgment, and tolerance, fails in a very vital spot.
III. History. References are given to books in which the biographies of the great men of science may be read, and the student is urged to read them and report. The arguments used by some of the great thinkers have been briefly sketched, and the methods devised by them for reaching conclusions have been given. The attempt has been made to present them as they live
PREFACE VU
in the ideas which they have handed down to us; to picture their mental processes and attitude, and to show how one thing leads to another as the subject develops in the discoverer's mind.
We wish to call the attention of our colleagues to several prac- tical points. In the first place, although each chapter is a con- tinuous argument, the paragraphs are headed in black type, so that the important steps are well marked; and a summary and |^ set of questions are added at the end of each chapter, to assist T the student in fixing the subject-matter in mind. The teacher will, we think, find these latter very helpful to his pupils in both advance and review work.
In the second place, the continuity of the treatment is not interrupted by the insertion of descriptions of laboratory and lecture experiments in fine type. Judged from our own experience, such experiments, thus inserted, confuse rather than assist the student. It goes without saying, that we expect both laboratory ( and lecture experiments to be given in connection with tliis book;y but every laboratory experiment made by the student, and* every experimental demonstration by the teacher should have a definite \ relation in time, place, and subject matter to the general argument \ as presented in the text. An experiment is simply an incum- brance and a source of distraction to the student unless its rela- { tion to the general scheme of the lessons in the classroom is per- ' fectly obvious. A detailed description of a lecture experiment which he has not seen is of relatively small value to the student, and ordinarily there is no interest or profit to him in obtruding on his attention the distracting details of setting up and manipu- lating the apparatus. If such description of an experiment occurs in the text book, while the teacher chooses to make it with some other style of apparatus, different in its details, his con- / fusion is all the worse, for his attention is distracted from the( principle to be illustrated, and lost in the details of the apparatus.
On the other hand, when the student is to make an experi- ment himself in the laboratory, he must be given many details in order that he may manipulate, observe, and record successfully and without loss of time. It is the province of the laboratory \ manual to give these details, for they can not be included in a text )
VUl PHYSICS
book without encumbering it to the exclusion of important theo- retical matter, and destroying its unity. We have therefore pre- ferred to leave the choice of illustrative experiments largely in the hands of the teacher, who may thus select them according to his individuality, his equipment, and the circumstances and limita- tions of his class and community.
I We have bnsed the argument wherever possible on the pupils' 1 experience, expecting this to be supplemented by the teacher with lecture demonstrations and laboratory experiments, chosen in accordance with the conditions which he has to meet and with his own taste and judgment. But when a particular kind of ex- perimental evidence is necessary to the argument, it has been used, without manipulatory details and in uniform type with the other subject matter.
In the third place, many of the old and familiar landmarks of the elementary physics text do not appear in these pages. Among
f these may be mentioned the division of levers into classes; the wedge; the classification of equilibrium as stable, unstable, and neutral; specific gravity as distinguished from density; the elec- trophorous and the electrostatic machine; the concave and con- vex mirrors; multiple reflection; and the formulas concerned with the radii of curvature of lenses. These have been omitted because they seem of less interest and importance than the' following new subjects which we have been able to introduce in the space thus saved: The use of graphical methods and of vectors; the discussion of efficiencies of engines, both prac tical and theoretical; the relations among electrostatic charge, current, and magnetic field; the meaning of harmony; the nature of spectra; the reasons for the electromagnetic theory of light; and the electron theory of matter. We also believe that the presenta- tion of the subjects of rotary motion and of optical instruments will be found much simpler and more satisfactory than those usually given. / The problems are also an innovation. They include no J cases of forces a, 6, and c, meeting at a point q, etc., but are, . as far as possible, real, concrete cases, such as occur in actual practice, and which every boy or girl ought to know how to meet.
PREFACE IX
( They also contain many of the subjects usually placed in the text I and there explained; for example, the pulleys, distillation, and the Wheatstone bridge. We hope that this form of problem will Unterest the student, as most of them are problems in whose solu- tion he can see some use.
Other devices for catching and holding the interest are the questions and the suggestions to students at the end of each chap- ter. We hope that these latter will be stimulating to the students and serve as hints which will lead them to suggest for them- selves other home experiments. Are not such experiments, clumsy though they be, yet made with a genuine interest in finding out something — in getting the answer from Nature herself — far more useful than many that are made in some laboratories?
The illustrations are also a novelty. Great pains have been
'^(taken to have every picture a photograph of a real thing, for a
photograph is always more interesting than a woodcut. It is
believed that these will add much to the interest of the work.
We have been favored with the original photographs for many of these illustrations, by the firms and individuals men- tioned on page x, whom we wish to thank for their courtesy.
We also desire to express our thanks to Professor R. D. Salisbury of the University of Chicago, Editor-in-Chief of the Lake Science Series for many valuable suggestions, and to Messrs. A. A. Knowlton of the Armour Institute of Technology, J. H. Kimmons of the Austin, Chicago High School, and C. Kirkpatrick of the High School, Seattle, Washington, for aid in the reading of the proof.
Many of the line diagrams are new and have been designed and executed with much thought and care, so as to present the essential ideas without complication by unnecessary details.
That great difficulties are involved in the working out of a method of instruction differing in principle from that in general use must be apparent to every one. We know better than any one else can that we have not produced a perfect book. This might be approximated by the concerted action of all teachers of physics. We therefore hope that members of the teaching fraternity will regard the result of our work as a first approximation, and will
X PHYSICS
join with us in making a united effort to lift our subject up to its proper place, and to inspire our young friends with an adequate appreciation of its interest, its majesty, and its grandeur. To this end we appeal to our colleagues to give us the benefit of their experience by sending us suggestions and criticisms, which will be gratefully received and carefully considered.
Charles Rtborg Mann, George Ransom Twiss.
ACKNOWLEDGMENT OF ILLUSTRATIONS
^ Plate I. The Lake Shore and Michigan Southern Railway. Large
copies of this picture in color may be obtained for 50 cents, by
applying to Mr. A. J. Smith, General Passenger Agent, Cleveland,
Ohio. Fig. 11. The Electric Vehicle Co., Hartford, Conn. Figs. 16, 17, 18. The Eastman Kodak Co., Rochester, N. Y. Fig. 19. Pawling, Harnischfeger & Co., Milwaukee, Wis. Plate II, and Figs. 70, 156, 157, 158. The Niles-Bement-Pond Co.,
New York. Fig. 31. The Manitou and Pike's Peak Railroad Co,, Manitou, Col. Fig. 51. Crowe Bros., House Movers, Chicago, 111. Plates III, IV, VI. The AUis-Chalmers Co., Milwaukee, Wis. Fig. 61. The Bausch and Lomb Optical Co., Rochester, N. Y. Figs. 65, 72. The Ingersoll-Sargeant Drill Co.. New York. Fig. 74. The Chicago Bridge and Iron Works Co., Chicago, 111. Fig. 77. The Century Co., New York. Figs. 91, 92. The Whitlock Coil Pipe Co., Hartford, Conn. Fig. 100. The Otto Gas Engme Co., Philadelphia, Pa. Fig. 101. Mr. Alfred Stieglitz, New York. Plate VII, and Figs. 102, 131, 132, 133, 141, 142, 143, 144, 147,
148, 151. The Westinghouse Electric Co., Pitlsburg, Pa. "The Electric Spark in Nature," page 205. Mr. M. I' Anson,
Newark, N. J. Fig. 162. The Electric Controller and Supply Co., Cleveland, Ohio. Figs. 168, 169, 170. ' The Electric Storage Battery Co., Philadelphia,
Pa. Plate VIII. The University of Chicago, Chicago, 111. Figs. 236, 237. Wm. Scheidel & Co., Chicago, lU.
TABLE OF CONTENTS
PAGE
Introduction 11-14
CHAPTER I Motion, Velocity, Acceleration —
Motion of a train — How velocity is measured — Units — Graphical representation of velocity — Analytical repre- sentation of velocity-^Slope — Changing velocity — Accel- eration— Graphical and analytical representation of accel- eration— Measurement of acceleration — Summary — Ques- tions— Problems — Suggestions to students 15-32
CHAPTER II Mass and Energy —
Production of acceleration —Acceleration and force — Dif- ferent bodies having the same acceleration — Mass — Masses compared by forces — Relation of force, mass, and acceleration — Unit mass — Weight — Galileo*? experiment — Weight and mass — Density — Work, force and distance — Unit work — Energy, how measured — Efficiency — Kinetic and potential energy — Newton's laws of motion — Power — Engineering units — Summary — Questions — Problems — Suggestions to students 33-56
CHAPTER III Composition and Resolution of Motion —
Up grade — Composition of motions — Vectors — Motions at right angles — Motions not at right angles — Vector solu- tions— Analytical solution — Traveling crane — Resolution of motions — Force vectors — Balanced forces — Mechanical advantage — Summary — Questions — Suggestions to stu- dents 57-72
5
6 CONTENTS
CHAPTER IV
PAGE
Moments —
How rotation is produced —Moment of force — The lever — Work done by the lever — The lever principle — Parallel forces — Weight and center of mass — Equilibrium — Sta- bility, how measured — Determination of the center of mass — Mechanical advantage of a composite machine — The law of machines — The screw — The equal arm bal- ance — Review — Summary — Questions — Problems — Sug- gestions to students 73-96
CHAPTER V Rotation —
Flywheels — Angular measurement and units — Correspond- ence with linear measurements and units — Moment of in- ertia and mass — Determination of moment of inertia — Conditions for circular motion, centripetal force— Burst- ing wheels — Distribution of mass — Moment of mass — Railroad curves — Spinning tops — Summary — Questions — Problems — Suggestions to students 97-111
CHAPTER VI Fluids —
Pumps — Air has weight — Torricelli's experiment — Pascal's experiment — Mercurial barometer — Characteristics of fluids — Pascal's principle — Hydraulic machines — Free level surface of a liquid — Gases — Air pump— Guericke — Density of air — Theory of pumps — Archimedes' prin- ciple— Flotation — Buoyancy — Determination of density — Boyle and his law — Summary — Questions — Prob- lems— Suggestions to students . 112-136
CHAPTER VII Heat —
Heat and work — Thermometers — Temperature scale — Gases — Change of volume at constant pressure — Change of pressure at constant volume — Air thermometer — Ab- solute temperature — Expansion of solids and liquids — Heat quantity — Gram calorie — Specific heat — Steam — Evaporation — Pressure and temperature of saturated vapor — Boiling point — Superheated vapor — Critical tem- perature— Humidity — Dew — Latent heat — Water and climate — Summary — Questions — Problems — Suggestions to students 137-157
CONTENTS 7
CHAPTER VIII
PAGE
Transfer op Heat —
Conduction and convection — Applications — Radiation —
Diffusion — Evaporation — Gaseous pressure — Effect of
heating — Kinetic hypothesis — Radiation — The ether —
Prevost's theory of exchanges — Absorption— Absorbing
power of water vapor — Radiation and absorption — Heat
and light — Summary^-Questions — Problems — Suggestions
to students 158-170
CHAPTER IX Heat and Work —
Mechanical equivalent of heat — Gas is heated when com- pressed— Gas cools when it expands and does work — Liquid air — Cooling by evaporation — Manufactured ice — The steam enghie — Work done by the steam — The pres- sure-volume graph — Back pressure — Lower pressure at exhaust — Condensers — Higher boiler pressure — Heat energy consumed — Efficiency and temperature — Com- parison of efficiencies — The triple expansion engine — The gas engine — The steam turbine — Summary — Ques- tions— Problems — Suggestions to students 171-189
CHAPTER X Electricity —
Transmission of power — Generators — Early knowledge of electricity — Gilbert — Electrification — Conductors and in- sulators — Repulsion — Discharge — Electroscope — Both . bodies equally charged — Two kinds of charge — Polariza- . tion — Charging by influence — Charge on the outside of a conductor — Coulomb's law — Leyden jar — Condensers — . Operation of a condenser — Discharge of condenser is oscil- latory— Lightning — Summary — Questions — Problems — Suggestions to students 190-210
CHAPTER XI Magnetism —
Lodestone and compass — Magnetic curves — Magnetic field — Like poles — Unlike poles — Permeability — Magnetic circuit — Earth's magnetism — Unit pole — Law of magnetic force — Chief characteristics of magnets — Electric currents — Voltaic cell — Electromagnetism — Magnetic field of the current — Electromagnets — ^Telegraph — Relay — Grounded
CONTENTS
PAGE
wires — Electric bell — Galvanometers — A suggestive ex- periment— Motors — Motor parts — From toy to practical machine^ Ampere's theory of magnetism — Magnetic field of moving charges — Energy of a magnetic system — Sum- mary— Questions — Problems — Suggestions to students. . . 211-243
CHAPTER XII Induced Currents —
Sources of current — Current and magnetic field — Faraday's discovery — Current induced by a moving magnet — Num- ber of lines of force changed — Currents induced by cur- rents— Iron core — Laws of induced currents — ^The dynamo principle — The dynamo — Magnetos — Alternating current dynamos — The induction coil — The transformer — Alter- nating current motors — The telephone — Summary — Questions — Problems — Suggestions to students 244-264
CHAPTER XIII The Electric Current at Work —
Pressure and current in the arc lamp — Current strength — Resistance — Laws of resistance — Ohm's law — Ammeters and voltmeters — Electric power — Watt meters — Arc light plant — Incandescent lamps, parallel distribution — Incan- descent light plant — Heating effects of the current — Joule's law — Heat loss in transmission — Three wire system — Al- ternating current transmission — Divided circuits — Shunts — Arc lamp regulation — Lifting magnets — Voltaic cells — Energy of the cell — Polarization of cells — The ion hypoth- esis—Commercial cells — Electrolysis — Faraday's laws — Electroplating — Storage batteries — ^^ Retrospect — Sum- mary— Questions — Problems — Suggestions to students . . . 265-299
CHAPTER XIV Wave Motion —
Water waves — Origin of waves — Characteristics of waves — What waves tell us — Wave motion — Wave length — Period — Phase — Velocity of propagation — Waves of sim- ple shape — Complex waves — Waves of different shapes — Stationary waves — Summary — Questions — Problems — Suggestions to students 300-316
CONTENTS 9
CHAPTER XV
PAOB
Simple Harmonic Motion —
Uniform circular motion — Displacement and force — The sine curve — Period, mass, and force constant — Pendu- lum— Uses of the pendulum — The Foucault pendulum- Summary — Questions — Problems — Suggestions to stu- dents 317-327
CHAPTER XVI
Sound —
Sources of soimd — Soimd a wave motion — Soimd waves longitudinal — Velocity of sound — Resonance — Noise — The piano — Pitch — Musical intervals — Laws of strings — Vibrating rods — Tuning forks — Organ pipes-^Air columns as resonators — Intensity — Summary — Questions — Prob- lems— Suggestions to students 328-341
CHAPTER XVII The Musical Scale —
Development of the musical scale — The related triads — The vibration numbers — The major scale — The complete scale — The tempered scale — Standard pitch — Forced vi- brations— The ear — Beats — Discord due to beats — Sum- mary— Questions — Problems — Suggestions to students . . . 342-356
CHAPTER XVIII Harmony and Discord —
Wave shape and tone quality — The vibrating flame — Mus- ical tones complex — How musical tones are possible — Fun- damental and overtones — Overtones of strings — Reson- ators— How the ear perceives a complex tone — Related tones — Chimes — Summary — Questions — Problems — Sug- gestions to students 357-368
CHAPTER XIX Light —
What does light do for us — Direction — Image by a pin . hole — Image by a lens — The eye — How light is changed in direction — Refraction — Index — How the lens forms the image — Reflection — Diffuse reflection — Summary — Ques- tions— Problems — Suggestions to students 369-383
10 CONTENTS
CHAPTER XX
PAGE
Optical Instruments —
Principal focus — Image of a point source — Construction of the image — Lens angle — Size and distance of the image — Virtual image — How the eye is focused — Spectacles — The simple microscope — The camera — Stops — Spherical aber- ration— The telescope — The concave lens — The opera glass — The compound microscope — Magnification — Reso- lution— Smnmary — Questions — Problems — Suggestions to students ' 384-402
CHAPTER XXI
Color —
Newton's experiment — Interference fringes — Wave length and color — Interference in white light — Dispersion — The spectrum — Bright-line spectra — Measurement of disper- sion— Achromatic lens — Spe.ctrum analysis — How the eye perceives color — Mixing colors — Colors of ordinary ob- jects— Paints and dyes-^Mixing pigments — Summary — Questions — Problems — Suggestions to students 403-422
CHAPTER XXII Velocity op Light —
. What we can learn from the velocity of light — Galileo's method — Fizeau's method — The velocity determined — Velocity of electric waves — Wireless telegraphy — Light and electricity operate through the same medium — The complete spectrum — Summary — Questions — Problems — Suggestions to students 423-434
CHAPTER XXIII Electrons —
Light waves start at an electrically charged particle — Other such particles — Cathode rays — Action of magnet — deter- mination of ~ — Comparison with the ion of electroly- sis— The electron — Radioactivity — The X-rays — The na- ture of white light — Conclusion 435-446
Index 447-453
PHYSICS
INTRODUCTION
It has been said that man made his sta,rt on the long road toward enlightenment when he learned how to make a fire. For many centuries, our ancestors groped at a snail's pace along this road where we of the twentieth century are advancing by leaps and bounds.
By slow and painful steps, prehistoric man learned to use fire in order to keep himself warm, to cook his food, to get metals out of their ores, and to forge them into rude tools and weapons of de- fense. By means of signal fires on the hilltops, he sent his first wireless messages across the valleys. The magnetic force of the lodestone and the electric attraction of amber were known to the ancients, and the fact that steam pressure can be made to produce motion was known in the early centuries of our era. Why was it that so many centuries elapsed before man learned to subdue these forces of nature and make them do his will? Now we have the steam engine, the electric dynamo and motor, the power printing press, the power loom, the telephone, the wireless telegraph. By means of these and countless other inventions, one man can do the work of hundreds, the continents are linked together, darkness is turned into light, time and space are vanquished.
We can best realize how important are these inventions when we try to think how we should get on without them. And yet this great development of miracle working machinery has come within the space of three centuries, and the greater part of it within seventy- five years 1
The stories of how these marvelous inventions came to be, of the struggles of the men who brought them into being, and of the pa- ll
12 PHYSICS
tient researches and brilliant discoveries of the men of science who established the foundation principles upon which all these inven- tions rest are among the most important and most interesting chap- ters of history.
In the studies which follow, we shall endeavor to get an understanding of some of these principles, to gain at least a slight acquaintance with some of the great discoverers who formulated them, and to get some insight into the kind of thinking and the methods of experimentation by which their truth has been made plain. Such studies are of interest not only to those who expect to make practical use of them, but also to those who, in the pursuit of a liberal education, wish to learn how to think clearly, to ex- press themselves precisely, and to test their conclusions accurately, as well as to get a properly balanced view of human life and ac- tivity.
The principles of physics are most easily understood by the beginner, and are also most interesting, when they are studied in connection with his own experiences. For no one can live long in this scientific age, surrounded as he is on all sides by the fruits of discovery and invention, without having a large amount of experi- ence with the forces of Nature and without obtaining therefrom a large fund of general information.
A rapidly-moving railway train is certainly a familiar object to every one. Even a small child would not have to be told that Plate I is the picture of such a train. Moreover, we all know that the locomotive causes the train to move, and that it can not do so unless it has a fire in it. We are also familiar with the fact that the locomotive must be supplied with water, and that in the boiler this water is converted into steam, which somehow makes the big driving wheels turn. That such an engine warns us of its approach by means of a whistle and a bell, and that it lights its own path in front of it at night by means of a brilliant headlight, are well- known facts.
Now, although these and many other things about the locomo- tive are matters of everyday knowledge to most of us, how many of us can tell exactly how the steam makes the engine's driving wheels turn? And why is steam used at all? Why are some loco-
INTRODUCTION 13
motives large while, others are small? How does the whistle work, and how does its sound get to us? How is the headlight made to send its light forw^ard on the tracks?
A ride in a steam or trolley car is one of the most common of our experiences, and we all know that the car has many different kinds of motion. Wherein do these motions differ? How are speeds measured and compared with one another? What sort of velocity has the car while it is starting or stopping? Why are we thrown against the side of the car when it rounds a curve? How is it that some engines can go faster than others?
The picture of the Twentieth Century Limited (Plate I) was taken while the train was running at full speed. How do cameras and lenses work?
We can obtain the answers to these and to other similar questions without great difficulty, if we are willing to devote to the subject some careful study and thought. When we have done this, w^e shall find that the knowledge thus acquireti gives us a greater con- trol over the forces of Nature, and that the training thus obtained is of great service to us in everything we may wish to do.
CHAPTER I --X;.
MOTION, VELOCITY, ACCELERATION ' ' y
• •
1. The Motion of a Train. In order to find the answe^^fci
some of the questions just asked, let us suppose that a locomdtk'f " stands with steam up, ready to make the run to the next station/ When it starts, we notice that at first it moves slowly, and that its velocity gradually increases until it has attained **full speed," when it runs for some time at a rate that is nearly constant. As the next station is approached, the speed gradually decreases; and the train comes to a full stop. What can we leam of its motion, of the way in which it rounds curves, of how it gets up speed, and of how it stops? How shall we describe and measure its velocity, and how take accbujit of the energy that it must expend in order to move its load?
2. How Velocity is Measnred. Since all motion implies both distance and time, and since distances and times must be meas- ured in order to be compared, it is necessary to have units of length and of time in terms of which the measurements can be expressed.
The units adopted in all scientific work are purely arbitrary, and are chosen simply for con- £~\ r~~]^,
venience. The unit of length is the centimeter, \ \ / / g which is the one-hundredth part of the dis- \ V — / / J tance between two lines on a certain bar of plati- / / V \ ^ num-iridium when the bar is at the temperature p / \ n j
of zero degrees Centigrade. This bar is care- fig. i
fully preserved at Paris, and is called the Inter- Standard Meter^ national Prototype Meter. The symbol for centimeter is cm, and that for meter is m. The unit of time is the SECOND, which is the one-eighty-six-thousand-four-hundredth of the mean solar day. Its svmbol is sec.
Now if a train, moving uniformly, attains in one minute a
15
16 y *. tPHYSICS
distance of 150,00g cpi*from a given post in a certain direction, then in one sepgft>3L*tfce change in its distance in that direction from the post*. wiH *be -^^ of 150,000 or 2500 cm. Therefore it travels at gti^e rate that its distance from the post changes 2500 cm eveiy'spftond. This rate of change of distance is called linear
VELOCITY. •
^ .. K'^without changing the direction of its motion, a body trav- . if^ equal distances in equal times, no matter how small the time • .in'tervals are taken, its velocity is uniform or constant. The unit •^•'of velocity used in physics is the velocity of a body moving uni- formly over one centimeter each second. If the body traverses 2 cm in each second, its velocity would be two units, or 2 centi- meters per second, and so on; therefore, a unijoria velocity is measured by the number of centimeters passed over in one second,
3. Comparison of Velocities. Let us now compare the veloc- ity of our train with that of a fast freight which passes over a distance of 90,000 cm from a given post in onfe minute. Its veloc- ity is then ^V of 90,000 = 1500 cm per second, which is evidently I of 2500, the velocity of the express. Similarly, a rifle ball that passes over 240,000 cm in 3 seconds, has a velocity of 80,000 cm per second. In all of these cases we obtain the number that expresses the velocity by dividing the number of centimeters in the distance by the number of seconds in which that distance is traversed. Since the expression for velocity is thus obtained, an appropriate sjrmbol for linear velocity is ~. Symbols made in this way will be found very useful because they show at once how a quantity like velocity is expressed in terms of the fundamental
. units.
4. The Analytical Method. Uniform velocity may therefore be measured by dividing the distance passed over in a given time by the number of seconds in that time. Since this expression is rather cumbersome, it is more convenient to state it in an abbreviated form by means of algebraic symbols. This is done by letting v represent the linear velocity, I the number of cm in the distance traversed, and t the number of sec in which the
MOTION, VELOCITY, ACCELERATION
17
distance is traversed. We may then write the expression
... Distance passed over in a ffiven time . . „
uniform velocity = — .r^ — -, — ^—^ i — : — " , .. m the form
•^ Number oi seconds m that time
(1)
This is the equation for uniform motion. This method of express- ing relations by means of an algebraic equation is called the ANALYTICAL METHOD. This method is extensively used in physics and engineering, and has the advantage of great conciseness.
|
•9 |
— |
â– |
_„. |
|||||||||
|
Sgo |
||||||||||||
|
- - |
V |
f^- |
=^ |
: |
z\ |
|||||||
|
V |
- - |
■— |
||||||||||
|
O |
||||||||||||
|
^ |
||||||||||||
|
•-7o |
||||||||||||
|
^^ |
^ |
a1 |
- |
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|
k |
||||||||||||
|
— |
||||||||||||
|
-- |
_ |
- |
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£ |
- |
-' |
" - |
- |
1 |
^ |
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E |
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4> |
'^ |
y |
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|
^5« |
1 |
1 |
4 6 8 Time In Hours
4 6 8 10
Fig. 2. Variations of Temperature During One Day
5. The Graphical Method. There is another very convenient method by which relations of this kind are expressed. This method is familiar to every- body, since it is very gener- ally employed to picture the relative variations of two quantities, both of which are continuously changing in value. Thus Fig. 2 repre- sents the variations of tem- jjerature during a day. The time intervals are repre- sented by horizontal dis- tances, and the corresponding temperatures by vertical distances. A single glance at the diagram tells us whether the range of temperature on that day was large, when it was highest, when lowest, and how hot or cold the air was. This method of present- ing relations is called the graphical method.
Let us then apply this method to the train mentioned in Art. 2. Since in this case the two quantities that vary are time and the distance of the train from the given post, we must let one of the quantities be represented by horizontal distances, and the other by vertical distances. We may choose freely what scale to use, i.e., how great a length on the diagram shall represent a sec or a cm. In this case we shall get a drawing of convenient size if we let 1 cm in the horizontal direction OX represent 1 sec, Fig. 3; and 1 cm in the vertical direction OY represent 2000 cm.
18
PHYSICS
If now we begin to consider the motion at the instant when the front of the engine, going at the rate of 2500 ~, passes a cer- tain post: then at that instant, since no time has elapsed and no distance been passed over, i.e., the time is zero and the distance also zero, the corresponding point on the diagram will lie at 0, Fig. 3. At the end of one second the train is 2500 cm from the post. Therefore the point that corresponds to this condition
must represent a time of 1 sec and a distance of 2500 cm, and so must be 1 cm from OY in the direction OX, and 1.25 cm from OX in the direction OY, To locate this point we lay off 1 cm along OX to a?!, and draw from x^ a dotted line parallel to OY. We then lay off 1.25 cm along OY to y^ and draw through y^ a dotted line V parallel to OX, The intersection p^ — ^of these two dotted lines will then be the point sought, since it is 1 cm from OY and 1.25 cm from OX. In like manner, at the end of the second second the train is 5000 cm from the post. So we lay off 2 cm along OX to the point X2 to represent 2 sec, and 2.5 cm along OF to the point ^2* *o represent 5000 cm. We then draw the dotted lines as shown in the figure, and find the point p2, which therefore represents the conditions at the end of 2 sec. Similarly, the point pg, distant 3 cm from OY and 3.75 cm from OX^ represents the conditions at the end of the third second; and so on. Note carefully that the line obtained does not represent the path of the train.
We next draw the straight lines Op^, p^ p^, pa Vzj ^tc. Is the resulting line Opg straight? Do the points that represent the con- dition of the train's motion at 0.5, 1.7, 2.2, 2.5 sec also lie on this line? Is there on the line a point corresponding to every possible instant of time? Does every such point also represent a distance from the post? Does the time Opg completely represent the motion of the train with respect to both distance and time?
Fig. 3.
1 \2 3
Uniform Velocity
MOTION, VELOCITY, ACCELERATION
19
Since we shall often use the graphical method, we shall need to know the names of the lines and points. The two lines OX and OF are called coordinate axes. The distances Ox^, Ox^y Ox^j etc., are called abscissas. They may be measured from any point on OF along a line parallel to OX; thus y^ pi, = Ox^, 2/2 P2i = 0^2> ^t^- The distances Oy^, Oy^y Oy^, etc., are called ORDi nates, and may be measured from any point on OX along a line parallel to OF. OX is called the axis of abscissas, and OY the AXIS OF ORDINATES. 0 is Called the origin of coordinates. The line O/jg representing the relations considered is called a graph.
Fig. 4. Slope Indicates Velocity
6. What the Slope Indicates. Let us now add to our diagram graphs for two other trains, one of which is a fast freight F traveling uniformly at the rate of 1500 ^, and the other an express E at the very high speed of 3000^. The result is sbo\\Ti in Fig. 4. In what respect are the second and third graphs like A, the first? Which graph has the steepest SLOPE, or in other words which makes the greatest angle with the axis of abscissas? Would the graph for a slow freight traveling at a rate less than 1500^ have a greater or a less slope than that of the fast freight? Would the graph RB for a rifle ball having a speed of 80,000 ^ make a greater or smaller angle with the axis of abscissas than does that for the express? What charac- teristic of the motion is indicated by the steepness of the slope?
It thus appears that in the graphical method of representation the velocity is represented by the slope, while in the analytical
method (c/. Art. 4) it is measured by the ratio — . The slope and
the ratio — then serve the same purpose. But on the graph / is
represented by the vertical distance pn (Fig. 5), and t by the horizontal distance On. Therefore the slope of the line Op may
be appropriately measured by the ratio ^. But pn is the side
20
PHYSICS
opposite the angle of slope, and On is the side adjacent to it in the right triangle pOn; and in a right triangle this ratio of the side opposite the angle to the side adja- cent to it is called the tangent of the ANGLE. It is clear that the tangent of a given angle has always the same value no matter what the size of the triangle is. Thus, since the triangles pOn and pqm
. ., pn pm
are similar, f—-=^—.
On qm
Hence the appro-
priate measure of slope is the tangent of the angle that the graph makes with the axis ^^°- ^' of abscissas.
Measurement of Slope
7. Increasing Velocity. Thus far we have considered the motion of the train only when it is uniform. What now are the characteristics of the motion just after the engineer has opened the throttle, so that the train is getting up speed; and what of the motion when he has shut off the steam and applied the brakes, so that the train is slowing down?
Since, now, the velocity is changing at every instant, it can not be measured by the distance traversed in one second. Therefore the velocity at any instant is measured by the distance which would be traversed in one second, provided that throughout that second the rate were to continue the same as it was at the given instant. Suppose now that the train starts from rest, and that at the end of the first second it has gained a veloc- ity of 50 ^, that at the end of the second second its velocity is 100 g^, and at the end of the third second 150 ^^; i.e., suppose that the velocities at the end of successive seconds are as fol- lows:
Fig. 6. Ready to Start
MOTION, VELOCITY, ACCELERATION 21
|
lec 0 |
cm sec 0 |
sec 5 |
cm sec 250 |
|
1 |
50 |
6 |
300 |
|
2 |
100 |
7 |
350 |
|
3 |
150 |
8 |
400 |
|
4 |
200 |
etc. |
etc. |
Is the change of velocity for any one second the same as for any other second, i.e., is the change of velocity constant? If during the interval between the end of the eighth second and the end of the twelfth the velocity changed uniformly from 400 to 600 ^, what was the rate of change of velocity, i.e., the change of velocity for any one second?
8. Decreasing Velocity. Again, let us suppose that when the train is slowing down, its velocity changes in the first second from 2500 to 2400 ^, and that at the ends of the successive seconds the velocities are as follows:
|
sec |
cm ' sec |
sec |
cm sec |
|
0 |
2500 |
5 |
2000 |
|
1 |
2400 |
6 |
1900 |
|
2 |
2300 |
7 |
1800 |
|
3 |
2200 |
8 |
1700 |
|
4 |
2100 |
etc. |
etc. |
What is now the rate of change of velocity? Since this rate of change is the ratio of the change of velocity to the time, it is expressed as a number of ^^ per second. Thus if the rate of change of velocity is such that 75 ~ is gained or lost each second, then this rate of change is expressed as 75 centimeters per second per second. It is customary to write this 75 ^^2-
9. The Name Given to Eate of Change of Velocity is Ac- celeration. When the velocity is increasing, the acceleration is positive; and when the velocity is decreasing the acceleration is negative. When the acceleration is constant, as in the examples just givQn, the motion is called uniformly accelerated motion.
22 PHYSICS
It is to be noted that the expression for linear acceleration is obtained by dividing a number of units of velocity by a number of units of time, ' Since velocity is length divided by time, it is plain that acceleration is length divided by the square of time. Hence the symbol for the unit of acceleration is ^j-
10. The Analytical Expression for Acceleration. The ana- lytical expression for acceleration may be found as in Art. 8, except that we now represent the related quantities by letters instead of by numbers. Thus, if a represent the acceleration, V the velocity at the end of a number of seconds denoted by t, and Vq the velocity
V — V at the beginning of this time, then the acceleration is a = — - — ^.
This equation is simply the definition of acceleration written in algebraic shorthand.
It is often necessary to find the change in velocity in terms of the acceleration and the time. In order to do this, we multiply both members of our equation by <, thus obtaining the result V — Vq = at, i.e., the change of velocity is equal to a, the rate of that change, multiplied by t, the time.
If we wish to find the value of the final velocity V when the other quantities are known, we add Vq to both members of this equation, which gives us
V = Vf^-{-at (2)
i.e., the final velocity is equal to the initial velocity 'plus the cJiange in velocity.
11. Relation of Distances to Times. It will be interesting to know what sort of lines we shall get if we plot graphs that repre- sent the relations of distances to times while our train is starting and stopping. In order to do this we must first know the distance of the train from a given point at the end of each second.
At the beginning of the first second, since the train is at rest, the velocity is zero: and the final velocity is this initial velocity plus the change, or F = i^o + ^^> ^s stated in equation 2, Art. 10. Now a, the acceleration, is 50 ^2 J hence the final velocity for the time 1 sec isF = 0 + 50X 1 = 50 ^. Since the velocity begins atO and
MOTION, VELOCITY, ACCELERATION 23
ends at 50 ^, it must, during that first second, have all values from zero up to 50 ^. Which of these values may we use in calcu- lating the distance traversed in that second? Since according to our supposition the velocity increases imiformly, the train will traverse in a given time with the uniformly accelerated motion the same distance that it would have traversed during that time with uni- form motion at the average speed. The average or mean velocity, then, is that which we must choose.
Since the velocity changes at a uniform rate, the average velocity may easily be found by taking the arithmetical mean of the initial and final velocities; and therefore, for the first second, if we represent
this mean velocity by v, we have v = — ^ — ^ ^^ s^-
Solving equation (1) for /, we have / = vt; and substituting, we get / = 25 X 1 = 25 cm, the distance of the train from the starting point at the end of the first second. Similarly for the time two seconds (since for each time period we must consider the motion from the beginning y in order to get the average velocity), the initial velocity is 0; and the final velocity is F= 0 + 50 X 2 = 100 ^-^.
Whence the average velocity 1;= = 50£I?; and the whole
distance traversed up to the end of the second second is again the mean velocity multiplied by the time, i.e., I = vt =50X2 = 100 cm. In like manner, for 3 sec, we get F = 0 + 50 X 3 = 150 ~,
0 4- 1 ^0
and V = \ = 75^, therefore / = 0;^ = 75 X 3 =f 225 cm.
^ sec
By the same method of calculation, we find that the distances for the first eight seconds are as follows :
|
sec |
cm |
sec |
cm |
|
0 |
0 |
5 |
625 |
|
1 |
25 |
6 |
' 900 |
|
2 |
100 |
7 |
1225 |
|
3 |
225 |
.8 |
1600 |
|
4 |
400 |
etc. |
etc. |
12. The Graph for Distance and Time. We now have the data that we need, and can proceed to construct our graph. Let
24
PHYSICS
Fig. 7. Graph for Single Seconds
US choose our scales so that for the abscissas 1 cm represents 1 sec, and for the ordinates 1 cm represents 100 cm. We locate the point corresponding to each second (Fig. 7) and find them to be 0 for the beginning of the first second, pi for the end of the first second,
P2 for the end of the second
.„ . ^ second, p^ for the end of the
third second, and so on. If, as before, we should connect the points in succession by straight lines, would the resulting line be straight? Does the velocity of our train change abruptly at the end of each second or is it increasing uniformly at every instant? Does the broken line connecting O, Pi,p2f etc., change its slope at every point or only at the points that we located? Then does such a line properly represent the uniformly accelera- ted motion of the train?
It ought now to be clear that the graph must change its slope at every intermediate point as well as at the few points that we located, ij it is to represent properly the uniformly increasing velocity of the train.
If we should locate the points for the intermediate half seconds, in addition to the points already placed, thus reducing our time interval to 0.5 of its former value, and if we should connect all the points successively as before, would the broken line thus ob- tained more nearly fulfill the condition of changing its slope at every point?
Suppose now that we were to reduce the time interval to 0.2 sec and to plot the corresponding broken line (Fig. 8) ; would this line approach more nearly than did the other to the line that would represent exactly the uniformly increasing velocity of the train? It must be clear that by continually diminishing our time intervals we shall get broken lines that more and more nearly fulfill the con- dition of changing slope at every point, and thus more and
MOTION, VELOCITY, ACCELERATION
25
Fia. 8 Graph for Fifths of a Second
more nearly approach to the graph that we want. It is obvious,
however, that, in a practical problem, it is useless either to ca/rry
the subdivision of the time inter-
vol beyond the point at which
the difference between the broken
line and a smooth curve is no
longer perceptiUe in the drawing,
or to use smaller time intervals
than we are able to measure by
means of the timepiece used in
making our observations.
In general when we wish to make a graph that corresponds to a series of observations, we locate the points corresponding to each of these observations, and then draw the smooth curve that most nearly passes through all of the points,
13. Slope of a Curved Graph. As long as \he line is a broken one, the slope of the portion between any two consecutive points is that of the straight line joining those points; but when we pass to a graph that changes its slope at every point, it must be evident that the slope at any point is approximately that of a straight line joining the given point with a nearby point. The nearer we take this point to the given one, the more nearly does the slope of the straight line represent that of the curve.
Since, for this graph, the ordinates and the abscissas represent respectively distances and corresponding times, just as they did in the graphs for uniform motion, the slope at any point of this graph must represent the velocity at the corresponding instant of tinier just as it did in their case.
14. The Train is Stopping. In order to construct the graph that will represent the relation between distance and time when the train is slowing down, we must again calculate the distances of the- train, at the ends of the successive seconds, from the j)oint
26
PHYSICS
at which the engineer applies the brakes. The velocity at this instant is the initial velocity and is 2500 ^ (c/. Art. 8). As the speed is decreasing, the acceleration is negative; and so (ef. Art. 8), its value is a = - 100 ^2- Hence the final velocity
for the first second is F = and the average velocity v
Vo-\- at = 2500 - 100 X 1 = 2400
2500 + 2400
= 2450^. Multiply.
ing the average velocity by the time as before, we get for the distance traversed in the first second, I = vt = 2450 X 1 = 2450 cm. like- wise for the second second we get V = 2300, v = 2400; so that / = 4800 cm. The values for succeeding seconds are as follows: sec cm sec cm
0 0 5 11 250
1 2450 6 13 200
2 4800 7 15 050
3 7050 8 16 800
4 9200 etc. etc.
15. Graph for Negative Acceleration. Choosing scales such that for the abscissas 1 cm represents 10 sec, and for the ordinates 1 cm represents 10,000 cm, and plotting precisely as before, we obtain
the graph shown in Fig. 9. In what way is this graph for the case of negative ac- celeration like that for the case of positive acceleration (Fig. 8)? In what way do these graphs differ? At the end of what second does the train come to rest? As- suming that the train then remains at rest, add to the diagram the points corre- sponding to the next five seconds. At the end of what second does the graph be- come parallel with the axis of abscissas? What, then, is the slope at the end of the 25th second? At the end of the 28th? the 30th? What velocity is represented by a slope of zero?
16. The Entire Motion Bepresented. We have now the graphs for the uniform motion of the express train going at full
ZO ZT> 36
Fig. 9 The Train is Stopping
MOTION, VELOCITY, ACCELERATION
27
speed, and for the uniformly accelerated motion with positive and negative accelerations while getting up speed and slowing down. In order that all of these motions may be represented by a single diagram that will go on a page, a smaller scale must be used. The complete graph appears in Fig. 10 (1 cm = 30 sec, 1cm =30,000 cm). Describe in succession the changes of slope.
140
uo
lOO
eo
CO
40
:io
17. Equations for TTniformly Accelerated Motion. Passing now to the anal3i;ical method of repre- senting uniformly accelerated mo- tion, let us develop an algebraic expression that will generalize the calculations of Art. 11 and Art. 14. If Vq represent the initial velocity, a
. , 1 i' a *u *• T7 J.U Fig. 10. The Complete Graph
the acceleration, t the time, V the
final velocity, and / the distance, then by equation (2), Art. 10,
V = ^0 + at Also, the average velocity, v is found by taking half the sum of the
6 9 lO
initial and final velocities; therefore, v =
^0 + (^0 + «0
at
On multiplying this average velocity by the time t to get the dis- tance /. we have
l = Vot + — .
(3)
Equations (2) and (3) are the equations for uniformly accelerated motion.
The laws of uniformly accelerated motion expressed by these equations may be stated as follows:
1. The final velocity is equal to the initial velocity plus the product of the acceleration and the time,
2. The total distance traversed is equal to the product of the initial velocity and the time, plus half tJie product of the accelera- tion and the square of the time.
28
PHYSICS
18. When the Moving Body Starts from Eest. In the cases thus far considered the initial velocity was zero. On substituting this value in the general expression, the term involving Vq vanishes
and the equations become V = at and /
— , which
express
the
relations when the moving body has started from rest.
It should not be forgotten that when the velocity is decreasing y a, the acceleration , is negative.
19. Acceleration is Not Necessarily Uniform. Throughout the preceding discussion we have assumed that the acceleration of the train was constant. In reality the case is not quite so simple, because the engineer at first puts on the steam pressure gradually, and because the acceleration is diminished by the resistance of the air, which increases very rapidly when the speed is increased. The acceleration which we assumed to be uniform was the average acceleration during the time considered.
20. Determination of Acceleration. The actual experiment of determining acceleration is made by observing distances and
corresponding times, sub- stituting their values in equation (3), and solving for a.
21. Translatory and Eotary Motions. Thus far we have considered only motion in a straight line. We are now ready to define motion in general, and to distinguish be- tween translatory and ro- tary motion. Fig. 11. Translation and Rotation p^ y^^^^, j^ ^.^y ^^ ^^ j^
MOTION with reference to a given point when it is changing either its distance or its direction from that point.
MOTION, VELOCITY, ACCELERATION 29
When a rigid body moves in such a way that all its p)oints describe equal and parallel paths, its motion is called translation.
When the motion of a body is such that its points describe cir- cumferences about some point or line, its motion is called rotation. The point or the line about which the body rotates is called the center or the axis of rotation. The planes in which the particles move are all parallel to one another, and the axis is necessarily per- pendicular to these parallel planes.
A sled going down a hill has translatory motion only, provided there are no turns in the road; for then all of its points describe equal and parallel paths. The same is true of a sail boat when it is making a straight course. On the other hand, the buzz saw and the grindstone are familiar examples of bodies that have rotary motion only. Every point on the grindstone, for example, de- scribes a circle about a point in the center of the axle on which the stone is mounted. The centers of all the circles described by the points lie on a line which is perpendicular to the planes of all the circles. When an automobile is traveling along a straight road, the body of the car has translatory motion only, while the wheels, considered with respect to their axles, have rotary motion only; but the wheels have both translation and rotation with reference to a point on the road.
SUMxMARY
1. The units of length and of time are the centimeter and the second. Their symbols are cm and sec.
2. Motion may be either translatory or rotary.
3. Linear velocity is the rate of change of distance in a given direction.
4. Uniform linear velocity is measured by the distance traversed in one second. Its symbol is g^.
5. Acceleration is the rate of change of velocity.
6. Uniform linear acceleration is measured by the change of velocity in one second. Its symbol is ^.
7. Acceleration may be either positive or negative.
8. The distance traversed by a body having uniformly acceler- ated motion is foimd by multiplying the average velocity by the time.
30
PHYSICS
9. The three methods of representing these relations are:
Analytical
I
(Equation 1)
V = Vo+ at (Equation 2)
of
(Equation 3)
Verbal Uniform or average velocity equals distance divided by time.
With uniform acceleration, final velocity equals initial velocity plus acceleration multiplied by time.
Distance traversed with uniform acceleration equals initial veloc- ity multiplied by time plus half the acceleration multiplied by time squared.
Time
QUESTIONS
1. Define the scientific unit of length, and give its symbol. Define the unit of time and give its symbol.
2. Define the term, linear velocity. What is meant by a constant linear velocity? What two things must be stated in order that the velocity of a movmg body may be fully described?
3. How is the numerical value of a imiform velocity found? What is the unit of velocity? What is its symbol?
4. ' Explain how to represent a constant linear velocity by the graph- ical method. In connection with the diagram, point out and name the co6rdinate axes, the coordinates, and the origin.
5. What characteristic of the motion of a body is shown by the slope of the graph that represents it?
6. When the velocity of a moving body is changing, how can we express numerically the velocity that it has at any instant?
7. Define acceleration, and illustrate by a numerical example.
8. When is an acceleration positive, and when negative? What is meant by uniformly accelerated motion?
9. Draw the graphs that represent the relation of distance to time for a positive and a negative acceleration. In these graphs, what does the slope represent?
MOTION, VELOCITY, ACCELERATION 31
10. What changes of slope occur in the graph when the acceleration is positive? When the acceleration is zero? When it is negative?
11. When a graph is curved, what line will represent approximately the direction of its slope at any point?
12. Define motion, and distinguish between translatory and rotary motion, illustrating by examples.
PROBLEMS ,
Note. 1 m = 100 cm = 39.37 inches.
1. A runner passes over 100 yards in 10 sec; what is his speed cmp
sec'
2. What is the speed of a race horse that covers a mile in 2 min-
utesWin^^ (2) in 52?
3. What is the speed of an automobile that runs a mile in 55 sec. W<^^ (2)i„|E?
4. Sound, at 0° Centigrade, travels 1090 feet in one sec. What is the speed of sound in ^? How many seconds would it take to trav- erse 1000 m?
5. What was the average speed of a railroad train that traveled 134 mUes in 115 minutes (1) in 2^|~? (2) in ^?
6. Express the velocity of 1 ^^ in ^*, and in ^.
7. A sled, started from rest and going down a hill of uniform slope, traverses 900 cm from the starting point in 3 sec. What is the accelera- tion, and what the final velocity?
8. A wagon starts down a hill with a velocity of 30 — and its
acceleration down-hill is 80 — ^? What is its velocity at the end of 5 sec? What is the total distance traversed in the same time?
9. A wheelman, starting from rest, had attained at the ends of the first three seconds the following distances: 1 sec, 90 cm; 2 sec, 360 cm; 3 sec, 810 cm. Supposing the acceleration to remain con- stant during that time, what is (1) the acceleration? (2) the velocity at the end of 6 sec? (3) the distance traversed at the end of 5 sec? (4) the distance traversed during the 5th sec? [For (4), subtract the distance attained in 4 sec from that attained in 5 sec]
10. The results of experiments show that the acceleration of a body allowed to fall freely is 980 -^j- (") Calculate the distances attained by the falling body when given an initial velocity of 10 cm vertically downward, making a table of distances and times up to 10 sec. (b) Choosing a convenient scale, plot a graph representing the motion.
11. Calculate the velocities of the falling body for the times given In problem 10.
32 PHYSICS
SUGGESTIONS TO THE STUDENTS
1. Which of you can make the longest list of the motions with which you are familiar, classified under the headings: Uniform, Uniformly Accelerated Positive, Uniformly Accelerated Negative, Translatory, Rotary?
2. Mark your height on a door-post; measure it in inches and. in centimeters. From these measurements can you find out how many centimeters are contained in one inch?
3. In ypur debating society, choose for a question the following: Resolved: That the general adoption of the metric system of weights and measures is advisable. For data write to the National Bureau of Standards, Washington, D. C.
4. Which of you can find out the most interesting facts about Galileo and his knowledge of falling bodies?
CHAPTER II MASS AND ENERGY
22. The Production of Acceleration. In the preceding chap- ter, we attempted to get clear notions about uniform and accelerated motions, without considering the factors upon which their production and variation depend. What are the relations that determine whether the motion shall be uniform or accelerated ? What relations determine the amount of the acceleration? We can most easily find the answers to these questions by again studying the train.
Let us first suppose that the train drawn by the engine consists of six cars all alike. Let us also suppose that the engine, using its full power, can impart to this train an acceleration of 50 |^. Now, if this engine be replaced by a smaller one having less power, will the acceleration that this smaller engine can impart to the same train be greater or less than 50 ^? Must the engine that can impart to this
train an acceleration of 60 ^^^2 have greater or less power than the first engine?
Those who can not answer these questions from observations made upon the train itself, will readily answer them by inference from similar cases. Thus, everybody knows that more force is required for imparting to a ball a great velocity in a given time than for imparting to it a small velocity in the same time, that two oarsmen can impart to a boat a greater velocity in a given time than can one, that greater effort is required by a bicyclist to attain a great velocity in a given time than to attain a small velocity in the same time. Observation and experience lead us habitually to associate a greater acceleration with a greater effort or force.
23. Acceleration and Force. Although common experience gives us this general information, it does not give us the specific numerical relations. This information can be obtained only by making careful measurements of the quantities involved.
33
34
PHYSICS
Thus, if we measure the pulls of different sized engines, having different powers, and observe the corresponding accelerations, arid if we make proper corrections for friction of the moving parts and for air resistance, we shall find that the numbers representing the pulls are directly proportional to the numbers representing the accelerations imparted to the train.
Many experiments of this sort have been devised and carried out in physical laboratories to test the validity of this conclusion, and they all tend to establish the truth of the general principle that when different accelerations are given to the same body, the ratio of the numbers by which we express the forces to those by which we express the corresponding accelerations is constant.
Fig. 12. Eight-Oared Shell
Another illustration will help to make this clear. When only two of the crew of an eight-oared shell row, they can impart to the boat a certain acceleration. After making proper allowance for the increased resistance of the water and air, it will be found that when four row, they can impart an acceleration twice as great, six an acceleration three times as great, and so on.
24. Different Bodies Having the Same Acceleration. We have thus far considered how the forces vary when different accelerations are given to the same body. Let us now consider how the forces vary when the same acceleration is given to different bodies.
If an acceleration of say 50 ^ can be given by a certain engine to a train of five empty cars, must the engine that can give the same acceleration to a train of ten similar cars be more or less powerful than the first? Again, if the same acceleration is to be
MASS AND ENERGY
35
given to the train of five cars loaded with passengers, can the same engine do the work?
Common experience again gives us qualitative answers; for everybody knows that an engine that can easily move a short train may fail to move a long one, so that another engine must be added. Likewise it re- quires greater effort on the part of a bowler to give a large ball a certain velocity than to give a small ball the same velocity in a given time; and it requires the efforts of more oarsmen to give a certain acceleration to a big boat than to a little one. The student can recall many similar facts from his daily observation.
It appears, then, that the more we increase the size of a body, the substance remaining the same, the greater is the force required to give it a certain acceleration.
Fig. 13. Small Engine: Short Train
25. Mass. In order to get quantitative relations, experiment is necessary. If we measure the pull of an engine when it is im- parting an acceleration of 50 ^ to a train of five empty cars, and again when it is imparting the same acceleration to a train of ten similar cars, we shall find that the pull in the second case is twice that in the first. Likewise we shall find that the pull for a train of fifteen 6ars is three times that for five cars, and so on; i.e., when the acceleration is the same, the numbers representing the forces are directly proportional to the corresponding numbers of cars.
The matter appears very simple as long as the cars are empty and all alike. But although we know from experience that more force is required to impart a given acceleration to a loaded train than to an empty one, yet it is impossible to determine how much force, until we have adopted a means of comparing the loaded train with the empty one.
36 PHYSICS
These differences in the make-up of the trains, whether in the number of cars or in the load, are differences in mass.
26. Masses Compared by Forces. It is easy to see that when two trains consist of precisely similar cars, all of them empty, the train of ten cars has twice the mass of the train of five cars, be- cause it is made up of just twice as many units of the same kind. But it has just been shown that to impart a certain acceleration to a train of ten empty cars the force is twice as great as that for a train of five empty cars; so that we may compare the masses of the two trains not only by the numbers of cars, but also by the forces re- quired to give them the same acceleration.
When the differences in the trains are differences in the loads, we can not compare their masses by comparing the number of cars, because the units are not alike. Therefore we must resort to the other method, that of comparing the masses by the forces that can impart to them the same acceleration. This method is applicable to all bodies, whether composed of like or unlike kinds of matter. Therefore, in general, tvx) masses are equal when, under the same conditions, equal forces can impart to them equal accelera- tions.
Applying this method to the cars, it appears that when an engine can give two empty cars the same acceleration that it can give to a single loaded car, the combined mass of the single car and its load is equal to that of two empty cars ; and therefore in this case the mass of the load is equal to the mass of one of the cars.
For a given acceleration, then, since the masses are equal when the forces are equal, it follows that if one of the masses be doubled, the corresponding force is doubled ; if the mass is made thrt^ times as great, the force is also three times as great; and so on. In general, then, when the acceleration is constant, the forces are proportional to the masses.
27. Force, Mass, Acceleration. Since we have shown in the preceding paragraphs that when the mass is constant, the force varies directly as the acceleration, and also that when the acceleration is constant, the force varies directly as the mass, it follows that, in
MASS AND ENERGY
37
general, the force must vary directly as the product of the mass and the acceleration. If we choose our units of force appropriately, and if we let / represent the force, m the mass, and a the acceleration, we may write
/ = ma (4)
This equation defines the force in terms of mass and acceleration.
In connection with this equation, it is to be noted that if m and / are constant, a must be constant also; i.e., if a body be acted on by a single or an unbalanced constant force, its motion will be uniformly accelerated.
Besides magnitude, every force has two other characteristics, namely, its direction, and its point of application. When these three characteristics are specified, the force is fully described.
28. Unit Mass. Thus far we have taken an empty car as the unit of mass; but it is manifest that accurate measurement necessitates the establishment of a unit that is fixed and at the same time more convenient. Therefore, just as we have a standard of length, the meter, we have also a standard of mass. The interna- tional STANDARD OF MASS is a cer- tain piece of platinum which is carefully preserved at Paris along with the standard meter, and is called the kilogram. The unit of mass employed by all scientists is the gram, which is the one-thou- fio. 14. sandth part of the mass of the standard kilogram. The abbreviation for gram is gm. To express very large masses, the kilogram is a more convenient unit. The abbreviation for kilogram is Kg.
Smce we are now able to express both mass and acceleration in terms of grams, centimeters, and seconds, we may also express force in terms of these same fundamental units. Thus in the equation
Standard Kilogram: Actual Size
38 PHYSICS
/ = ma, if we substitute m = 1 gm, and « = 1 ^2> ^'^ obtain / = 1 X 1 = 1 ^^^> i^7^ic^ defines the scientific unit of force as thai force which can impart an acceleration of one centimeter per second per second to a mass of one gram. This unit of force is called the DYNE. Note that the number of units of force is obtained by multiplying together the numbers representing mass and accelera- tion. This operation gives us gm x ^2- Hence the symbol for the dyne is ^^.
V—v.
By definition (c/. Art. 10), a = — - — ^, therefore f ^ ma = The product m{V — v^), or mass X change of veloc-
•miV-Vo)
711 ( V — V ^
ity, is called change of momentum; — ^^ — - — — is therefore the
rate of change of momentum; and since it is equal numerically to may it is also a measure of the force to which it corresponds.
29. Weight. Since we have learned to state the relation of force to mass and acceleration, we are in a position to get some definite ideas concerning a subject about which there are many common misconceptions. We all learned in early childhood that bodies, including ourselves, fall to the earth when unsup- ported. We are accustomed to associate this motion with a force called gravity, which we conceive acts so as to attract all bodies toward the earth. The attraction between the earth and any par- ticular body is called its weight, and tends to give the body an acceleration vertically downward. This fact is also a familiar one, for everybody knows that a body falling from a great height acquires a greater velocity than does one falling from a less height. It is our knowledge of this fact, acquired from very early expe- rience, that impels us to avoid a high fall.
Now, what is the relation between the weights of bodies and their masses? Equation (4) will give us the answer. Thus^ the weight /, in dynes, of any body whose mass is m, is equal to this mass multiplied by a, the acceleration that this weight will give it if it is allowed to fall freely; i.e., / = ma. Similarly, the weight f of any other body having a mass m', and receiving an acceleration a\ is /' = m'a'. In order to find the ratio of the two
MASS AND ENERGY
39
weights in terms of their corresponding masses, we must divide one of these equations by the other, thus: j = . We therefore see
TTia
that if both bodies have the same acceleration when falling freely, i.e., if o' = o, then their weights are proportional to their masses.
30. Gktlileo's Experiment. The question to be answered now is, When two bodies have different masses, does the attrac- tion of the earth give them equal accelerations? From the time of Aristotle to the end of the sixteenth century, this was a much disputed question. Aris- totle (384-322 B.C.) taught that if t\/o bodies of unequal .mass were dropped from the same height at the same in- stant, the heavier body would reach the earth first; and his followers defended this opinion by his authority and by argu- ments based upon what they thought ought to be the nature of things. Galileo (1564-1642 A.D.) was the first to recognize that svxih a dispute can be set- tled only by experiment. Ac- cordingly, about the year 1590, he performed the experiment of dropping at the same instant a small cannon ball and a large bomb from the top of the Leaning Tower of Pisa. They reached the ground at very nearly the same instant; so he came to the con- clusion that if it were not for the resistance of the air, they would have fallen in exactly the same time. The fact still remained, how- ever, that a body with a large surface in proportion to its mass, such as a feather, was known to fall very much more slowly than
Fig. 16. Leaning Toweb op Pisa
40 PHYSICS
a piece of metal. After the invention of the air pump, in 1660, it became possible to settle the dispute finally. This was done by showing that when a feather and a coin were dropped simul- taneously in a long tube from which the air had been pumped, they fell side by side and reached the bottom at the same time.
31. The Belation between Weight and Mass. Reasoning from these experiments by means of equation (3), Art. 17, it follows that the accelerations of all freely falling bodies are equal. For if two bodies fall simultaneously through a distance I in time <, then for the first, since the weight and hence the acceleration is
constant, I = -—; and likewise for the second, V = —^, But
since the distance is the same for each, as is also the time, I = V and t = t', whence a = a'.
Thus it has been proved that at any given place, the accelera- tion due to the earth's attraction is the same for all bodies and that therefore, so long as they are compared at the same place, tlie weights of all bodies are proportional to their masses,
Galileo, wishing to prove this statement with greater accuracy, devised experiments with pendulums of different mass. These experiments verified more accurately the same conclusion. Re- peated with greater refinement by Sir Isaac Newton and others, they have given convincing evidence of the truth of this statement.
From what has just been stated, it follows that v)e can cmtt- pare vmsses by comparing their weights. This is the method in common use; but it must be noted that, since the attraction of the earth for a given body is different at different places, the weights of the two masses that are to be compared must in gen- eral be determined at the same place. For the comparison of masses by means of their weights, the equal arm balance is gen- erally used.
32. Density. In connection with the masses of different bodies, we have seen that bodies having equal volume may differ greatly in mass. Thus, one cubic centimeter of lead has a much greater mass than has one cubic centimeter of water; while the latter has a greater mass than has one cubic centimeter of wood. The
MASS AND ENERGY 41
appropriate measure of the density of any substance is the mass in unit volume at a temperature of zero degrees Centigrade. Thus, if the mass of a specimen of a certain kind of glass is found to be 25 gm, and its volume 10 cm', the average density of the glass is 0 . 1 of 25, or 2 . 5 grams per cubic centimeter.
If D represent the density, m the mass, and V the volume, these relations are stated analytically by the equation
D = -
As defined by this equation, the density of a substance is its mass per unit volume. The unit of density is one gram per cubic centimeter, and its symbol is ^^, Since the gram was intended to be the mass of 1 cm' of water, and since it is so, very nearly, the number of cm' in the volume of a quantity of water is the same as the number of gm in its mass. The density of water, there- fore, may be taken as 1 ^^.
33, Work. •, In Chapter I we have studied the motion of a rail- road train and seen how that motion is produced by the engine. Why does the engine move at allf Must more steam be used to move a train of large mass than to move a train of small mass? Must more steam be used to move a given train over a long distance than over a short one? Other conditions being the same, does it require a larger amount of coal to generate a larger amount of steam? The student probably knows the answers to these ques- p tions, and also in a general way that, other things being equal, the amount of coal required is proportional to the amount of work to be done. Since most kinds of work, like that done by the locomotive, consist in putting bodies into motion, and in maintaining tlieir motions in opposition to resistances of some sort, and since some- body always has to pay for getting work done, it becomes necessary to know definitely just what an amount of work depends on, and to have a unit in terms of which all kinds of work may be measured.
34. Work, Force, Distance. If an engine or a horse or a man is doing any kind of work it is evident that, other thuigs being
42
PHYSICS
Fig. 16. Plowing Work is proportional to force and to distance.
equal, the amount of work done is directly proportional to the push or pull, i.e., to the force exerted by the agent that does the work. Thus, if each of two engines pulls its train on a straight and level track for the distance of a mile, and if the second engine has to pull with twice the force of the first, it is clear that the second engine
must do twice the work that the first does. Again, suppose that the second en- gine pulls for . one mile, and then con- tinues to pull with the same force for another mile, it must again be clear that in pulling the train two miles it does twice the work that it did in pulling it one mile. It follows, then, that if the second engine, exerting twice the force of the first, and hence doing twice as much work per mile as does the first, should continue pulling with this force through a distance of two miles, it would do four times as much work as the first engine did in pulling its train one mile.
Since, then, the amount of work done by an agent is directly proportional to the force and also to the distance through which the agent acts, and since the amount of work depends on these two factors only, it follows that when the units are properly chosen, the measure of the work done is the jyroduct of the numbers repre- senting the farce and the distance. In symbols, if / represent the force of the agent, and I the distance through which it acts, and if W represent the work done, then
W = fl. (5)
This is the equation for work.
35. TTnitWork. The unit in terms of which work is meas- ured may easily be defined with the help of the equation W = //, for if / = 1 dyne and 1=1 cm, we have W =1x1 = 1.
MASS AND ENERGY
43
Therefore, since the equation gives unity for the work when the force is one unit and the distance one unit, it is most con- venient to define the unit of work as the amount of work that is done when a force of one dyne acts through one centimeter. This is the unit of work adopted by physicists, and it is called the erg. Since the symbol for the dyne is ^^^, and since the number of ergs is obtained by multiplying together the number of dynes and the number of centimeters, it follows that the symbol for the ergis«^Xcm,orSMm?.
36; Unergy. We now come to the question of the relation betweeWlhe^ amount of coal burned and the amount of work done. It is generally recognized that a water wheel, in order to move machinery continuous- ly, must be continuously sup- plied with water, which must be allowed to fall upon it from a higher level; that a windmill will not continue to pump water unless the wind continues to blow against its blades, that a horse or a man can not con- tinue to do work unless he regularly consumes food, and that an engine of any sort must continuously consume
coal in order that the steam may be kept up at the necessary pres- sure while it is doing its work.
For centuries the most careful thought of philosophers and the greatest genius of inventors were employed in trying to think out and construct some device for obtaining perpetual motion, i.e., a device which would continue to move indefinitely without a con- tinuous external supply of energy. Since every such attempt has been unsuccessful, scientists have become convinced that a per- petual MOTION MACHINE is impossible. Thus, if any machine be
Fig. 17. Haying A man can not work unless he consumes food.
44
PHYSICS
at rest, it can not start itself; and if it be in motion, the greater the friction of its moving parts the sooner will it stop. If it be harnessed to other bodies and made to do work in moving them,
it will come to rest all the sooner. It can be made to work continuously only by supplying it continuously with ENERGY from some external source. Energy y then, represents ability to do work. In the case of the water wheel the energy is derived from the motion of a mass of water; in the case of the windmill, from the motion of a mass of air; in the case of the horse or man, from the consumption of a quantity of food; and in the case of a steam or gas engine, from the consumption of a quantity of fuel.
Thus it becomes evident that to do work energy must be expended, and that to store up this energy work o{ some sort must have been done. Now, many careful experiments with all forms of energy have shown that a given amount of energy always corresponds to the same amount of work, whether that energy be expended in doing the work, or the work be done in storing the energy.
Fig. 18. The Windmill
It will not go when there is no
wind.
37. Energy Measured by Work. Since the energy of a body is equivalent to the work it can do, and also to the amount that had to be done an it in order to impart the energy to it, we may measure this energy by measuring either of these amounts of work. Some- times one of these methods is more convenient, sometimes the other. For example, let us consider the energy necessary to run an eight- day clock. Such a clock is usually operated by a spiral spring, or by a weight which is raised by winding up the cord upon which it hangs. Suppose that the weight has a mass of 5000 gm. Then, since / = ina, the force with which it pulls on the cord is ma, or 5000 gm multiplied by the acceleration that it would have if allowed to fall freely in consequence of the earth's attraction. This
MASS AND ENERGY
45
acceleration, which we have learned is the same for all bodies, is found by experiment to be, at sea level and in the latitude of New York, 980^. Therefore, the force with which we must pull in order to lift this mass is 5000 X 980 = 4,900,000 dynes. To avoid the repetition of zeros it is convenient to write this 49 X 10^.
If the distance through which the mass is lifted is 100 cm, then from Art. 34, W = fl = 4 900,000 X 100 = 49 X 10^ ergs. Since this is the amount of work done in winding up the clock, it represents the energy stored in the lifted weight. Likewise when the weight descends, it does work in running the clock and this work is again f X I = 5000 X 980 X 100 = 49 X 10^ ergs. Since this is the work done by the energy stored in the lifted weight, it also is a measure of that energy. Thus, in general, if we can measure or com- pute the work done on a body in imparting energy to it, or the work that it does when it parts with its en- ergy, we can determine the amount of energy that it had.
It will be noted that in the exam- ple just given a small amount of use- less WORK, done in overcoming fric- tion while winding the clock, was neglected. In every case in which energy is transformed or transferred some of this useless work is done. If the amount of useless work is at all comparable with that of the useful work, allowance must be made for it. The ratio of the useful work done to the total amount of energy expended is called the EFFICIENCY of the machine by which the transformation or transference i$ accomplished.
Suppose now that in the clock just considered we were to replace the weight by a spiral spring. How much energy must the spring have when wound up in order that it may be able to run the clock
Fig. 19. Hoisting Coal Work equals force times distance.
46
PHYSICS
for as long a time as the weight ran it? How much work would have to be done in winding up the spring?
38. Energy is Potential or Kinetic. In the cases that we have considered, energy has been stored in a lifted weight, a
coiled spring, unbumed coal, and unconsumed food. En- ergy of this kind that a body has because of its position or internal condition, so that it tends to move and do work, is called potential energy. In the case -of the windmill or the water wheel, the energy of the air or water is due to the fact that it is in motion. Likewise a base ball or can- non ball does work while it is being stopped. Hence it also possesses energy; and it must be quite clear that it has this energy because of its motion. The energy that a moving body has because of its motion is called kinetic energy.
|
J |
'/I |
|
|
1 |
||
|
.** |
Xi |
J |
Fig. 20. Pile Driver
The weight has potential energjr when it is
raised, kinetic when it strikes.
39. The Kinetic Energy of a moving body being evidently due to its mass and velocity, it is often more convenient to measure it in terms of these quantities than in terms of work done. This may readily be done with the help of equations (3), (4), and (5). Thus W = fl, in which / is the average force used in imparting to the moving mass its velocity, and I the distance through which this force acted. Also this force / = ma, in which m is the mass of the body, and a its acceleration while acquiring its full velocity. Therefore the work done in giving the body its kinetic energy is W= fl = mal. Again, by equation (3), / = ^af, and by equation (2) V = at, in which I is the distance traversed in the time t while acquiring the velocity V with an acceleration a. Since we do not
MASS AND ENERGY 47
care to know the time t, we may eliminate it by substitution. Thus,
V P
from (2) / = — ; then f = —j. Substituting this value for t in
equation (3), we have I = -^ = — . Finally, by substituting this value for I in the equation W = mat,, we find that the energy is e = TF = — ^ — . Simplifying, we have
c == — - ergs. (6)
Since the symbol for mass is gm, and that for velocity is ^, the symbol for kinetic energy is ^^^^ . Note that this is the same symbol as that for the unit of work, as it should be, because en- ergy is measured by work.
The advantage of deriving the equation e = --r— is mani- fest when we apply it to the case of throwing a ball or firing a shot. For while it would be very difficult to measure t and a correspond- , ing to the distance I through which the force of the hand or the powder was exerted, it is not so very difficult to measure F. Hence it is often desirable to have an equation in which a and t are not in- volved. In many cases, however, the kinetic energy of a body can be measured with convenience by the work that it does when it gives up its energy in stopping,
40. Newton's Laws of Motion. From all that has been said it must be apparent that a body can not of itself start, or stop, or otherwise change either the rat6 or the direction of its motion. This fact is often expressed by saying that every body has inertia.
The relations of the phenomena with which we have become familiar in this chapter were described tersely by Sir Isaac Newton in the following statements, which first appeared in his celebrated Principia in 1687. They are known as Newton's Laws of Motion.
1. Every body continues in its state of rest or of uniform motion in a straight line, except in so far as it is compelled by force to change that state.
2. Change of motion is proportional to the force impressed
48 PHYSICS
and takes place in the direction of the straight line in which the force acts.
3. To every action, there is always an equal and contrary re- action, or the mutual actions of any two bodies are always equal and oppositely directed.
41. Illustrations. All these laws are illustrated in the train. The train can not start unless pulled by the engine, which exerts force upon it, i.e., imparts energy to it. Once started, the train can not stop itself. If brought to rest it gives up its energy in over- coming the resistance of the air, the friction of the moving parts, and the friction of the brakes when they are applied to the wheels. If there were no such resistances, the train, once set in motion with a certain velocity, would continue to move without change either of speed or direction. Again, while starting, if there were no friction, and if a constant force were applied, there would be an increase of velocity the same for each second, i.e., the rate of change of motion, as measured by the product of the mass and the accelera- tion, is proportional to the force; and it is in the direction of the straight track along which the engine pulls.
When the brakes are applied, their force, and therefore the corresponding acceleration, is in a direction opposite to that of the motion; and if this force remains constant, there is a decrease of velocity the same for each second. Here again the total change of motion is proportional to the force impressed and takes place in the direction in which this force acts.
But why is it that the train when under full head of steam does not continue with uniformly accelerated motion, and therefore increase its speed indefinitely, instead of reaching a certain speed, which it can not surpass? The answer is that the resistance of the air increases very rapidly, and therefore the engine soon has to use all its energy against air resistance and internal friction ; so that there is no excess left to do the work of increasing the motion of the train. The total external force opposed to the motion of the train is exactly equal to the total external force urging it forward; and therefore the result is the same as if no force were a^cting at all — namely uniform motion in a straight line.
MASS AND ENERGY
49
Fig. 21. Diving
The boat has an acceleration in the opposite
direction.
Now where are we to look for the application of the third law? We have seen that the engine can not move the train if the driving ivheels sHp; therefore it appears that the force of the engine is applied at the place where the drivers bear upon the track. The engine, then, tends to push the track backward, and would do so if the track were free to move. But the track is made fast to the earth, and therefore the engine tends to push the whole earth backward. The force of the engine, the action, is equal to ma, i.e., to the total mass of the engine and train mul- tiplied by the accelera- tion that they acquire. Also the resistance of the earth, the reaction, is equal to mV, i.e., to the mass of the whole earth multi- plied by the acceleration that it receives. This force and accel- eration are oppositely directed with respect to those of the train. Why does not the earth move? The answer is that it does move, but so little that the motion is imperceptible. This will easily be understood when we remember that the two forces are equal, i.e., mV= ma. But, dividing both members of the equation by m'a, so
as to get the ratio of the accelerations, we have —7- = -7-, whence ° ma m'a
— — — ;, which tells us that when the forces are equal the accelera- a m'
tions are inversely as the masses. Since the mass of the earth is
very large compared with that of the train, it is evident that the
acceleration of the earth must be very small. If the masses were
more nearly equal, the accelerations would be more nearly equal.
Every one knows that when he dives or jumps from a small boat
it has a perceptible acceleration in the direction opposite to that in
which he jumps; and if he jumps from a larger boat, the accelera-
50 PHYSICS
tion of the boat is smaller; while if he jumps from a big ship, the acceleration of the ship is imperceptible. So it is with the engine and the earth.
42. Kate of Boing Work. In connection with work and energy there remains another important question to be considered. How are we to measure the rate at which energy is supplied; or, what comes to the same thing, how are we to measure the rate at which work is done? With the units we have adopted this is very simple. We have only to calculate the number of ergs of work done per second. Thus, if 120,000,000 ergs are done in 60 sec, then in 1 sec there will be perfonned one sixtieth of 120,000,000 or 2,000,000 ergs. The rate, then, is 2,000,000 ^^|. The rate at which any agent does work is called its power or activity, and the power is measured by the work that it can do in one second ; i.e.
Power = -^^ . seconds
43. Engineering Units. For measuring force, work, energy and power, engineers use a system of units based on the pound weight, the foot, and the second. These units are not nearly so convenient as those based upon the centimeter, the gram, and the second; but since they are so widely used in engineering practice they are here described for reference. Those students who expect to prepare themselves for engineering, should master these defini- tions and be able to apply them in numerical problems.
Since the foot is equal to 30.48 cm, the numerical value of the
980 ft
acceleration of gravity in this system is q, or 32.2 — g (nearly).
oU.4o sec
This quantity is usually denoted by g.
Instead of deriving their unit of force from a unit of mass and a unit of acceleration, as the physicists do, engineers use as their unit of force the weight of a pound mass at sea level and in the lati- tude of New York, and call it the pound-force.
Whenever in an engineering equation the mass of a body ap- pears— as in the case of kinetic energy — it should be noted that we must eliminate it from this equation with the help of equation (4),
MASS AND ENERGY 51
which expresses the relation between the mass of a body and its weight. Thus, / = ma, whence m = — . But / is expressed in
pounds-weight, and o, the acceleration in this case, is 32.2 — -^;
, . ., p , 1 pounds-weight of the body, therefore the mass of a body m = ^^-^ ~
which expression must be substituted for the mass in the given equation.
The amount of work done or of energy expended when a jxmnd- force is exerted through the distance of 1 foot, is called one foot-pound. By equation (5), Art. 34:
W (in f oot-pounds)\ = // = pounds-force X feet.
To get the measure of the kinetic energy of a body, in terms
of its weight and velocity, we must resort to the equation,
mV'^ e = — ^— . Since in engineers' units the mass m of the body is
~ ^^ — , and since the velocity V is expressed in feet per
second, the equation becomes^
pounds-weight (feet per second)^ e = - X 2 ' ^^
.... , . pounds-weight X (feet per second)' e (m foot-pounds) = ^ 32 2 X 2
The engineers unit for power or activity is tlte horse-power, which is the rate at which work is done or energy expended when 550 foot-pounds of work are done in ea^h second. Hence -J. _ pounds-force X feet
^ 550 X seconds
One horse-power is found to be equal to 746 X 10^ ^^.
On the continent of Europe, engineers use a system of units based upon the kilogram, the meter, and the second. These units are defined or derived in a similar manner.
Thus, the kilogram-meter is the work done, or energy expended, when a force that is equal to the weight of a kilogram mass is exerted through the distance of 1 meter. Hence,
W (in kilogram-meters) = fl = kilograms-force X meters.
52 PHYSICS
One kilogram-meter equals 980 X lO"* ergs.
Since g, the acceleration of a freely falling body, expressed in meters and seconds, is 9.8 ^, the equation for kinetic energji in kilogram-meters is
_ mV^_ kilograms-weight (meters per second)'
^ - "2" - g ~ ^ ] 2 ' ""^
,. , ., , V kilofframs-weiffht X (meters per sec*. )* e (m kilogram-meters) = .
To solve problems in which the relations are expressed by these equations, it is necessary only to substitute the known values for the quantities represented in the equations, each expressed in its appropriate units; and then the unknown quantities can be found, provided, of course, that in the statement of the problem, one equa- tion can be formed for each of the unknown quantities.
SUMMARY
1. To describe a force completely we must state: 1, its point of application ; 2, its direction ; 3, its magnitude.
2. Two bodies are said to have equal masses if equal forces give them equal accelerations.
3. The unit of mass is the gram, and its symbol is gm.
4. The unit of force is the dyne and its symbol is ^â„¢^.
5. Force is measured by the product of the mass and the acceleration, i.e., / = ma,
6. If a body that is free to move be affected by an unbalanced constant, force, the motion will be uniformly accelerated.
7. At any given place, all freely-falling bodies have the same acceleration. At sea level in the latitude of New York this acceleration is 980 ^; therefore, since / = ma, a mass of 1 gm has a weight of 1 X 980 = 980 dynes.
8. At any given place the weights of bodies are proportional to their masses; therefore the masses of two bodies may be com- pared by comparing their weights.
9. The density of a substance is its mass per unit volume. Its symbol is ^,.
10. When masses are moved, or when their motions are changed,
MASS AND ENERGY 53
work is done; and the measure of the work done is the product of the force and the corresponding displacement, i.e., W = fl,
11. The unit of work is the erg, and its symbol is ^^^^ .
12. Scientists are convinced that a perpetual motion machine is impossible.
13. The doing of work implies the transfer of energy, and in every such transfer two bodies are equally and oppositely affected.
14. The energy transferred is measured by the work done, i.e., e = W,
15. The ratio of the useful work done to the total amount of energy expended is the efficiency of the machine.
16. Energy is either potential or kinetic.
17. Kinetic energy may also be measured in terms of mass and
velocity, i.e., e = W= -r— .
18. Activity is the rate of doing work, and is measured by the number of ergs done per second.
19. Engineering units are, for force, the pound-force; for work or energy, the foot-pound; and for activity, the horse-power.
QUESTIONS
1. In what two ways may bodies differ in mass?
2. When different forces act on the same mass, what is the relation of the forces to the corresponding accelerations? Illustrate by exam- ples.
3. When the same acceleration is imparted to different bodies, what is the relation between the forces and the corresponding masses? Illustrate.
4. What kind of motion results from the action of a single or an unbalanced constant force?
5. Define the cm-gm-sec unit of force and give its name and sym- bol.
6. What is meant by the weight of a body?
7. What is the use of the equal arm balance, and why can it be employed for this purpose?
8. Of what does work consist, and what is the numerical meas- ure of an amount of work? Write the equation for work.
9. Name and define the cm-gm-sec unit of work. Give its symbol. 10. When is a body said to possess energy?
54 PHYSICS
11. What is meant by the term perpetual-motion machine? What reason have we for believing that no such machine can be made?
12. What are some of the sources from which our supplies of en- ergy ordinarily come?
13. When a body possesses energy, is all of it available for the doing of useful work? Illustrate by some examples.
14. Define the terms kinetic energy and potential energy, and give some examples of each kind.
15. What is the advantage of having an equation for kinetic energy in terms of mass and velocity?
16. Explain the application of each of Newton's laws to the cases of a rimner, a bicyclist, or an automobile.
17. If all the moving bodies on the earth, such as railroad trains, steamships, and animals, were to travel eastward at the same time, and continue to do so indefinitely, what would be the ultimate effect upon the eastward velocity of the earth's rotation?
18. Define the engineer's units of force, work, energy, and activity.
19. What expression should be substituted for mass when engi- neer's units are employed?
PROBLEMS
1. The masses of two loaded cars are 40,000 and 50,000 lb. re- spectively; If a locomotive engine, exerting 2000 pounds-force on the first car, gives it an average acceleration of 2.0 — j, what acceleration would it give to the second car? What force would give the second car the same acceleration as was given the first? Note: Friction is not here considered. Each car would require a certain force to overcome this, in addition to that required for the acceleration.
2. Five men, rowing a boat, give it in 30.0 sec a velocity of 15.0 — . What is the average acceleration? All otlier things remaining the same, what velocity would be given the boat by three men? What is the average acceleration in this case? What in each case is the dis- tance traversed in the 30 sec?
3. A base ball has a mass of 140 gm, and is thrown from home base to first, a distance of 2743 cm, in 0.90 sec. What is its velocity? If the catcher applied the throwing force during 0.10 sec, what was the average acceleration during that time? What was the force in dynes? If it was stopped by the first baseman in 0.05 sec, what then was the amount and sign of the acceleration? What force did it exert on his hands?
4. What velocity and acceleration are given to a mass of 500.0 gm by a force of 50,000 dynes applied for 10.00 sec? Tlirough what dis- tance does the force act? How many ergs of work are done? How many ergs of kinetic energy are stored in the moving mass?
MASS AND ENERGY 55
5. A base ball, mass 1^0 gm, was thrown vertically upward and is caught by the thrower at the er i of 5.8 sec. Find the height to which it rose, the velocity with which t, was thrown, its weight in dynes, the work done on it, and the energy stored in it. If the force of the thrower was applied during 0.05 sec, what was its amount, exclusive of that re- quired to overcome the weight?
6. A block of marble, 1.00 X 0.50 X 3.00 m, has a mean density of 2.70 Q^. What is its mass? Express its weight in kilograms, and in dynes. Express in ergs and in kilogram-meters the amount of work that would be needed to lift it 5 m from the ground.
7. A brass cylinder has a mass of 122.50 gm, a diameter of 1.90 cm, and a length of 5.10 cm. What is its density?
8. What is the volume of a copper ball whose mass is 130.0 gm, and whose density is 8.87 gm?
9. A pound = 453.6 gm. How many dynes does its weight equal? Find the weight of a 130 lb. boy in grams and in dynes.
10. To how many dynes is the weight of a kilogram equal? How many ergs equal a kilogram-meter?
11. The pile driver. Fig. 20, has a mass of iron weighing 3500 lb. This mass is raised by a steam hoisting engine to a height of 45 ft. and dropped upon the head of a pile. Calculate the work in foot-pounds required to raise the mass of iron to position. How much potential energy has it when lifted, and how much kinetic energy when it strikes? How much work does it do?
12. In the case of the pile driver, problem 11, calculate the time of falling and the final velocity of the iron mass. From the weight and velocity, calculate the energy when striking, and compare the result with that calculated from the weight and the height. Which method of calcu- lation for the energy would you choose if- both weight and height were given, as in this problem? Which if the velocity were known, but not the height?
13. How many pounds of water can be pumped per minute from a mine 600 ft. deep by a 75 horse-power pump?
14. It is desired to raise ore from a mine 550 feet deep at the rate of 3 tons per minute; what horse-power must the hoisting engine be able to develop? How many foot-pounds of work would it do per ton?
15. An automobile weighing with its load 2000 lb., starting from rest, requires 22 seconds to attain a speed of 88 — , when it continues at uni- form speed. Calculate its kinetic energy. What average horse-power was used in putting it into motion? What other work had to be done? How was the energy being expended after the speed became constant?
16. How many pounds of water must go over a fall each second in
56 PHYSICS
order to furnish 25 horse-power, if the fall is 10 ft. high and all the power is to be used? How many cubic feet of water were used each second if 1 cu. ft. weighs 62.5 lb.? What must be the cross-sectional area of the stream at the fall, if the speed of the water there is 3 — ?
SUGGESTIONS TO STUDENTS
1. Consult the libraries on the life of Sir Isaac Newton, and prepare a brief paper containing the facts that most interest you. This paper may be read before the Physics class, published in the school maga- zine, or offered as a theme in the English class.
2. Repeat Galileo's experiment, by throwing a block of wood and a brick from a third story window.
3. How high can you throw a base ball? Take the time with a stop watch, or with an ordinary watch, as accurately as you can; and use equation (3). Plot a graph for the complete motion of the ball, working out the distances and times for each of the seconds by equations (2) and (3). Let the best throwers plot on the blackboard, to the same scale, the graphs for their throws. Let the class compare and interpret the changes of slope.
4. Get the necessary data by trial; and calculate your horse-power (a) when going upstairs as fast as you can comfortably without a load; (h) when carrying the greatest load that you can.
5. Devise a method of measuring on the wall of the house the greatest height to which your lawn hose can throw water. Observe with a watch the number of seconds taken by it to fill a gallon jar. Allowing 8 lb. to the gallon, calculate the number of pounds of water thrown out in one sec. From this and the height, calculate the horse-power that this stream of water could be made to furnish to a small water motor.
CHAPTER III COMPOSITION AND RESOLUTION OF MOTIONS
44. Up Grade. In Chapter II we have learned that a train moves because the driving wheels of the engine are turned; and we have studied the motions when the track is straight and level. There still remain, however, many questions that need considera- tion. Why must the engine work harder in ascending a grade? How can we find the amount of this extra work? What is the relation between the extra pull of the engine and the weight of the train?
In all our previous study we have considered motions along a straight line, i.e., in one direction or dimension only. The questions just asked lead us to the consideration of what takes place when a body has at the same time two or more different motions. Since these motions may or may not be in the same direction, the resultant motion may take place in two or three dimensions, i.e., the path of the motion may be a plane curve, or it may be twisted like the thread of a screw.
45. The Composition of Motions. One of the simplest cases of two simultaneous motions of the same body is that of a man walking lengthwise in a car that is moving uniformly on a straight, level track. If the car is moving northward at the rate of 600 ^> and the man walks in the same direction at the rate of 150 ^» how far does the man travel northward in one second? In three seconds? If the man faces about and walks southward in the moving car at the rate of 150 ^, how far northward will he travel in one second? In ten seconds?
From these examples it must be evident that in considering motions we must take account of two characteristics of the motion, namely, direction and magnitude. For this reason it is very convenient to represent a motion by a straight line whose length
57
58
PHYSICS
and direction correspond to the direction and magnitude of the motion. Thus, for the first case just considered, let ab, Fig. 22, represent the motion of the car northward: cd, which has the same direction and is one-fourth as long, will then represent the motion of the man with reference to a point in the car; and ad, which is obtained by adding together ah and cdy will represent the resultant motion of the man in both direction and magnitude. 7
Similarly, in the second case, if ab, Fig. 23, represent the motion of the train, then efy which has the opposite di- rection and is one-fourth as long, will represent the motion of the man with reference to a point in the car. Hence a/, which is obtained by adding together the two oppositely di- rected lines, will represent the resultant motion of the man in both direction and magnitude.
It is to be noted that in both cases this addition of the lines is performed by drawing the first line with its proper direction and magnitude, and then from the end of the first line drawing the second with lis proper direction and magnitude. Then the line drawn from the beginning of the first line to the end of the second represents the resultant motion.
Fig. 22. Vectors
Fig. 23
46. Vectors. In order to indicate clearly the direction that such a line represents, it is usually tipped with an arrow point as in the figures. A line may be used in this manner not only to rep- resent motions, but also to represent any sort of physical quantity that has both direction and magnitude. A line that is used to represent both the direction and magnitude of a physical quantity is called a vector.
47. The Motions are at Bight Angles. Suppose now that instead of walking northward or southward in the moving car,
COMPOSITION AND RESOLUTION OF MOTIONS
59
the man walks eastward across it. If the velocity of the car is 600 ^ northward, and that of the man 150 ^ eastward, what is the resultant motion during two seconds? Simple arithmetic can not give us a solution that will determine the resultant both in direction and magnitude. Therefore let us see what the graph- ical method will do for us.
• In Fig. 24 let the distances northward be represented by the ordinates, and the distances eastward by the abscissas, the scale being 1 cm = 200 cm for each motion. Plot- y ting the graph in accordance with the method learned in Chapter I, we find that pi and 'p^ rep- resent the positions of the man at the ends of the first and second seconds respectively. Will the points that represent his position at the end of 0.5 sec and 1.5 sec also lie on the line O'p^, Will this line include the points corresponding to his position -at the end of 0.1, 0.3, 1.9 sec, etc.? If we further subdivide the time unit and locate points corresponding to any of the hundredths of a second, will these lie on the line Opz? Is it necessary to subdivide the time unit further in order to show that the line O'p^ represents the path of the man's motion as accurately as is possible in the drawing?
Fig. 24 gives us the clew to an easy method of finding the resultant of any two uniform motions; for it is clear that the resultant is represented by the concurrent diagonal Opj o^ the parallelogram Oy^ Ji^ ^2> whose adjacent sides 0x2 ^^d Oz/j represent the two component motions in both their directions and their magnitudes.
Fig. 24
Parallelogram op
Motions.
48. The Motions are not at BigHt Angles. Furthermore, a little careful thought will make clear the fact that whatever may he the angle between the component motions, the resultant is com- pletely represented by the concurrent diagonal of the parallelogram whose adjacent sides represent the two component motions both in direction and magnitude. Thus, in Fig. 25, Ox represents one
60
PHYSICS
of two uniform motions, Oy the other, and the diagonal Op the resultant. This construction is called the parallelogram
OF MOTIONS.
49. A Shorter Method. It may already have occurred to the reader that the process of finding the resultant may be very (/ _-yO much abbreviated. For it is evi- dent that we can determine the resultant Op2 (Fig- 24) just as defi- nitely by means of the triangle OX2P2 as by the whole parallelogram. In order to do this we have only to draw the vector 0x2, representing the first motion, and from its end Xj to draw the vector a^jPa representing the sec- ond motion; and then the line Op2> which joins the beginning of the first vector with the end of the second, is the vector that represents the resultant.
This method of construction is called the vector method. Fig. 26 is the diagram for a problem similar to that just considered. Since the vector method is simpler than the parallelo- gram method and is employed by physicists and engi- neers, it will be used in the discussions that follow. When we have found the vector that represents the resultant, the actual magnitude of the resultant can readily be found either from the diagram or by the analytical method. --Thus, in Fig. 24 we can measure the resultant vector Opj and we find its length to be 6.15 cm (nearly), and since in this case 1 cm of the vector represents 200 cm traversed, the resultant dis- tance is 6.15 X 200 = 1230 cm, the result by construc- tion and measurement
Fig. 26
60. The Analytical Solution. To obtain the analyt- ical solution we note that, since the two component motions are at right angles to each other, the resultant is the hypothenuse of a right triangle; and therefore, since the square of the hypothenuse is equal
COMPOSITION AND RESOLUTION OF MOTIONS 61
to the sum of the squares of the other two sides, the square of the re- sultant is equal to the sum of the squares of the two components. Hence, iii the example represented in Fig. 24, since one component is 1200 cm, and the other, at right angles to it, is 300 cm, the result- ant = ^1200H 300* == 1236 cm, the result by the analytical method. In general, i: R represent the resultant and A and B the two components, then R = ^/Al^ + B\ provided that the com- ponents are at right angles with each other.
Since the two numerical results just obtained represent the same distance,, why are they not identical? Would the agreement be closer if the diagram were constructed more carefully and on a larger scale?
51. When the Angle between the Components is Oblique. In this case the magnitude of the resultant can be obtained graph- ically by the addition of vectors in the way just described. Having drawn the resultant vector, we measure it in centimeters, and multi- ply its length by the number of units that 1 cm represents on the scale used in the diagram.
When the vector triangle is oblique, a purely analytical solution is impossible without the use of the elements of trigonometry. With a very little knowledge of trigonometry the solution is simple, but those who have not this knowledge can always find the result- ant by construction and measurement. In fact, it is generally more convenient to get the resultant in this way; so that this method of solution is very generally used by engineers.
52. Traveling Crane. The composition of three motions is illustrated by a device used in shops where heavy castings or other weights have to be lifted and carried from one position in the shop to any other. This device, Plate II, is called a traveling crane and consists of a steel bridge whose ends rest on little motor cars which run on tracks supported by the side walls of the shop, so that the crane can traverse the shop from one end to the other. The bridge also carries another track along which another motor car can run across the shop from one side to the other, while the weight to be carried may be lifted to any desired height by means
62 PHYSICS
of a pulley hanging from the bottom of this car. The motor cars and pulley are operated by electricity, steam, or compressed air, and the operator controls them by levers sp that the car carry- ing the pulley may be made to move either across or along the shop while the weight is being raised or lowered by means of the pulley. Thus the weight may move vertically while the car carries it horizontally across the shop, or the crane may also at the same time carry the weight horizontally along the shop. Therefore, with this device it is possible to combine motions in three direc- tions at right angles to each other.
53. Besolution of Motions. We have just seen how two com- ponent motions may combine to make a single resultant motion. In obtaining the solutions of engineering problems it is often con- venient to conceive that an observed motion is the resultant of two other motions that may have combined to produce it. Thus, when by means of the traveling crane a casting is made to move diagonally across the shop, it is clear that its actual motion is the resultant of two> motions, one across and the other along the length of the shop. In a similar way the motion of a railroad train up a grade may be conceived as the resultant of two component motions, one horizontal and the other vertical.
This separation of the actual motion into two conceived motions leads us at once to the solution of several interesting problems connected with the motion of bodies "up hill''; for let us suppose that a train, running with uniform speed, is just beginning to ascend a grade. How much more work must the engine do in pulling the train up the grade than in pulling it for the same distance and at the same speed along the level track? It is evident that the amount of this extra work depends only on the steepness of the grade. Suppose that the grade is 1 : 10, i.e., for every 100 cm measured along the track, the track rises 10 cm. In order to calculate the amount of extra work. done in pulling the train up the grade, we shall, as stated, conceive the motion along the track as the resultant of two component motions, one horizontal and the other vertical. In Fig. 27, ac is the vector representing the motion of the train while passing over 100 cm of track, and ab
Fig. 27. Resolution of Motions
COMPOSITION AND RESOLUTION OF MOTIONS OJ
and be are the vectors representing respectively the horizontal and vertical motions of which we conceive ac to be the resultant. Now, since ab is horizontal, no extra work is done by the engine in imparting to the train the motion represented by that vector. But be is vertical, and it is clear that the engine can not impart a vertical motion to the train without doing the work of lifting the weight of the train. Hence the extra work done by the engine in pulling the train up grade is the work done in imparting to the train the motion represented by the vector be. But since the grade is 1 : 10, the work done when the train traverses 100 cm of track is that of lifting the train through a vertical height of 10 cm. Thus, if the mass of the train is 2 X 10^ gna, then, since the acceleration of gravity is 980 ^, we find by equation (4) that the weight of the train is / = ma = 2 X 10* X 980 = 196 X 10* dynes. Since the vertical displacement is Z = 10 cm, we have from equation (5) for the work done JF = /Z = 196 X 10* X 10 = 196 X 10^' ergs. This, then, is the extra work done on the train by the engine for every 100 cn;i up grade along the track.
54. The Engine is Stalled. If the engine is not able to supply the extra energy necessary to do this amount of work, it will be stalled on the grade. Let us suppose that this has just happened. What ten4ency to motion down grade has neutralized that due to the engine pulling up grade? How does the magnitude of this tendency depend upon the steepness of the grade?
55. Force Vectors. In these questions we are dealing with forces, not motions; but forces have both direction and magnitude; and therefore they can be represented by vectors, provided they act at the same point. Thus, in Fig. 28 let the vector Om rep- resent the weight of the train, which acts vertically downward. This weight produces both a pressure against the track, and a tend- ency to move downward along the incline. Hence to answer our questions, we conceive the vector Om to be resolved into two components, one perpendicular to the track and the other parallel
64 PHYSICS
to it. The first vector Op will then represent the pressure against the track, in both direction and magnitude, and in like manner the vector jmi will represent the tendency to move down the incline. Now, since the pressure of the train on the track produces no
motion in the direction of the vector Op, it must be evident that this pres- sure is balanced by an equal and opposite pres- sure. This equal oppos- ing pressure will at once be recognized as the reac- tion of the track and earth. But if the component represented by the vector pm were not balanced by an opposing force, the train would move down the incline, i.e., in the direction of pm. At the instant when the train is stalled, it is not moving either upward or down- ward along the incline, and therefore what must be the direction and magnitude of the opposing • force that prevents the down- ward motion? How should the vector representing this oppos- ing force be drawn in the figure?
Fig. 28.
Resolution op Forces on an Inclined Plane
66. Balanced Forces. When two or more forces act simul- taneously on a body in such a way that no motion results, these forces are said to be in equilibrium. When forces are in equilibrium the vectors that represent them in the vector diagram, when added together, form a CLOSED FIGURE. Thus, in the example just dis- cussed there are three forces acting, namely, the weight of the train, represented by Ow (Fig. 28), the resistance of the track, represented by pO, and the pull of the engine, represented by mj). If these three vectors be added by laying off one from the end of an- other, each with its proper direction and magnitude, and in any order, they form a closed triangle as in Fig. 29. forces in
On the other hand, and in general, if we have any ' quilibrium number of forces not in equilibrium acting on a body, and wish
TO FIND THE FORCE THAT WILL HOLD THE SYSTEM IN EQUILIBRIUM,
COMPOSITION AND RESOLUTION OF MOTIONS 65
we add successively the vectors that represent the given forces, and draw a line from the end of the last vector to the beginning of the first. This line will then be the vector that repre- sents the force sought, in both direction and magni- tude. Thus, in our example, if we know the weight of the train and the resistance of the track, and if we wish to find by this method the pull of the engine which will hold the train stationary on the grade, we add together the vectors rs and st which repre- sent the two known forces as in Fig. 30; then the line tr is the vector sought.
This method of finding the force that is able to hold a system of other forces in equilibrium is very useful in engineering practice in connection with the design of bridges, roof trusses, and other structural work in which it is nec- essary to determine how strong a beam or tie must be in order to resist the given stresses and hold them in equilibrium.
57. The Pull of the Engine and the tendency down the incline are in equilibrium; therefore we can determine the magni- tude of either of them, either graphically or analytically. Thus, since the mass of the train is 2 X 10* gm, the vector Om (Fig. 28), 2 cm long, represents the weight, namely, 196 X 10' dynes; and hence 1 cm in the diagram represents 98 X 10' dynes. The length of the vector pm is found by measurement to be 0.2 cm, and hence it represents a force of 0.2 X 98 X 10' = 196 X 10* dynes, which is the magnitude sought.
68. To Get the Analytical Solution we must notice that the triangles .450 and Omp (Fig. 28) are similar. (Why?) Therefore
^^AC' (^^^^ • ) ^"* ^^^^^ *^^ ^^^^^ ^^ AC ^ To ^^'^yP^^^"
esis, it follows that ^Ic— = — . (Why?) Whence pm = ^V Om..
Since Om = 196 X 10\ pm = yV X 196 X 10' = 196 X 10' dynes, as in the preceding paragraph.
^^ PHYSICS
69. Less Force: Greater Distance. Now, we have learned in Art. 53 that the extra work done in pulling the train 100 cm along the incline is the work done in lifting the train through a vertical height of 10 cm, i.e., in the case there considered, it is W = Jl= 196 X 10* X 10 = 196 X 10'" ergs. But we have just seen that the pull' of the engine is 196 X 10* dynes; and therefore, when this pull is exerted through a distance of 100 cm, the work done is W = /7'= 196 X 10* X 100 = 196 X 10'* ergs, as it should be. It will be noted, however, that although the amount of work is the same as that previously calculated from the vertical lifting of the train, the force of the engine is only -^^ of that which would be required to lift the train vertically through the 10 cm. The advan- tage of using AN INCLINED PLANE is therefore apparent, since we see that by means of it we can do a given amount of work with a smaller force than would be required without it. Hence such an inclined plane is said to furnish a mechanical advantage. This mechanical advantage is defined as the ratio of the resistance overcome to the effort applied. In the case of the inclined plane, when the effort is applied parallel to the length of the plane, the measure of the mechanical advantage has been shown to be the ratio of the length to the height.
Thus, in general, if h represent the height of the plane, I its length, R the vertical resistance to be overcome, and / the force exerted parallel to the plane (c/. Art. 58 and Fig. 28), then
? = i
/ ~ h'
This is the analytical expression for the mechanical advantage of the inclined plane when the effort is applied parallel to its length. It may also be written Rh = fl, which expresses analytically the fact that the amount of work done by the force applied parallel to the plane is the same as that which would be done if the body were lifted vertically through a distance equal to the height of the plane.
It has probably occurred to the reader to ask, Since the engine
. is stalled part way up the grade because its pull is no greater than
the pull of the train dow^l grade, why does the train ascend the grade
at all? The answer is that when the train reached the grade it
COMPOSITION AND RESOLUTION OF MOTIONS
67
was moving with a uniform velocity; hence it had kinetic energy whose amount is determined by equation (6) as e — im F*. It was this kinetic. energy that did the work of lifting the train; and when this energy was expended, the unaided force of the engine could carry the train no farther.
60, Definitions. Some of the ideas considered in this chapter occur so frequently that » we shall do well to frame definitions for them.
The single motion that will produce the same effect as that produced by two or more motions is called a re- sultant MOTION.
The several motions that combine to produce the resultant are called
COMPONENT MOTIONS.
The process of find- ing the resultant of two or more motions is called
the COMPOSITION OF MO- TIONS.
The process of find- ing the components when the resultant is known is called the
RESOLUTION OF MOTIONS.
By substituting the word force wherever the word motion is used, we can frame a similar set of definitions for the composition and resolution of forces.
The single force that will hold two or more others in equilib- rium is called their equilibrant. The equilibrant of any set of forces is equal in magnitude to their resultant, and opposite in direction. The point of application of the resultant is identical with that of the equilibrant,
61. The Problem of the Besolution of a Motion, or of a force acting at a given point, into two components is indeterminate unless something more than the resultant is given. Stated
Fig. 31. Inclined Railroad, Pike's Peak
68 PHYSICS
geometrically, the problem is : given one side of a triangle, to find the other two. Evidently, we can construct any number of triangles that will satisfy this condition.
A little attention to the geometry of the triangle shows that in addition to the direction and magnitude of the resultant, we must know of the components either (1) both magnitudes (three sides) ; or (2) both directions (a side and two adjacent angles) ; or (3) one magnitude and one direction (two sides and an angle).
SUMMARY
1. Any linear motion may be represented in both direction and magnitude by a straight line called a vector.
2. The vector of a resultant motion is found by adding the vectors of the component motions.
3. If two component motions are at right angles to each other, the resultant motion is numerically equal to the square root of the sum of the squares of the two component motions.
4. Any motion may be resolved into two or more component motions.
5. In order to resolve a motion into two components, we must know of the components either (1) both directions; or (2) both mag- nitudes; or (3) one direction and one magnitude.
6. The mechanical advantage of an inclined plane is equal to the length of the plane divided by its vertical height.
7. The work done in moving a body up an inclined plane is equal to the work done in lifting the same body vertically through a distance equal to the height of the plane.
8. Forces that act at a given point may be represented by vectors.
9. When the vectors that represent any set of forces in equi- librium are added together in any order, they form a closed polygon.
10. The vector that represents the resultant of a number of forces not in equilibrium is found by adding in any order the vectors of these forces, and drawing a straight line from the beginning of the first vector to the end of the last.
11. The equilibrant of any set of unbalanced forces is equal to their resultant in magnitude, but opposite in direction.
COMPOSITION AND RESOLUTION OF MOTIONS 69
QUESTIONS
1. Explain what a vector is, and how it may represent completely any physical quantity that has direction and magnitude.
2. Explain how vectors may be added in order to find the resultant of two motions when these two components have: 1, the same direc- tion; 2, opposite directions; 3, directions that are neither the same nor opposite. How is the magnitude of the resultant motion found after the resultant vector has been drawn?
3. Explain the manner in which the analytical expression for the resultant of two motions may be found when the components have directions at right angles to each other.
4. Describe a traveling crane, and explain how with it a body may be given two or three different motions at the same time.
5. With the aid of a vector diagram, explain how the motion of a body up or down an inclined plane may be conceived as made up of two components, one horizontal and the other vertical.
6. When the weight of a body and the vertical height of an in- clined plane along which it is to be lifted are known, what is the amount of work done in lifting it along the plane?
7. How does it follow from the vector diagram in question 5 that the work done in lifting the body up the incline is equal numerically to the weight of the body multiplied by the vertical distance through which it is lifted?
8. Show by a vector diagram that the weight of a body is to the force necessary to hold it in equilibriima on an inclined plane as the length of the plane is to the height.
9. Write an equation which expresses this relation. How does this equation show that the inclined plane furnishes a mechanical ad- vantage?
10. By means of this equation, show that the work done by a force pushing the body upward along the incline is equal to the work that would be done if the body were lifted vertically through a distance equal to the height of the plane.
11. Show how the vector for the resultant of any set of forces acting at a point may be found.
PROBLEMS
1. A man rows a boat with a velocity of 200 ^"^ southward in a
"^ sec
stream that has a velocity of 100 — southward. Find the resultant
velocity of the boat by the vector method, and also by calculation.
2. Find the resultant velocity by both methods when the boat is rowed northward, the speeds remaining the same.
70 PHYSICS
3. Find the resultant velocity by both methods when the man keeps the boat headed due westward, and does not try to resist the current, but rows with the same speed as before.
4. A boy rows a boat with a velocity of 3 ^^^^, keeping it headed across the stream, and not attempting to resist the current. The ve- locity of the current is 4 E^. Find the resultant velocity of the boat by both methods.
5. Suppose that the width of the stream in problem 4 is } mile, how many minutes will it take to cross the stream? How far will the boat drift down stream? How far will it actually travel along the resultant path?
6. The boy wishes to cross the stream in the same time as in prob- lem 5, but intends to land at a point directly opposite the starting point. Show by vectors the direction in which he must keep the boat headed. By both methods find the speed with which he must row in order that the boat may move in a straight line from the starting point to the landing point. How far up stream would his row have taken him if there were no current?
7. If the traveling crane, Plate II, carries the pair of wheels across the shop at the rate of 1.2 — , while it moves along the shop at the
rate of 1.6 — » ^^ ^^® resultant velocity by both methods, sec
8. Suppose that in addition to the other two motions of problem 7
the crane pulley rises vertically at the rate of 0.5 — , what is the final resultant velocity of the pair of engine wheels?
9. A trolley car weighs 10 tons and moves 1000 ft. along a grade that rises 1 ft. in every 100; how much work must the motor do? What is the mechanical advantage of the plane? What is the amount of the force that moves the car up the grade?
10. If in problem 9, the speed was 50 — , what was the horse-power?
11. The height of an inclined plane is 2 m and its length 10 m; the weight of a barrel that is rolled up this plane is 150 Kg; required the mechanical advantage of the plane, the number of kilograms-force exerted, and the number of kilogram-meters of work done.
12. A ball rolls down a smooth inclined plane whose length is 10^
cm and whose height is 10* cm. If the ball had fallen vertically, its
acceleration would have been 980 -^. Conceive this acceleration to
sec2
be made up of two components, one along the plane, and the other perpendicular to it. Determine both graphically and by calculation the acceleration of the ball down the incline.
13. The weight of a kite is 2X10^ dynes; the pull on the string is 4X10^ dynes and makes an angle of 60° with the vertical. Find the resultant pull on the kite. What must be the direction and magnitude of the force that keeps the kite in equilibrium?
COMPOSITION AND RESOLUTION OF MOTIONS
71
14. In Fig. 32, ab represents the direction of the keel of a boat, and the line d the direction of the sail, and / is a vector that repre- sents the effective pressure of the wind, 200 kil- P ograms-force. By the vector _
method, find the force that urges the boat forward, and also that which urges it sideways. How is sideways motion pre- vented?
15. By the vector method, find the nimiber of kilograms- force with which the beam or strut ab, Fig. 33, must push, and that with which the tie rod cd must pull in order to keep the 40 Kg ball in equilibrium.
Fig. 33
16. A ball is thrown upward with a velocity of 4900
For
how many seconds will it rise before its velocity is reduced to zero by the negative acceleration of 980 ^j? What is the distance to which it rises in this time? How long will it take to reach the starting point? Calculate the velocity at the instant of reaching the starting point and compare this with the velocity with which it is thrown.
17. Calculate the distances traversed by the ball of problem 12 at the ends of the successive seconds, and plot the graph for its motion. Describe the changes of slope. What is the slope at the maximum distance, or highest point? Compare this graph with the path described by a body thrown obliquely upward.
SUGGESTIONS TO STUDENTS
1. Point your lawn hose at an angle of 45° elevation; note the
path of the drops of water. Assuming 1000 — as the initial velocity
of the water, find its vertical and horizontal components. Assume that
the horizontal velocity is uniform and that the vertical velocity has a
negative acceleration of 980 ~. Calculate the distances traversed ver- sec2
tically and horizontally at the end of each fifth of a second. Plot a graph with the vertical distances for ordinates, and the horizontal distances for abscissas. Is this graph the same sort of curve as the actual path of the water? Are you justified in inferring from the com- parison that the vertical velocity was uniformly accelerated and the horizontal velocity uniform in the case of the water jet?
2. Bring a toy sail-boat to the class room to illustrate problem 14. With the aid of vectors, can you find an explanation of how such a bo»<^ can "beat against th^ wind"?
72 PHYSICS
3. Bring in sketches or photographs which show struts and ties used in ways similar to that mentioned in problem 15. You will find them on electric light poles, supporting signs, in the frames under cars, in roofs, in bridge trusses, in jib cranes, in locomotive cranes, in bicycle frames, etc. Try to draw the vector diagrams for each case brought in.
4. How high can you throw a ball? Note with a watch the total time taken by the ball in rising and falling. Also calculate the initial velocities (c/. problems 16 and 17). Place on the blackboard the names of the best throwers, with velocities and distances attained.
CHAPTER IV
MOMENTS
62. How Botation is Caused. Thus far^ we have consid- ered motion of translation only. We are now ready to take up some of the conditions under which rotary motion may occur; and the railroad train furnishes us with several questions whose answers will help us to describe accurately some relations about which we already have some general ideas. How is the translatory motion of the piston converted into rotary motion of the drivers? And why are the drivers of the fast passenger engine made large, while those of the freight engine are made small?
In order to find the answers to these questions, let us consider the diagram. Fig. 34. When the connecting rod pushes on the
Fig. 34
crank pin at nj* or pulls at n^, it is evident that it can not cause the wheel to rotate, but produces only a useless strain on the moving parts. When, however, the crank pin is anywhere above or below the line nn^, the pull or push of the connecting rod will cause the wheel to revolve. Furthermore, common expe;rience tells us that the force of the connecting rod is more and more effective as the distance from the center of the wheel to that rod increases. There is, then, some relation between the effectiveness of a force in pro-
73
74
PHYSICS
Fig. 35. The Moments are Balanced
ducing rotation, and the distance from the axis of rotation to the line of direction in which the force acts. How shall we measure the effectiveness of a force for producing rotation?
Let us suppose that the board in Fig. 35 is supported at the middle. It will then balance, so that its weight may be left out of
the problem. Sup- pose that a boy, whose weight is 20 kilograms, sits 100 cm from the axis. If a girl is seated 100 cm from the axis on the other side, the boy's weight can just hold the girFs in equilibrium, provid- ed her weight is also 20 kilograms. Now, if the boy's weight is 25 Kg and his distance from the axis is 100 cm, he can balance another at 100 cm whose weight is 25 Kg; and so on. Thus in general it appears that the effectiveness of a force at a constant distance from the axis of rota- tion, is directly pro- portional to the mag- nitude of the force. Again, suppose that a boy's weight is 20 Kg, and that he is distant 200 cm from the axis. He can now balance two children at 100 cm, each having the weight of 20 Kg (Fig. 36). If the 20 Kg boy is distant 300 cm from the axis, his weight will be as effective in turning the board as is a weight of 60 Kg at 100 cm; and so on. Thus, in general, if the distance from the axis varies, while the force remains constant, the effectiveness
Fig. 36. Moment Equals Force X Arm
MOMENTS
75
of the force in producing rotation about that axis is directly pro- portional to the ARM OF THE FORCE, i.e., to the perpendicular dis- tance between the axis and the line of direction of the force.
63, Moment of Force. The effectiveness of a force in pro- ducing rotation about an axis is called the moment of the force about that axis.
Since we have seen that the moment of a force is directly propor- tional to the magnitude of the force when the arm is constant, and directly proportional to the arm when the force is constant, it is clear that the appropriate numerical ineoMire of the moment of a force is the product of the force and its arm with respect to the given axis.
Returning to the case of the locomotive drivers (Fig. 34), we see that the measure of the turning effect is the force F^, applied to the crank pin, multiplied by the perpendicular distance from the center of the wheel to the middle line or axis nn^ of the connect- ing rod.
Since now we know how to calculate the moment of a force, we shall be able to consider a few problems that will lead us to the statement of some very important principles, and will also enable us to answer the questions that were raised concerning the relative sizes of driving wheels for passenger and freight engines.
64. The Lever. Suppose that the man in Fig. 37 is to do the work of lifting, Avith the lever, a stone which weighs 100 Kg. He pushes vertically downward at one end with a force which we will call /. The fulcrum, i.e., the axis p (Fig. 38) about which the lever turns, is distant 40 cm from the center of the stone and 200 cm from the man's hands. The moment of / with respect ^^°- '^- ^''^ ^^^«
to the fulcrum is / X 200, and that of the stone's weight is 100 X 40. If the moment of / is just sufficient to keep that of the stone's
76
PHYSICS
weight in equilibrium, then / X 200 = 100 X 40. Whence, finally, / = 20 kilograms-force = 20 X 1000 X 980 = 196 X lO' dynes.
The force that will move the stone must, of course, be somewhat greater than this, because some unbalanced force is required to produce the acceleration.
The equation may be written:
100 200 5 / " 40 "^ 1 ' which states that the mechanical advantage of this lever is 5 (cf. Art. 59).
65. The Work Done by the Lever is easily calculated. When the lev^r is moved. Fig. 38, the point s describes an arc with a radius
of 40 cm, and moves, say, from s to s', while the point h describes a similar arc with a radius of 200 cm, going from A to A'. Suppose the vertical distance sm through which the stone is lifted is 10 cm. The effort, at the same time, acts through the vertical distance hn. If the stone weighs 100 Kg, or 98 X 10" dynes, calculate how many ergs of work are done in lifting it through 10 cm.
Now, since the right triangles msp and nhp are similar (Why?),
— == -TTT = T- Since sm = 10 cm, what is the value of 5m 40 1
An? Thus it appears that, although by means of this lever we are able to dok the work of lifting a stone with a force that is only one- fifth of the weight of the stone, this force must be exerted through a distance or displacement five times as great as that through which the resistance is moved.
The work done by / is / multiplied by its displacement, or (196 X 10^) X 50 = 98 X 10^ ergs. How does this amount of work, done by the man, compare with that done on the stone as previously calculated?
40 jn ./
Fig. 38. The Lever Diagram
MOMENTS 77
A lever is often used in another way. as in Fig. 39, when the fulcrum is at one end, and the resistance between, — the effort being applied at the other end as before. In this case the application of the principle is entirely similar; but the possible mechanical ad- vantage is greater, because the / lever arm of the effort is longer, t The moment of the effort with respect to the fulcrum is now / X 240, and that of the resist- ance is 100 X 40 as before, the
... - ^ . .1 Fio. 39. Another Lever Diagram
mechanical advantage is there- fore found from the equation / X 240 = 100 X 40. Whence
40 / = 100 X oTrj = 16.66 Kg-force. The mechanical advantage
240 in this case, therefore, is — , or 6. The geometrical construc- tion by which the number of ergs of work are found and proved equal is much like the preceding, except that the similar trian- gles are differently placed. It is easily seen from the figure that
-f— — -777 = T> and that the effort X 60 = the resistance X 10. sm 40 1
66. The Lever Principle. In the examples just worked out, we have learned four things about the lever. Other problems involving levers can be solved in a similar manner. The four things that we have learned are:
1. The lever is in equilibrium when the moment tending to turn it in one direction is equal to that tending to turn it in the opposite direction.
2. The mechanical advantage of a lever is equal to the effort arm divided by the resistance arm.
3. The mechanical advantage of the lever may also be obtained by dividing the displacement of the effort by the displacement of the resistance.
4. The work done by the effort is equal to tlie work done on the resistance.
78 PHYSICS
These statements may all be verified by very simple experiments in which the forces and distances are measured when various kinds of levers are in equilibrium. In such experiments and problems, it must be noted that whenever the weight of the lever itself is at all comparable in magnitude with the other forces involved, it also must enter into the calculation.
67. Equilibrium of Parallel Forces. Another impoiiant fact concerning the lever (Fig. 38) is sufficiently obvious without argument. The two downward forces must produce a downward pressure on the fulcrum; this downward pressure is their resultant, and is equal in magnitude to their sum. Hence it is evident that the fulcrum must exert an upward resistance which is the equilibrant of this resultant, and which is therefore equal in magnitude to the sum of the downward forces. It is also clear from Art. 65 that the point of application of this equilibrant divides the line joining the points of application of the two forces into segments that are inversely proportional to the magnitudes of the forces. There- fore when a system of parallel forces in one plane acts on a body, the condition that must be fulfilled in order that no translatory motion may take place is that the sum of the forces acting in one direction be equal to the sum of those acting in the opposite direction.
Similarly, the condition that must be fulfilled in order that no rotation may take place is that there be no resultant nioment, i.e., that the sum of the moments tending to turn the system in one direction about any point be equal to the sum of the moments tending to turn it in the opposite direction about the same point.
It will easily be understood that these conditions for equilib- rium which we have seen apply in the case of three parallel forces, must hold for any number of such forces; for clearly if there is no unbalanced force, there can be no translation; and if there is no unbalanced moment, there can be no rotation.
68. Illustration by a Problem. If we wish to determine the single force that will hold a system of known parallel forces in equilibrium, we can do so with the help of these principles. For
-1Q.QL — »50.
^
MOMENTS 79
example, suppose it is required to lift the shaft with its pulleys. Fig. 40, by applying a single vertical force in such a way that the shaft will remain in a horizontal position as it rises. How great a force will be neces- sary, and at what point must it be ap- plied? The indi- .^_ M^_^_
cated weights of the
wheels and the shaft A
are the known forces, T \ ^o
and their respective So
*â– â– Fig. 40. The Shaft Remains Level
distances from A^
the end of the shaft, are the known arms. We may assume that the bar is uniform, sd that its weight acts at its middle point, as shown in the figure. The lifting force / and its arm r are to be deter- mined. From the dimensions on the diagram we see that the sum of the downward forces is 50 + 60 + 30 + 80 = 220. The required upward force, therefore, must be equal to this sum, or / = 220 Kg-force.
Since the condition for no rotation is that the moments, taken with respect to any point, be balanced, we may select the left end of the shaft as the most convenient point of reference. The sum of the moments with respect to this point is, then, evidently (50 X 25) + (60 X 200) + (30 X 250) + (80 X 350) = 48750. This moment must be counterbalanced by that of the upward force of 220 having the unknown arm r. Hence, (220 X r) = 48750. Whence r = 221.6 cm, i.e., the force necessary to hold the shaft in equilibrium is 220 Kg-force; and it must be applied at a point 221.6 cm from the left end of the shaft. Of course some addi- tional force will be required to produce the acceleration when the shaft is moved.
69. The Equilibrant of Any Number of Parallel Forces may be determined in a manner similar to that used in the example just given. Since we can form two equations in which all the forces appear, we may determine either one force and one arm, as in the example, or two forces whose arms are known.
80
PHYSICS
Fig. 41. A Fast Engine
70. The Locomotive Drivers. Let us apply the principles of the lever to the driving wheels of the locomotive. Let F^ repre- sent the pull of the connecting rod nn^ (Fig. 34); and let r^, which
is the perpendicular dis- tance from the center of the wheel to nn^, rep- resent the arm of this force. Also let Fj rep- resent the push mFj, exerted by the rim of the wheel along the track; and let rg, the radius of the wheel, which is the perpendicular distance from its center to mF2, repre- sent the arm of the push Fj. Then from the equation for the
F r .
mechanical advantage of the lever, -^r = — • This equation shows
^ t ^ that the horizontal push at the rim of • the driver is less than that exerted on the crank pin, in the same proportion as the distance T^ of the crank pin from the center is less than the radius ra of the wheel.
Now, an engine with large driving wheels can develop greater speed than can one with smaller ones, because the circumferences of the drivers are large; and the engine will go farther, for each stroke of the piston. But in this case our equation shows us that, other things being equal, the push that can be exerted on the track is proportionately less; because rj, the radius of the driving wheel, is increased in the same proportion as is the circumference.
On the other hand, an engine which is to haul a long and massive freight train, must be able to exert a very great horizontal push on the track, and therefore rg must be made smaller in proportion to r^. This necessitates smaller driving
Fig. 42. A Powerful Engine
MOMENTS
81
wheels, giving less speed. Also, since the aim is to get as much power as possible, the engine must not only have large and power- ful steam cylinders, but must also be very heavy, so as to exert sufficient pressure on the track; otherwise the driving wheels will slip, and the engine will not be able to move the train.
71. Weight and Center of Mass. Some very important applications of the principles pertaining to parallel forces are found in the action of gravity on bodies. For gravity tends to pull each particle of a body toward the center of the earth; therefore, the gravity forces that act on all the particles of a body are prac- tically parallel, and their result- ant is the weight of the body.
Now, when any body is acted on by a system of forces affecting all its particles, and all in the same direction, there is a point so situated that the moments of all those forces will be balanced about any axis that passes through this point. This point, which is evidently the point of application of the resultant of all the parallel forces acting on the particles of the body, is called the center of mass. The center of mass of a body is, therefore, the point of application of its weight, and hence it is often called the center of gravity.
72. Equilibrium. // a force acting vertically upward, and equal to the weight of a body, he applied so that its line of direction passes through the center of mass, the body will he in equilibrium under the action of this force and its weight
Thus suppose that c (Fig. 44) is the center of mass of a body sus- pended at a point s, about which it is free to rotate, as is the case with swing (Fig. 43). Then if the body has been slightly displaced
FiQ. 43. The Swing
82
PHYSICS
from the position wherein c is vertically below s, there is a moment which is equal to the product of its weight w and the distance sb, and which will return it to that position. In what position will such a suspended body be in equilib- rium? The vase and the pitcher, Fig. 45, are in equilibrium; what moment tends to return each of them, when it is slightly tilted?
73. The Stability of a Body like the vase or the pitcher, which rests on a BASE, is measured by the amount of work that must be done in overturning it. A little consideration will show that the amount of this work may be de- termined as follows: With o as a cen- ter and a radius equal to ac, describe an arc. This arc is the path that the cen- ter of mass c will describe when the body is overturned about the point or axis represented by a. When the cen- ter of mass c is in the vertical line that passes through a, the body will be in unstable equilibrium, and the smallest further displacement will overturn it. From c draw a horizontal line intersecting oc' at a point 6. Then &(/ is the ver- tical distance through which the center of mass must be raised in order to overturn the body; and the work done is found by mul- tiplying the. weight by this vertical distance.
Other things being equal, if the base of the body were smaller, or if the center of mass were higher, as in the case of the vase, what would be the effect on fee', and on the work done in over- turning the body?
Answers to questions like these lead to the general conclusion that, other things being eqical, the larger the base of a body, and the lower its center of mass, the greater is its stability.
Fig. 46 represents a sphere of uniform density, whose center of figure is therefore its center of mass. Show that it is in equi-
FiG. 44.
The Swing Dia'
GRAM
MOMENTS
83
Hbrium in any position on a level plane. Show also that when the plane upon which it rests is slightly tilted, there is a component
Fig. 45. Stability is Measured by Work
of force urging it down the plane, and also a moment of force tend- ing to rotate it.
74. Determination of Center of Mass. The foregoing prin- ciples of equilibrium enable us to find the center of mass of a body by experiment; for if the body be freely suspended from a point near one of its extremities, it will come to rest in the position wherein the arm of its weight becomes zero (cf. Fig. 44). This position is evidently that in which the center of mass is in the vertical line passing through the point of support. If this line be indicated by a plumb line, and marked on the body, we know that it contains the center of mass.
Fig. 46. The Ball Has No Sta- bility
84
PHYSICS
If, now, another point of suspension be selected, and a new vertical line marked in the same way, it must be apparent that the center of mass, since it is in both these lines, can be nowhere else than at their intersection.
Another way of finding the center of mass of a flat, thin body, such as a piece of pasteboard, is to balance it flatwise upon a straight- edge, and mark on it the axis upon which it balances. This axis, in accordance with the definition, must contain the center of mass. Therefore, if another axis about which the moments balance be located in the same way, the center of mass is the* point in which the two axes intersect, for it will be found that every other axis on which the body will balance passes through this point.
These experiments are of great con- venience in connection with certain engi- neering problems; for it is often neces- sary to find the center of mass of a part of a machine, or of a piece of some struc- tural work, in order that it may be de- signed so as to be in equilibrium under the given conditions.
For example, there must be placed on a locomotive driver (Fig. 47) a coun- terpoise having a moment exactly equal to that due to the con- necting rod or side rod used in turning the wheel. This is be- cause if the moments of all the rapidly rotating parts are not thus accurately balanced against each other, the system will wabble and produce a wasteful and even destructive strain on its axis of rota- tion. In order to place the counterpoise properly, its center of mass must be known ; and since it is not a regular body, this deter- mination can not easily be made by geometry. The usual practice, therefore, is to cut out a pasteboard model to a certain scale, and experimentally determine the center of mass of this model. The center of mass of the real object is then easily located, for it is the point of the real object that corresponds to the center of mass of the model.
Fio. 47. The Driver Has A Counterpoise
MOMENTS 85
The same method is used in order to get the position of the center of mass of half of a stone arch, so that the moment due to its weight can be calculated.
75. Mechanical Advantage of a Composite Machine. Before leaving the study of the- applications of the lever principle, let us consider how we can find the mechanical advantage of a contrivance like that in Fig. 48, in which the lever principle and that of the inclined plane are used simultaneously.
In pulling the safe up the inclined plane whose height is 100 cm and whose length is 400 cm, the mechanical advantage obtained
by means of the plane (c/. Art. 59) is t- = -jt^ = 4; i.e., the
weight of the body that can be moved along the incline is four times the pull of the rope. Fur- thermore, the effort, which is ap- plied to the end of the crank, has a greater lever arm than has the pull of the rope. The effort arm in this case is the length of the crank, and the resistance ann is the radius of the axle. There- fore we obtain by this device a /'«• f- Composite Machine . •' /PA ^ windlass and an inclined plane, mechanical advantage (c/. Art.
65), which is equal to the ratio of the length of the crank to the radius of the axle. If these two lengths are 50 cm and 10 cm
/ 50 .
respectively, then -^ = — = 5, i.e., the pull on the rope is five
times as great as the effort applied at the crank handle.
Now, if the man applies to the crank handle a force equal to the weight of 40 Kg, it is clear that since the mechanical advantage of the windlass is 5, the pull on the rope is equal to 40 X 5 kilograms- force. Furthermore, since the mechanical advantage of the inclined plane is 4, the weight that can be lifted along the incline by this pull of 40 X 5 is, (40 X 5) X 4 = 40 X 20 = 800 kilograms-weight. Thus we see that the mechanical advantage
of the combination is ;= — ; — = -77- = 20. This mechanical
\ effort 40
86 PHYSICS
advantage of the combination can be quickly obtained by multiply- ing together the mechanical advantages of the elementary parts; e,g,, 5 X 4 = 20.
Similar reasoning applied to other problems shows that this method of procedure will give the mechanical advantage of any composite machine, no matter how complicated it may be. Hence in general we can find the mechanical advantage of any composite rrmchine by multiplying together the mechanical advantages of the several elementary machines of which it is composed.
76. The Law of Machines. We have seen that for the inclined plane and for the lever, the work done by the effort is equal to the work done on the resistance. Let us now see if this is true for the combination of these two devices. In the case just discussed, the effort was supposed to be 40 kilograms-force = 40 X 1000 X 980 = 392 X 10^ dynes. The resistance was 800 kilograms-weight = 800 X 1000 X 980 = 784 X 10^ dynes. The distance Ij^ through which the effort acts in one revolution of the handle is Z^ = 27r X 50 cm; and the vertical distance ^ through which the weight is lifted may be found as follows : For one turn of the crank, the rope is drawn up a dis- tance equal to the circumference of the axle, or 27r X 10 cm. Evidently the safe moves the same distance up the incline. But since the height of the plane is one-fourth of its length, the vertical distance through which the safe is lifted is one-fourth of the corresponding distance that it moves along the incline;
, 27r X 10 cm . ^ _ _ I.e., I2 = T = 27r X 2.5 cm.
The work done by the effort, therefore, is found to .be /i^i = (392 X 10^) X (27r X 50) = 392 X 10^ X tt. That done on the resistance is f^k = (784 X 1(f) X (27r X 2.5) = 392 X 10^ X tt. Thus the two amounts of work are equal.
Similar reasoning proves that this principle, which we have demonstrated in the cases of an inclined plane, of a lever, and of a combination of the two, applies to all machines whatsoever. It is usually called the law of machines, and is stated as follows: The product of the effort and the distance through which it acts is
MOMENTS 87
equal to the jyroduct of the resistance and the distance through which it is overcome; or, the work done by the effort equals that done on the resistance. In symbols,
fA-hh- (7)
It should be noted that the distances l^ and l^ mu^st always be measured in the directions of the corresponding forces. Also, in applying this statement to any particular case, it should be bome in mind that the total work done invariably includes some useless work against such resistances as friction, rigidity of parts, inertia^ reaction, and resistance of the air; so that in order to make the state- ment precise and perfectly general, this useless work must be under- stood to be added in with the useful work. When this has been done, it is invariably found that the work done is the exact equiv- alent of the energy expended upon the contrivance (c/. Art. 36).
77. Efficiency. As the cost of the energy used in a manu- facturing plant or a system of transportation is a very large part of the operating expense, the efficiency of the machinery used is a feature of great importance. It often proves to be very poor economy to buy machinery of low efficiency simply because it is cheaper.
Since some useless work is always done, no machine has an efficiency of 100%. No machine can create energy (c/. Art. 36); it can only transfer or transform energy that is supplied to it from some external source. It enables the user to apply his energy in more convenient ways than would be possible without it; but the user is always taxed, as it were, a certain per cent of the energy for the convenience thus obtained.
78. Mechanical Advantage from Law of Machines. The law of machines enables us to find the mechanical advantage of any machine. For we may write equation (7) (Art. 76) in the form
T — ^y i-^-> *^^ mechanical advantage of any machine is
equal to the ratio of the displacement of the effort to that of the resistance. It is often more convenient to find the mechanical
88
PHYSICS
advantage of a composite machine by measuring these distances than it is to calculate it by multiplying together the mechanical advantages of the parts. This is the case with the screw.
79. The Screw. The thread of the screw is an inclined plane wrapped around a cylinder. Fig. 49 shows how the screw would look if one turn of the thread were unwrapped.
In order to turn the screw about its axis, a force /^ is applied at the end of the lever, or at the circumference of the head; and its displacement Z^, for one turn, is the circumference described by the point of application of this force.
When the screw is. rotated, either the screw itself or the nut in which it turns, moves in a direction parallel to the axis. Thus when the jack screw (Fig. 50) is turned once around, the stone, or whatever rests on the head of the screw, is lifted through the distance /j be- tween two adjacent turns of the thread. This distance, measured parallel to the axis, is called the pitch of the screw. Finally if /a represent the resistance to be overcome.
Fig. 49. Screw Thread Unwrapped.
we
have from equation (7), t ^ ^f which /I h
tells us that the mechanical advantage OF THE SCREW is nunierically equal to the circumference through v* which the effort is applied, divided by the pitch of the screw. Since the lever or head of the screw may b2 made very large, and the pitch very small, this equation shows that the mechanical ad- vantage may be enormous, and is limited only by the strength of the materials used in the construction of the screws. Thus, wagons, locomotives, and even large buildings are lifted by means of jack screws.
Fig. 50. Jack Screw
MOMENTS
89
Fig. 51 shows how a large house was lifted up a hill 100 ft. high with the help of such screws. They may be seen in the picture between the timbers and the house.
80. The Equal Arm Balance, which is generally used for comparing masses, is another important application of the law of moments. If we wish to weigh a certain quantity of some sub- stance, for example a pound of sugar, the mass, of the sugar in one
FiQ. 61. Jack Screws in Action
pan is assumed to be one pound when its weight balances a standard pound weight on the other pan. For if the balance comes to rest with the pointer at zero the opposing moments are equal.
Hence if /j represent the weight of the sugar, and /^ that of the standard pound mass, and if r^ and r^ represent the corresponding arms, then the equation for the balanced moments is ^rj = f^r^. But r^ = r^, therefore /j = /i, i.e., the weights are equal. This will be true if the arms are exactly equ^l, and if the balance comes to rest with the pointer at zero under no load.
And since it was shown in Art. 31 that the weights are piopor- tional to the corresponding masses, it follows that if the weights are equal the masses are also equal. Thus the law of moments shows us that we are correct in our habitual assumption that we can com- pare masses correctly by the process of weighing. Of course the
90 PHYSICS
accuracy of the comparison is dependent on the accuracy of the balance and of the masses in the set employed as standards.
8L Looking Backward. We have now arrived at a place in our studies in Physics where it will be well to pause and review what we have learned. The principles are really very few^ although, as we have begun to see, their applications to every- day life and to the devices of modem civilization are countless.
First, we have learned how uniform and uniformly accelerated motions may be accurately and concisely described, and especially so by the graphical and analytical Inethods. We then endeavored to gain clear notions of the relations of mass and acceleration to work, energy and activity, or power; and we found that by means of concise equations in which these relations are expressed, many important practical problems may be solved.
We then considered the behavior of bodies in motion and of bodies in equilibrium when acted on simultaneously by two or more forces; and we found that the resultant motions and the resultant forces can be represented with great ease and clear- ness by means of vectors. Further, we learned that many compli- cated machines are made up by combining the principles of two or more of the elementary mechanical devices known as the inclined plane, the lever, the pulley and the screw; and that for each of these devices a simple numerical relation between the effort and the resistance can be established. This is done by compound- ing or resolving forces or motions with the aid of vectors, or by taking the moments of all the fojces with respect to some conveniently chosen axis, and forming the equation for their equi- librium.
In conclusion, we found that whenever energy is expended upon any kind of machine for the purpose of doing work, the sum of the useful work and the inevitable useless work is the exact equivalent of the energy expended — no more, no less.
82. Our Future Study of the several forms of energy will show us that there is nothing in the study of Physics but the accurate description of relations that occur when energy is transferred from one portion of matter to another,, or changed from one form
MOMENTS 91
into another fonn. We can gain knowledge of these changes only through observation and experiment
Phenomena are thus learned and grouped into classes. The conditions under which they occur and their relations to each other are described in concise statements called laws. With the aid gf the reasoning powers and the trained imagination, HYPOTHESES are framed for the explanation of these laws. The hypotheses are then tested by deducing from them relations which follow as necessary consequences. Careful experiments are then devised and carried out in order to determine whether or not the relations thus deduced are verified — that is, whether they are true or not.
When a hypothesis is found competent to explain every known fact that must follow as a consequence of it, and is verified by every appropriate experiment that is made in order to test it, it takes rank as an established theory.
By deducing from a hypothesis or theory certain consequences, and then testing these deductions experimentally, most of the great scientific discoveries have been made.
The method of study here outlined is called the scientific METHOD. Since the history of great scientific discoveries, and of the inventions which have always followed in their wake, has plainly shown that this method is the only one by which such advances have been made, the great advantage of a study like Physics is manifest. It is only by training the powers of observa- tion and reasoning, and by developing the scientific imaginiation in as many people as possible, that individuals can be produced who shall continue the progress in discovery and invention which is now going on. For discoverers and inventors must not only be bom and educated, but they must be supported materially, and encouraged by an intelligent interest on the part of the great body of people among whom they live and work.
SUMMARY
1. In order that a body may be made to turn about an axis, it must be acted on by a force whose line of direction does not pass through the axis.
92 PHYSICS
2. The effectiveness of a force in producing rotation is its moment. Moment of force = force X ann of force.
3. The mechanical advantage of a lever may be found by equating the opposing moments taken with respect to the fulcrum.
T* • 1 * ru X- effort arm
It is equal to the ratio, — r-r .
^ resistance arm
4. The mechanical advantage of a lever is also equal to the
^ displacement of effort * displacement of resistance'
5. The resultant of two parallel forces having the same direc- tion is equal to their sum; it has the same direction as the two forces; its point of application lies on the line joining theirs, and divides that line into segments that are inversely as the magnitudes of the two forces.
6. In order that any system of parallel forces may be in equilib- rium, the sum of the forces in one direction must be equal to the sum of the forces in the opposite direction; also the sum of the moments tending to turn the system in one direction about any point or axis must be equal to the sum of those tending to turn it in the opposite direction about the same point or axis.
7. The equilibrant of any number of parallel forces may be fully determined by means of equations formed in accordance with this statement.
8. The center of mass of a body is the point of application of the resultant of any set of forces that act in the same direction equally on all its particles.
9. The center of gravity of a body is the point of application of its weight, and is identical with its center of mass.
10. The stability of a body is measured by the work that must be done in overturning it.
11. The mechanical advantage of a composite machine is equal to the product of the mechanical advantages of all its elementary parts.
12. In the case of every mechanical contrivance, the work done by it is equal to the work done upon it, or f^l^^ = /jZj.
MOMENTS 93
13. The work done by every machine includes some useless work.
14. The mechanical advantage of any machine is also equal
^, ^. displacement of the effort ,.,,.,
to the ratio, t^ r i-"j.^i ^i > both displacements
displacement of the resistance ^
being measured in the directions of their corresponding forces.
15. The mechanical advantage of the screw is equal to the . circumference described by the effort
' pitch of the screw
16. The equal arm balance is used for comparing masses by means of their weights.
QUESTIONS
1. How must force be applied to a body in order to make it rotate about a given axis?
2. What is meant by the moment of a force with respect to a given axis? What is its numerical measure?
3. Show how, by equating the moments about the fulcrum of a lever, we can find the equation for its mechanical advantage.
4. Show by geometry that for a lever the displacements are pro- portional to the corresponding arms. What, then, is the relation between the displacements and the corresponding forces?
5. Show in the case of the lever that the work done by the effort equals that done on the resistance.
6. What are the conditions that must be satisfied in order that any system of parallel forces may be in equilibrium? What kind of motion will result if each of these conditions is not satisfied? If neither is satisfied?
7. Explain what the center of mass of a body is.
8. Explain why the weight of a body may be supposed to be a single force, acting at its center of mass.
9. What is the measure of the stability of a body?
10. Show how the stability of a body may be determined graph- ically.
11. Show by a diagram that the stability of a suspended body is increased by increasing the distance between its center of mass and its point of suspension; and vice versa.
12. Show by diagrams that the stability of a body supported on a horizontal plane is increased (a) by increasing the area of the base, (6) by lowering the center of mass.
13. Explain how the principles of stability are applied practically m the construction and loading of buildings, wagons, and cars.
94 PHYSICS
14. Explain why a man on a step ladder is more easily overturned the higher he ascends, unless the feet of the ladder are put propor- tionately farther apart.
15. In what two ways may the mechanical advantage of a com- posite machine be determined?
16. State the law of machines, and write the equation that ex- presses it analytically.
17. What kinds of useless work are done by a machine?
18. Of what commercial importance is the efficiency of machines?
19. What are some of the uses of the screw? How may its mechan- ical advantage be determined?
20. From the law of moments, show why we can correctly compare masses by means of the equal arm balance.
PROBLEMS
1. A workman applies 75 Kg-force at one end of a crowbar 200 cm long; what weight may be lifted at the other end distant 25 cm from the fulcrum? What is the mechanical advantage? With the same ratio of the arms, what effort will be required to overcome a resistance of 800 lb.? In each case what was the amount and direction of the pressure exerted by the fulcrum?
2. A safety valve lever, Fig. 52, has its fulcrum at a, and is to push down on the valve rod at c, which is 2 cm from a. What must be the
weight of the ball, if it is to be applied at notch 5, which is 6 cm from a, and exert at c a 5 kilograms-force? What force will be exerted at c if the ball weighs 2 Kg and is placed at notch 10, which is 12 cm from o?
3. Suppose that a trunk 0.9 m long, 0.6 m high, and weighing 120 Kg is to be tipped over on one end by lifting at the other. The weight is uniformly dis- tributed: how much force is necessary to start it? Will this force increase or diminish as the trunk approaches the upright position? Rep- resent graphically the stability of the trunk, and express the value of the stability in kilogram-meters of work.
4. Devise a scheme for weighing a turkey, using only a stick of uniform density and cross-section, and 50 cm long, a cm scale, some strong cord, and a flatiron known to weigh 2.73 Kg (6 lb.). Be sure that your scheme provides for eliminating the weight of the stick. Illustrate your method by working out a numerical example.
5. A bridge whose weight is 3 X 10^ lb. rests on abutments 75 ft. apart. Assuming that the weight of the bridge is uniformly distrib- uted, what part of this weight is supported by each abutment? What
^
I
MOMENTS 95
additional pressure is applied when an engine that weighs 12 X 10* lb. stands with its center of mass 25 ft. from one end of the bridge?
6. With a single fixed pulley (Fig. 53) , what pull on one end of the cord will support a weight of 100 Kg at the other? Use the lever principle, assuming the radius of the pulley to be 5 cm.
In this case what is the pull on the support if the pulley i "^ i
itself weighs 2 Kg? When the resistance is overcome * — h ■'
through 1 m, through what distance does the force act? V
What is the mechanical advantage? x-
7. A movable pulley is arranged as shown in Fig. 54. — '
With what force /i, and in what direction, ^ J I must you pull in order to support a weight ,
^y^ /j, of 150 Kg? Would the force be the same t I if you pulled at /, making use of the fixed "^ pulley? Use the lever principle, taking q p
for one fulcrum and i for the other. Can you V y obtain th^ same result from the law of ma- j ^
chines [equation (7), Art. 76]? Express the mechanical advantage in terms of the lever Fig. 53 arms, and also in terms of the distances. ^iq. 54
8. Show that if the arrangement of pulleys in Fig. 54 were turned end for end, the mechanical advantage would be 3 for a pull at /. In arrangements of this sort, what is the relation between the mechanical advantage and the number. of parts of the cord that pull against the resistance? Is there any useless work done by the pulleys, so that the mechanical advantage actually obtained is less than that given by the calculation?
9. The screw of a cider press has a pitch of 0.5 cm, and is turned by a lever 50 cm long. What is its mechanical advantage? What pressure will be exerted on the apples when you apply a 30 Kg-force at the end of the lever?
10. Make a diagram of a combination of any two of the machines mentioned in this list of examples. Find the mechanical advantage of the combination (c/. Art. 75), and the effort necessary to overcome a resistance of 400 Kg-force.
SUGGESTIONS TO STUDENTS
1. Measure the lever arm and the pitch of the screw of a vise, and find its mechanical advantage.
2. If you are interested in turning lathes, examine one in a shop, and see how many of its parts have mechanical advantages. How is more force and slower speed obtained by shifting the belt from one pair of pulleys to another? When a screw is to be cut, how do you
96 PHYSICS
make the cutting tool travel along the lathe bed at the desired rate; — for example, to cut twice as many threads to the inch as there are on the lead screw? What other examples of the composition of mo- tions and of the lever principle does the lathe furnish?
3. What kinds of lever can you find in a sewing machine? in a typewriter? in a bicycle? Consult a book on physiology and see if you can find the lever principle in the human arm, foot, jaw, etc.
4. Can you solve the lever problems presented in rowing a boat? in using a nut-cracker? in the sugar tongs? in the scissors? in the gas tongs? in the claw hammer?
5. With a set of pulleys like Fig. 54, determine by experiment the number of gms- weight at / that will just lift a given weight at W with uniform speed. Measure the distance I through which / moves, while W is being lifted a distance /i = 10 cm. Calculate / X ^, the work done by the effort, and W y^h, the work done on the resistance; also calculate
the efficiency, . , . Now, by taking off weight at /, find the number
of gms-wt at / that will just allow W to descend with uniform speed; and also find the efficiency in this case as you did in the first. Take the average of these two efficiencies as the mean efficiency for the given load. In the same way, find the mean efficiencies for, say, 9 other loads, and choosing a convenient scale, plot a graph with efficiencies for ordinates, and loads for abscissas. Does the efficiency increase with the load? In direct proportion, or according to some other law? A set of pulleys can be bought cheaply at a hardware store, and will be all the more interesting if not too good. How can you determine the number of gms-force of the friction?
6. If mechanically inclined, you may find mines of interesting infor- mation about all sorts of mechanical devices, their mechanical advan- tages and efficiencies in Perry's Applied Mechanics (Van Nostrand, N. Y.), and in Pullen's Mechanics (Longmans, N. Y.). There is much in these books that you may not be able to understand; but you can read without difficulty enough to increase immensely the knowledge that you have thus far acquired. Lodge's Mechanics (Macmillan, N. Y.) is easier reading, and will also interest and help you.
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CHAPTER V
ROTATION
Note. The authors recommend that this chapter be used for informal dbcussion on the first reading. If time is limited it may be omitted.
83. Flywheels. In the preceding chapter we learned that, in order to cause rotary motion, an unbalanced moment of force is re- quired.
Another important case of the conversion of translatory motion into rotary motion is that of a stationary engine and its flywheel, Plate III. Here the relations involved in producing the rotary motion are the same as in the locomotive and its drivers. But the flywheel is large and has a very massive rim, and is designed to produce an effect which is not necessary in the locomotive drivers.
This effect is that of steadying the motion; for it is clear from what has preceded that the moment of force acting on the wheel is different in different positions of the crank pin, and this will cause a jerky motion of the machinery. But the big flyvv^heel receives and stores up energy of rotation when the crank pin is in the favorable positions, and faithfully pays it back again when the crank pin is in the unfavorable positions. Thus it prevents the sudden jerks which would be injurious to both the engine and the machinery which it runs.
Why is it that the flywheel has a massive rim and large diameter? How does this distribution of the mass make it more effective in storing up energy and paying it out again?
84. Angular Measures. These questions can not be answered by expressing the relations in terms of linear velocity, because it is clear that different portions of the mass, being at different distances from the axis of rotation, have different linear velocities Therefore we must have some other means of measuring this velocity. Now, it is evident that every spoke of the wheel, and
97
98 PHYSICS
in fact every radius, sweeps over the same angle in the same time, and therefore all the particles of the wheel have the same angular velocity. How, then, is angular velocity measured?
A convenient way to measure an angle is to find the number of times that the radius is contained in the corresponding arc; i.e.,
. _ length of arc ^ length of radius'
If in this equation we make