MECHANICS

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3

MECHANICS

A TEXT-BOOK FOR ENGINEERS

JAMES E^^^OYD, M.S.

PROFESSOR OF MECHANICS, THE OHIO STATE UNIVERSITY

AUTHOR OF "strength OF MATERIALS," "DIFFERENTIAL EQUATIONS FOR

ENGINEERING S'TTDENTS," ETC.

First Edition

McGRAW-HILL BOOK COMPANY, Inc.

NEW YORK: 370 SEVENTH AVENUE

LONDON: 6 <k 8 BOUVERIE ST., E. C. 4

1921

V^'. '■ ^ '^^.

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Copyright, 1921, by the McGraw-Hill Book Company, Inc.

THB MA7I.X FXBB8 TOXX P Jk.

PREFACE

This book is intended to give a working knowledge of the principles of Mechanics and to supply a foundation upon which intelligent study of Strength of Materials, Stresses in Structures, Machine Design, and other courses of more technical nature may rest.

In the development of this subject, emphasis is put upon the physical character of the ideas involved, while mathematics is employed as a convenient tool for the determination and expres- sion of quantitative relations. Analytical and graphical methods are given together and each is interpreted in terms of the other. While the principal stress is placed upon Mechanics as a science, considerable attention is given to Mechanics as an art. In the text, in some of the problems, and in many of the illustrative examples, methods of calculation are suggested by means of which accurate results may be most readily obtained.

The definitions of work and potential energy, together with the solution of problems of statics by the method of virtual work are given early. In the treatment of dynamics, the definitions of kinetic energy and its application to the conditions of variable motion are introduced as soon as possible.

In equations involving acceleration or energy, the common commercial units are employed — the pound mass as -the unit of mass and the weight of the pound mass as the unit of force. In order to clear up the confusion which results from the fact that physicists use one set of units while some engineering writers use another, Chapter XVIII is devoted to a discussion of the various systems.

The author acknowledges his obligations to many of his col- leagues who have assisted in the preparation of this book. P. W. Ott of the Department of Mechanics checked the problems of several chapters. S. A. Harbarger of the Department of English read all the manuscript and assisted in the final revision. Professor O. E. Williams of the Department of Engineering

vi PREFACE

Drawing supervised the preparation of the drawings, and Professor Robert Meiklejohn of the same department made many valuable suggestions.

J. E. Boyd. CoLtTMBXJs, Ohio, May 13, 1921.

CONTENTS

Paob Preface v

Notation • xiii

CHAPTER I

Fundamental Ideas 1

Mechanics — Illustrations — Fundamental Quantities — Standards and Units — Length — Time — Matter and Mass — Force — Weight — Relation of Mass to Weight — Variation of Weight with Latitude and Altitude.

CHAPTER II

Quantity and Calculations 9

Representation of Quantity by Numbers — Representation of Quantity by Lines — Vectors — Addition of Vectors — Components of a Vector — Computation of the Vector Sum — Vector Difference — Vectors in Space — Vectors by Direction Cosines — Addition of Vectors in Space — Graphical Resolution of Vectors in Space — Graphical Determination of the Vector Sum — Product of Two Vectors — Summary.

CHAPTER III

Application of Force 29

Tension, Compression, and Shear — States of Matter — A Rigid Body — A Flexible Cord — Equilibrium — A smooth Surface — A Smooth Hinge — Resultant and Equilibrant — The Force Circuit — The Free Body — Application of Forces — Resultant of Concurrent Forces — Resolution of Forces — Work — Classes of Equilibrium.

CHAPTER IV

Concurrent Co-planar Forces 43

Resultant — Calculation of Resultants and Components — Equi- librium— Equilibrium by Resolutions — Trigonometric Solution — Number of Unknowns — Moment of a Force — Moment of Resultant — Equilibrium by Moments — Conditions for Independent Equa- tions— Connected Bodies — Bow's Method — Summary — Miscellane- ous Problems.

CHAPTER V

Non-concurrent Co-planar Forces 72

Resultant of Parallel Forces in the Same Direction — Resultant of Parallel Forces in Opposite Directions — Equilibrium of Parallel Co-planar Forces — Condition of Stable Equilibrium — Resultant of

viii CONTENTS

Page

Non-parallel Forces Graphically — Calculation of the Resultant of Non-parallel Forces — Equilibrium of Non-concurrent Forces — Condition for Independent Equations — Direction Condition of Equilibrium — Trusses — Method of Sections — Jointed Frame with Non-concurrent Forces — Summary — Miscellaneous Problems.

CHAPTER VI

Couples Ill

Moment of a Couple — Equivalent Couples — Algebraic Addition of Couples — Equilibrium of Couples — Reduction of a Force and a Couple to a Single Force — Resolution of a Force into a Force and a Couple — Summary.

CHAPTER VII

Graphics of Non-current Forces 120

Resultant of Parallel Forces — Resultant of Non-parallel Forces — Parallel Reactions — Non-parallel Reactions.

CHAPTER VIII

Flexible Cords 129

The Catenary — Deflection in Terms of the Length — Deflection in Terms of the Span — Solution by Infinite Series — Cable Uni- formly Loaded per Unit of Horizontal Distance — Summary — Miscellaneous Problems.

CHAPTER IX

Concurrent Non-coplanar Forces 141

Resultant and Components — Resolution of Non-coplanar Forces — Calculation of Resultant — Equilibrium of Concurrent, Non- coplanar Forces — Moment About an Axis — Equilibrium by Moments — Summary.

CHAPTER X

Parallel Forces and Center of Gravity 149

Resultant of Parallel Forces — Equilibrium of Parallel Forces — Center of Gravity — Center of Gravity Geometrically — Center of Gravity of a Triangular Plate — Center of Gravity of a Pyramid — Center of Gravity of Integration — Combination Methods of Calcu- lation— Center of Gravity of Some Plane Areas — Liquid Pressure — Center of Pressure — Summary.

CHAPTER XI

Forces in Any Position and Direction 173

Couples in Parallel Planes — A Couple as a Vector — Resultant Couple — Forces reduced to a Force and Couple — Equilibrium of Non-concurrent, Non-coplanar Forces — Summary.

CONTENTS ix

Page CHAPTER XII

Friction 189

Coefficient of Friction — Angle of Friction — Cone of Friction — Bearing Friction — Rolling Friction — Roller Bearings — Belt Friction — Summary — Miscellaneous Problems.

CHAPTER XIII

Work and Machines 205

Work and Energy — Equilibrium by Work — Machines — Inclined Plane — Wheel and Axle — Pulleys — The Screw — Differential Applicances — The Lever — Virtual Work — Character of Equi- librium— The Beam Balance — Surface of Equilibrium — Summary.

CHAPTER XIV

Moment of Inertia of Solids 228

Definition — Moment of Inertia by Integration — Radius of Gyration — Transfer of Axis — Moment of Inertia of a Thin Plate — Plate Elements — Moment of Inertia of Connected Bodies — Summary — Miscellaneous Problems.

CHAPTER XV

Moment of Inertia of a Plane Area 241

Definition — Polar Moment of Inertia — Axis in Plane — Relation of Moments of Inertia — Product of Inertia — Transfer of Axes for Product of Inertia — Change of Direction of Axis for Moment of Inertia — Change of Direction of Axes for Product of Inertia — Maximum Moment of Inertia — Center of Pressure — Moment of Inertia by Moment of a Mass — Summary.

CHAPTER XVI

Motion 259

Displacement — Velocity — Average and Actual Velocity — Velocity as a Derivative — Acceleration — Acceleration as a Derivative — Acceleration as a Vector — Dimensional Equations — Summary.

CHAPTER XVII

Force and Motion 273

Force and Acceleration — Constant Force — Integration Methods — Connected Bodies — Velocity and Displacement — Energy — — Potential Energy — Motion Due to Gravity — Projectiles — Summary — Miscellaneous Problems.

CHAPTER XVIII

System of Units 294

Gravitational System — Absolute Systems — The Centimeter — Gram — Second System — The Foot — Pound — Second System — The Engineer's Unit of Mass — Summary.

X CONTENTS

Page CHAPTER XIX

Force Which Varies AS THE Displacement 301

Force of a Spring — Potential Energy of a Spring — Velocity Pro- duced by Elastic Force — Vibration from Elastic Force — Sudden Loads — Composition of Simple Harmonic Motions — Correction for the Mass of the Spring — Determination of g — Determination of an Absolute Unit of Force — Positive Force which Varies as the Displacement — Summary.

CHAPTER XX

Central Forces 327

Definition — Work of a Central Force — Force Inversely as the Square of the Distance — Equipotential Surfaces — Summary.

CHAPTER XXI

Angular Displacement and Velocity 334

Angular Displacement — Angular Velocity — Kinetic Energy of Rotation — Rotation and Translation — Translation and Rotation Reduced to Rotation — Summary.

CHAPTER XXII

Acceleration Toward the Center 342

Components of Acceleration — Acceleration toward the Center — Centrifugal Force — Static Balance — Running Balance — Ball Governor — Loaded Governor — Fly-wheel Stresses — Summary.

CHAPTER XXIII

Angular Acceleration 354

Angular Acceleration — Displacement with Constant Acceleration — Acceleration and Torque — Equivalent Mass — Reactions of Supports — Reaction by Experiment — Reactions of a System — Summary.

CHAPTER XXIV

Angular Vibration 368

Work of Torque — Angular Velocity with Variable Torque — Time of Vibration — The Gravity Pendulum — Simple Pendulum — Axis of Oscillation — Exchange of Axes — Summary.

CHAPTER XXV

Momentum and Impulse 379

Momentum — Impulse — Action and Reaction — Collision of Inelastic Bodies — Collision of Elastic Bodies — Moment of Momentum — Center of Percussion — Summary.

CONTENTS xi

Page

CHAPTER XXVI

Energy Transfer 395

Units of Energy — Power — Mechanical Equivalent of Heat — Power of a Jet of Water — Work of an Engine — Friction Brake — Cradle Dynamometer — Transmission Dynamometer — Power Transmission by Impact — Summary.

Index 413

NOTATION

Symbols frequently used in this book are:

a = linear acceleration; apparent moment arm; length of balance beam; radius of circle; distance on figure.

a = a vector of length a. A = Area; a force (in few cases).

b = breadth; base of triangle; a distance.

b = a vector of length b.

J5 = a force (in few cases).

c = a distance; a constant of the catenary; distance of center of gravity of balance beam below central knife-edge.

c = a vector of length c.

C = integration constant; a force (in few cases).

d = diameter; a distance; pitch of screw; distance of central knife-edge above end knife-edges; distance between parallel axes.

d = a vector of length d.

e = base of natural logarithms; a distance. E = electromotive force; modulus of elasticity.

/ = coefficient of friction.

F = force; total force of friction. Fw = weight.

g = acceleration of gravity; a constant, 32.174.

h = height; height of triangle. hp = horsepower. H = product of inertia.

Ho = product of inertia for axes through center of gravity. H, H, Hx, Hz = horizontal force; horizontal vector.

/ = moment of inertia; electric current.

/o = moment of inertia for axis through the center of gravity. Ix = moment of inertia with respect to the X axis. ly = moment of inertia with respect to the Y axis. I max, Imin = maximum and minimum moments of inertia.

J' = product of inertia.

k = radius of gyration; a constant.

fco = radius of gyration for axis through the center of gravity. K = force which deforms a spring 1 foot; a constant; integration constant; coefficient of discharge.

I = length; length of simple pendulum, m = mass in pounds, grams, or kilograms. m.e.p. = mean effective pressure. M = moment.

Xiii

xiv NOTATION

N = force normal to surface, p = a small weight. P, P = a force; force which causes an acceleration.

Q = quantity of liquid. Q, Q = a force.

r = radius; amplitude of vibration. r = radius to center of gravity. R = resistance in ohms. R, i2 = resultant force; reaction of support.

s = length; length of catenary from lowest point. S, S = equilibrant; a force; unit stress. t = time; time of vibration; thickness. tc = time of a complete period. T = torque. Tf T = tension; tension in a cord. U = kinetic energy; work. V = linear velocity. V = volume. V, F = a vertical force.

w = weight per unit length. w' = weight per unit of horizontal distance. W = weight. X, y, z = distances.

X, y, z = coordinates of center of gravity. X, Y, Z = coordinate axes.

yc = distance to center of pressure.

a = angular acceleration; any angle; angle with X axis; angle

of contact with belt. 13 = angle with Y axis; any angle. 7 = angle with z axis; any angle. dx, dy, dz = small increments of x, y, and z. p = density.

4) = an angle; angle of friction. 6 — &n angle; angular displacement. S = a summation, w = angular velocity.

MECHANICS

CHAPTER I

FUNDAMENTAL IDEAS

1. Mechanics. — Mechanics is the science which treats of the effect of forces upon the form or motion of bodies. The science of mechanics is divided into statics and dynamics. When the forces which act on a body are so balanced as to cause no change in its motion, the problem of finding the relations of these forces falls under the division of statics. When the forces which act on a body cause some change in its motion, the problem of finding the relation of the forces to the mass of the body and to the change of its motion falls under the division of dy- namics. The division of dynamics is frequently called kinetics.

2. Illustrations. — Fig. 1 is an example of a problem of statics. The figure shows a 10-pound mass which is supported by two spring balances. In Fig. 1, I, the balances are nearly vertical. One balance reads a little less than 5 pounds and the other balance reads over 6 pounds. In Fig. 1, II, each balance makes a large angle with the vertical. The right balance, which is nearly horizontal, reads 13 pounds and the left balance reads 15 pounds. In this position, the reading of each balance is greater than the entire weight which is supported by the two balances. In each position, the balance which is the more nearly

1

MECHANICS

[Art. 3

vertical gives the larger reading. The problem of finding the relation between the pulls which these balances exert and the angles which they make with the vertical is a problem of statics.

In Fig. 2, the mass of 10 pounds is supported by one spring balance and by a cord which runs over a pulley and carries a mass of 8 pounds on its free end. This system will come to rest in a definite position. If moved from this position, the system will return to it after a few vibrations. Thfe problem of finding this position and the tension in the spring balance is a problem of statics.

If the cord which runs over the pulley of Fig. 2 is cut or broken, the 10-pound mass will swing back and forth as a pendulum and will finally come to rest with the balance in a vertical position.

-&

Fig. 2.

Fio. 3.

This final position is shown in Fig. 3. The 8-pound mass will fall vertically downward with increasing speed. As the 10- pound mass swings back and forth, the pull on the balance will change. The problem of finding the position of either body at any time after the cord has been severed, and the problem of finding the pull of the spring balance are problems of dynamics.

3. Fundamental Quantities. — The problem of Fig. 2 involves position and direction. These are properties of space. The problem also involves the pull of the cord and of the spring balance. These are forces. Every problem of statics involves the two fundamental ideas of space and force. If a body of different mass were used in place of the 10 pounds, the force and space relations would be changed. In this problem, the forces depend upon the masses. All problems of statics involve force and space. Most problems of statics involve also mass.

A problem of dynamics differs from one of statics in the fact that it involves the element of time.

Chap. I] FUNDAMENTAL IDEAS 3

The four fundamental quantities of mechanics are space, mass, force, and time. A problem of dynamics includes all four of these quantities. A problem of statics may include all except time.

Space, mass (or matter), force, and time are elementary. None of them may be reduced to anything more simple. Con- sequently, it is useless to attempt to define them. On the other hand, everyone has a clear knowledge of these quantities. This knowledge has been gained through one or more of his senses as a part of the experience of his lifetime.

4. Standards and Units. — While space, mass, time, and force can not be defined, they may all be measured. To measure any quantity, it is necessary to have a unit of measure. If measurements are to be taken at diverse times and places, it is necessary to have some standard unit to which all other units are referred. These units of measure were originally arbitrarily chosen. Other units might have been selected just as well. When a particular unit has once been adopted, however, it is important that its value be preserved without change, in order that physical measurements separated by wide intervals of time may be accurately compared.

6. Length. — Space in one direction is length. There are two official standards of length preserved by the Bureau of Standards at Washington. These are the Standard Yard, which is practi- cally equal to the British Imperial Standard Yard, and the Stand- ard Meter, which is a copy of the International Standard.

The length of the International Meter in terms of the wave length of cadmium vapor light has been carefully determined by Michelson. If this standard bar and the copies preserved by various nations should undergo any change, the magnitude of the variation may be found by a new comparison with the wave length of this light.

The foot is the unit of length which is commonly used by American engineers. A foot is one-third the length of the Standard Yard. Physicists use the centimeter as the unit. In countries where the metric system has been adopted, engineers employ the meter as the unit of length.

Length measurements are generally made by means of the sense of sight. Sometimes the sense of touch, the sense of hear- ing, or the muscular sense is used in comparing two lengths. The vision, however, is employed to get the actual reading.

4 MECHANICS [Art. 6

Space in one dimension is length; space in two dimensions is area; space in three dimensions is volume.

The idea of space, including length and direction, is gained by the child through the sense of touch, the muscular sense, and the sense of sight. The sense of hearing also assists in deter- mining direction.

The relation between the metric system and the inch is given with sufficient accuracy for most purposes by

39.370 inches = 1 meter. 2.540 centimeters = 1 inch.

Problems

1. Calculate the length of a foot in centimeters and memorize the result.

2. Calculate the length of a meter in feet and memorize the result.

3. Find the length of a kilometer in feet and in miles and compare the results with some reference book.

4. Express 100 meters in yards and 440 yards in meters.

5. By logarithms find the number of square inches and square feet in one square meter.

6. A hectare is 100 meters square. Find the value of a hectare in acres.

7. Using five-place logarithms, find the number of cubic centimeters in one cubic inch. Compare with some handbook.

8. A liter is a cubic decimeter. Find the relation between the liter and the U. S. liquid quart.

6. Time. — The standard of time measurement is the mean solar day. The unit commonly employed in problems cf mechanics is the mean solar second. The subdivision of the solar day into hours, minutes, and seconds is made by means of the vibration of pendulums or other mechanical devices. In making time meas- urements, the senses of sight and hearing are used in connection with these mechanical timepieces.

The child gains his ideas of time from the succession of events as revealed through any or all of his senses.

7. Matter and Mass. — From the mechanical standpoint, at least, time and space are simple and easily measured. Time possesses a single property, that of extent; space has extension or length in three dimensions. Matter, on the other hand, pos- sesses many properties, some one of which must be selected and defined as measuring the amount of material in a given body. Volume might be chosen as the measure of the quantity of mate- rial. It is found, however, that the amount of a given material in a given volume may be greatly changed by pressure. It is

3 C

1 [

Chap. I] FUNDAMENTAL IDEAS 5

also found that equal volumes of different materials differ greatly in their mechanical effects. It is evident, then, that some other property must be selected to designate the amount of matter.

In Fig. 4, A represents a block of soft rubber resting on some convenient support. In Fig. 4, II, a body B has been placed on this block of rubber. The length of the block is found to have been shortened. If B is removed, the block A returns to its original length. If a body C is now placed on A , there is again a change in length. If the change in length of A due to the body C is the same as that due to the body B, the bodies B and C are said to con- tain equal amounts of material. The amount of material (or matter) in a body, as thus defined is called the mass r of the body. Two bodies have equal masses if they produce equal deforma- tions in a third body when they are applied to it in exactly the same way. The ordinary spring balance is a common form of a third body for the comparison of masses.

Instead of being supported on an elastic body, the bodies B and C may be carried on the hand or shoulder of the observer. The deformation of his muscles is accompanied by a sensation, called the muscular sense, which enables him to judge roughly which body has the greater mass.

A second method of measurement of mass is by means of inertia. This involves the conditions of change of motion and will be considered in Chapter XVII.

The child gains an idea of mass in the mechanical way by means of the muscular sense and the sense of touch as experienced when he supports bodies free from the earth, stops them when moving, or otherwise changes their motion. The concept of matter in general is gained through all the senses.

The pound is the common unit of mass. In the metric system the unit is the kilogram. Physicists and chemists use chiefly the gram. Units of mass are generally called "weights." A so-called 10-pound weight as used on a beam balance is a 10- pound mass.

To convert from the metric system to the avoirdupois system, the relations are,

15432 grains =» 1 gram, 453.6 grams = 1 pound.

6 MECHANICS [Art. 8

The official Standard Pound and Kilogram are preserved by the Bureau of Standards.

Problems

1. Find the number of pounds in 1 kg. correct to four significant figures.

2. The mass of one cubic centimeter of water at 4°C. is practically one gram. Find the mass of one cubic foot of water in pounds.

8. Force. — In Fig. 4, the rubber block A is shortened when the body B is placed on it. The body B is said to exert a force on A at their surface of contact. The block A is also said to exert a force on B in the opposite direction. Force is some- thing which may exist between two bodies or between two parts of the same body. Forces occur in pairs. There is a force from the first body to the second body, and an equal and opposite force from the second body to the first body. Newton stated this fact in what is called the Third Law of Motion: "Action and reaction are equal and opposed to each other."

A force always causes some change in the dimensions of a body. A force always tends to produce some change in the motion of the body upon which it acts, and does cause some change unless it is balanced by an equal and opposite force acting on the body, or by a number of forces equivalent to an equal and oppo- site force.

Force is recognized and measured by means of the change in the dimensions and form of elastic bodies, by the muscular sense, and by the change in the motion of bodies of known mass.

9. Weight. — When the masses of two bodies are compared by means of the muscular sensation experienced in lifting them, the observer really gets a comparison of two forces. The comparison of mass is indirect. If the observer were at the center of the Earth, there would be no muscular sensation so long as there were no change in the motion of the body. The Earth exerts a force on all bodies. This force is in the form of a pull directed toward the center of the Earth. This pull or attraction is called the weight of the body. When a body is supported and thus pre- vented from moving toward the earth, the support exerts a force upward which is equal to the weight of the body.

When a physicist speaks of the weight of a body, he always means the force with which the Earth attracts the body. In the com- mon use of the word, weight generally means mass. When one says that the weight of a bar of iron is 16 pounds, he is usually

Chap. I] FUNDAMENTAL IDEAS 7

thinking of the amount of iron and not of the force required to lift it. Much confusion has resulted from the failure to designate clearly which of these two meanings is intended.

10. Relation of Mass to Weight. — The definition of mass in Art. 7 may now be extended. Two bodies have equal masses if, at a given point, they are attracted toward the Earth with equal force. The determination of mass by means of a spring balance or a beam balance is accomplished indirectly by a comparison of forces, with the tacit assumption that equal forces produce equal effects. The first definition of mass is: The mass of a body is pro- portional to its weight. If F^ is the weight of the body in some convenient unit, and m is its mass, the definition may be ex- pressed algebraically by the equation,

F^ = km, (1)

in which kisa constant. The numerical value of k depends upon the units used in expressing Fu, and m. These units may be so chosen that k is unity. If m is expressed in pounds of mass and Fu> is in pounds of force, then k = 1, and

F^ = m. (2)

Equation (2) states that the mass of a body in pounds is equal to its weight in pounds.^ The word pound has two meanings in mechanics. It may be used to designate the amount of material (mass) or to express the force of attraction toward the Earth (weight as meant by the physicist). In a similar way, the weight of one kilogram of matter is one kilogram, and the weight of one gram of matter is one gram. With the systems of units in everyday use, k is unity. In some systems, k is not unity. In the absolute system of units k = g, and Fy, = mg. The weight of a mass of m grams in that system is mg dynes.

The absolute systems of units are not used by engineers in the solution of problems of statics. In all such problems, Equation (2) applies. The weight of a body is numerically equal to its mass. The absolute systems of units and a second

^ A formula is merely a brief statement of the relation of quantities. TKe letters of a formula represent the number of units which express the magnitude of the quantity. In the above equations, m represents the num- ber of pounds, the number of kilograms, the number of tons, or the number of grams of material in the body under consideration. Similarly F„ repre- sents the number of units of force in pounds, kilograms, tons, grams, poundals, or dynes.

8 MECHANICS [Art. U

definition of mass will be considered in this book in Chapters XVII and XVIII.

11. Variation of Weight with Latitude and Altitude. — The

weight of a body at any given position varies as its mass. It has been shown by experiment that the weight of a body varies inversely as the square of its distance from the center of the Earth. If the Earth were a sphere and did not rotate on its axis, the weight of a body would be the same at all points at the same level on its surface. Since the Earth is a sphoroid with its polar radius about 13 miles shorter than its equatorial radius, the weight of a body increases with latitude. While a pound mass is an invariable quantity, the weight of a pound mass, as measured by a spring balance, varies with the latitude. If two masses have equal weights at one locality, their weights will be equal at any other locality. Masses may be compared by weigh- ing at any point. A spring balance, however, which has been cahbrated by means of a standard weight at one locality, can not be used for the accurate determination of mass at another locality. (No one would do so on account of the variation of the spring, even if the force of gravity were constant.)

Since the practical units of force are determined from the weights of the standard units of mass, it is necessary to choose some standard location for the definition of these units of force. The sea level at 45° latitude is taken as this standard location.

A pound force is defined as the weight of a pound mass at the standard location. A pound mass will weigh 0.997 lb. at the Equator and 1.003 lb. at the Poles on a spring balance which is correct at the standard latitude. This difference, while impor- tant in the determination of physical constants, is usually neg- lected in engineering calculations.

Problems

1. Taking the equatorial diameter of the Earth as 8000 miles and the polar diameter as 26 miles less, and neglecting the effect of the rotation of the earth, what is the weight at the Pole of a body which weighs 1 pound at the Equator, if both weighings are made on the same spring balance?

2. What is the relative change in the weight of a body when it is taken from a point at the sea level to a point one mile higher?

CHAPTER II QUANTITY AND CALCULATIONS

12. Representation of Quantity by Numbers. — There are several ways of representing the magnitude of a quantity. The most common method is by means of numbers, as 6 feet, 8 pounds, 10 seconds. A number expresses the magnitude of the quantity in terms of the unit and means Httle to one who does not possess a definite idea of the magnitude of the unit. Two such numbers give a clear notion of the relative size of quantities without conveying any information as to the actual size of either. Any- one will know that 20 dekameters is twice 10 dekameters, without having any idea as to the size or nature of a dekameter. If he has learned that a dekameter is 10 meters and that a meter is 3.28 feet, he will calculate that one dekameter is approximately 2 rods and that 20 dekameters is nearly 40 rods, or he may reduce to yards and think of 10 dekameters as a little over 100 yards. A Frenchman, on the other hand, who thinks in the metric sys- tem, must translate rods and yards into meters before he can have a real idea of their meaning.

13. Representation of Quantity by Lines. — The relative magnitudes of several quantities are frequently represented to the eye by means of straight lines as in Fig. 5. Economic data, such as the population and area of countries and cities, the production and con- sumption of commodities, etc., are commonly shown in this way. These lines may be hori- zontal, as in Fig. 5, or vertical with their lower ends on the same horizontal line.

For most purposes such lines are merely used to express magnitude to the eye. The necessary calculations are made by means of numbers. The operation of «;ddition, however, may be performed conveniently with lines. In Fig. 6, it is desired to find the sum of the quantities rep- resented by the lines ab and cd. The lines are placed together so as to form one continuous line without overlapping. The total

9

Fig. 5.

10 MECHANICS [Art. 14

line thus formed is the sum of the hnes. As may be seen from Fig. 6, it is immaterial in what order the hnes are placed together. For subtraction, especially when the remainder is negative, it is desirable to adopt some convention as to positive and nega- tive direction. Horizontal lines extending toward the right and vertical lines extending upward are regarded as positive. In Fig. 6, the line ah runs from a to h. The left end, a, is called the origin, and the right end, 6, is called the terminus. To find ah + cd, the origin of the second line is placed at the terminus of

3 2 I O

0 1 234-567 89 lO

7-3 Fig. 8.

d

r-a S '

I ab+cd

I a b

,d + ab h-ab-cd-«J«— ^cd — J

cd + ab Fig. 6. Fig. 9.

the first line. The sum of the two lines extends from the origin of the first line to the terminus of the second line. This cor- responds with ordinary addition of numbers. To get the sum of 7 -j- 3 begin at 7, which is the terminus of the first number, and count forward 3 steps. This is shown graphically by Fig. 7.

Subtraction is the addition of a negative quantity. To get 7-3, begin at 7 and count backward 3 steps. This is shown by Fig. 8. To get ah — cd, begin at the terminus of ah and measure the length of cd toward the left. This is shown in Fig. 9. The arrows in Fig. 9 give the direction of the motion.

14. Vectors. — A quantity which has both magnitude and direction is called a vector quantity. Force is an example of this kind of quantity. In Fig. 10, ah, cd, and e/are vectors in the plane of the paper. The vectors ah and ef are equal, since they have the same direction and equal length.

Chap. II] QUANTITY AND CALCULATIONS 11

In the vector ab, the point a is the origin and the point 6 is the terminus. The vector is considered as extending from the origin to the terminus, as is indicated by the arrow. The arrow points from the origin and toward the terminus. The direction of the arrow is the positive direction of the vector.

When a vector is represented by a

single letter, that letter is usually 2 >

printed in black face type. In Fig. ^j

11, a and b represent two such vectors. Si^^^""^

The arrow shows the origin, terminus, e ^ f

and direction. „

Fig. 10.

A vector is described m words by giving its direction and its length. When the plane of the vector is known, a single angle is sufficient to designate its direction. For instance, a given vector is 8 feet in length and makes an. angle of 35 degrees with the horizontal. When the vectors under consideration are not all in the same plane, two angles are required to express the direction of each vector. For instance, a vector is 8 feet in length and makes an angle of 40 degrees with the vertical in a vertical plane which is north 25 degrees east.

When it is desirable to distinguish a quantity which has magnitude but not direction from a vector, such a quantity is called a scalar quantity. The mass of a body or the number of individuals in a group is a scalar quantity. In an algebraic formula in which only the magnitude of a vector is represented by a letter while its direction is expressed in terms of angles, the

letter is printed in Italics in- stead of in black face tjrpe. In this book, a letter (such as P or Q) is used to represent a force, which is a vector. When emphasis is put on both the direction and mag- nitude of the force, the letter is printed in black face type. When only the magnitude is stressed, it is printed in Italics.

15. Addition of Vectors. — The addition of vectors is defined in the same way as the addition of fines (Art 13). The origin of the second vector is placed at the terminus of the first vector. The line which extends from the origin of the first vector to the terminus of the second vector is the sum of the two vectors.

12 MECHANICS [Abt. 16

Fig. 11, II, shows the addition of the vectors a and b to get their sum a + b. Fig, 11, III, shows b + a. The vector a is first in Fig. 11, II, and the vector b is first in Fig. 11, III. Fig. 11, IV, shows both additions in one diagram. The additions begin at a common origin 0. Since the two lines a are equal and parallel and the two lines b are also equal and parallel, the four lines form a parallelogram. The diagonal of this parallelogram is the vector sum and it is immaterial in what order the two vectors are added. The sum of three vectors is found in the same way. Fig. 12, II, repre- sents the sum of three vectors. Fig. 12, I, may be regarded as a graphical statement / y / of the vectors which are to be added. In

'aF ^y^ / ^y^ t,his figure all the vectors start from a com- ^*^ i^ mon origin. In Fig. 11, I, on the other

Piy 12. hand, the vectors a and b start from differ-

ent origins. There are two methods of finding the vector sum. These are the graphical method in which the lengths and angles are measured, and the trigonometric (or algebraic) method in which the lengths and angles are computed.

Problems

1. Given two vectors, a = 15 ft. at 0 degrees, b = 12 ft. at 40 degrees. Solve graphically for the vector sum, a + b. Use the scale of 1 inch = 5 feet. Measure the vector sum and express the result in feet. Measure the angle of the vector sum with the first vector and express the result in degrees.

First construct the statement to scale as shown in Fig. 13, I. Then draw a in Fig. 13, II, equal and parallel to a of the statement. From the terminus of a draw b equal and parallel to b of the statement.

2. Solve problem 1 for the vector sum b + a«

3. Find the sum of three vectors: 20 ft. at 0 degrees, 15 ft. at 45 degrees, and 10 ft. at 110 degrees. Use the same scale as in Problem 1.

4. Find the vector sum of 16 ft. at 10 degrees and 20 ft. at 70 degrees. Construct the 10-degree angle by means of its tangent. Construct the 60-degree angle by means of its chord. Measure the angle of the vector sum by means of its chord and check by means of the sine of the angle which the vector sum makes with a line at 90 degrees.

16. Components of a Vector. — The sum of two or more vectors is frequently called the resultant vector and the vectors which

CHAP.n] QUANTITY AND CALCULATIONS 13

are added are called components of the resultant. The process of finding the resultant is called composition of vectors. The process of finding the components is called resolution. In Fig. 13, the vector a + b is the resultant of vectors a and ^ ^ > '-"^ & b, and the vectors a and b are com- ^ ponents of a + b. ^'°- ^^•

Problems

1. A vector of 25 ft. at 30 degrees is made up of two components. One of these components makes an angle of 5 degrees with the reference line and the other makes an angle of 45 degrees with the reference line. Find these components graphically.

2. A vector of 20 ft. at 45 degrees is made up of a vector a at 20 degrees and a vector b which is 12 ft. in length. Find the magnitude of a and the direction of b.

The most important kind of resolution of vectors is that in which each com- ponent is formed by the orthographic