A brief course in elementary dynamics

In preparing this text book it has been the aim to present a clear, consecutive, and consistent development of those laws and principles of dynamics which are of most frequent occurrence in the ordinary affairs of life and of widest application in the arts. In carrying out this purpose the following points have been kept in view: (1), all technical terms should be accurately and succinctly defined; (2), the number of propositions derived from experience and experiment that are to be taken as fundamental should be as few as possible; (3), the laws of the subject should be deduced from the definitions and fundamental principles by rigid methods whenever the advancement of the student will justify it — when tnis is not feasible the weakness of the treatment should be indicated; (4), the laws should be enunciated in such terse and inclusive statements that they can be easily assimilated and readily applied; (5), the laws should be vivified in the mind of the student by numerous illustrations drawn from applications with which he is familiar and in which he is interested; (6), since mistakes are due not only to failure to comprehend laws but also to faulty reasoning from them, the student should be trained in solving numerical problems. For engineers, especially, is facility in solving numerical physical problems of enormous importance.

It will be noticed that the body of the text is in large type, while problems, examples, and all illustrative material has been put into smaller type. This arrangement assists in obtaining a proper perspective of the various parts, and in a review permits the "story" to be followed without the attention being distracted by unessentials. A few articles containing long proofs, which might with propriety be omitted in the first reading, have been marked with an asterisk. In all cases, however, the conclusion printed in italics should be learned.

The problems at the ends of various sections have been selected out of a card index which has been collected from various sources through a number of years. Few of them are original, but the sources from which they were obtained have been lost.

I.    FUNDAMENTAL NOTIONS.
Stress and Force. Reaction. Representation of Forces.  Illustrations of Force and Reaction. The Effects of Force. Equilibrium defined. Measure of a Force. The Gravitational Units of Force. Inertia, and Newton's First Law of Motion. Newton's Second Law of Motion. The Principle of the Independence of Forces. The Principle of the Transmissibility of Forces. The Foot. The Centimeter. Work defined. Energy defined. The Principle of the conservation of Energy. Measure of Work. The Work Principle. Dynamics defined. The Basis of Dynamics


STATICS, OR THE LAWS OF EQUILIBRIUM

II.    COMPOSITION AND RESOLUTION OF FORCES.
Translation and Rotation defined. Moments of Forces. Illustrations of Moments of Forces. Definitions. The Parallelogram of Forces. Computation of the Resultant of two Concurrent Forces. Resultant of Any Number of Concurrent Forces. The Triangle of Forces. The Polygon of Forces. Definitions. Rectangular Components of a Force. The Component, in any direction, of the Resultant of a System of Concurrent Forces. Illustration of Forces in the sailing of a boat. Resultant of Two Parallel Forces. Resultant of a System of Parallel Forces. The Force Couple defined. Conditions of Equilibrium. Stability of Equilibrium 15

III.    CENTROIDS.
The Centroid defined. Location of the Centroid of a System of Parallel Forces. The Center of Gravity defined. Location of the point of Application of the Weight of a Body in a few simple cases

IV.    FRICTION BETWEEN SOLIDS.
Reaction at a Smooth Surface. Coefficient of Static Friction. Limiting Angle of Friction. Coefficient of Kinetic Friction. Rolling Friction. Definitions. Mechanical Advantage of the Wedge. Mechanical Advantage of the Screw


KINETICS, OR THE LAWS CONNECTING FORCE AND MOTION.

V.    THE MOTION OF A BODY UNDER THE ACTION OF ZERO FORCE.
Linear Velocity and Speed defined. The Unit of linear speed. The Composition of Uniform Linear Velocities. Computation of the Resultant of two Uniform Linear Velocities. Resolution of Uniform Linear Velocities. Measurement of Angles. Angular Velocity. Representation of Angular Velocity. Composition of Angular Velocities. The Relation between Angular and Linear Speed. Instantaneous Axis of Rotation

VI.    THE MOTION OF A BODY UNDER THE ACTION OF A CONSTANT FORCE.
Linear Acceleration. Acceleration due to Gravity. Acceleration produced by a Uniform Force acting in the direction of Motion. Acceleration produced by a Uniform Force acting perpendicularly to the Direction of Motion. Mass and Inertia. Weight proportional to Mass. Change of Apparent Weight due to Acceleration. D'Alembert's Principle. Density and Specific Gravity. Linear Momentum. Conservation of Linear Momentum. Forces in Circular Motion. Fundamental and Derived Units. The Centimeter-Gram-Second Absolute System of Units. The Foot-Pound- Second Gravitational System of Units

VII.    CENTER OF MASS OR CENTER OF,INERTIA.
The Center of Mass defined. Location of the Center of Mass. The Centroid coincident with the Center of Mass. Motion of the Center of Mass

VIII.    THE MOTION OF A BODY UNDER THE ACTION OF A CONSTANT TORQUE.
Angular Acceleration. The Relation between Angular and Linear Acceleration. Angular Acceleration produced by a Uniform Torque. Inertia and Moment of Inertia compared. Illustration of Moment of Inertia. The Falling Cat

IX.    ENERGY.
Definitions. Work done, and Power developed, by Forces and Torques. Kinetic Energy. Potential Energy. The Potential Energy of a system tends to become a Minimum. The Sum of the Kinetic and Potential Energies is Constant. Ill

X. MOTION UNDER THE ACTION OF A VARIABLE FORCE.
Simple Harmonic Motion of Translation defined. Relation between Uniform Circular Motion and Simple Harmonic Motion. Velocity and Acceleration of a Particle moving with Simple Harmonic Motion of Translation. Displacement of a Particle moving with Simple Harmonic Motion of Translation. Simple Harmonic Motion of Rotation defined. Angular Velocity and Angular Acceleration in Simple Harmonic Motion of Rotation. Displacement of a Point moving with Simple Harmonic Motion of Rotation. Period of a Simple Pendulum. Sympathetic Vibration. Waves. Transverse Wave Motion. Longitudinal Wave Motion. Speed of a Wave Motion. Flow of Energy. Phase. Interference of Wave Motions. Standing or "Stationary" Waves. Polarization

XI.    STATICS OF FLUIDS.
Definitions. General Properties of Fluids. Fluid Pressure due to Weight. Atmospheric Pressure. The Barometer. The Open Manometer. The Common Pump. Upward Pressure in Fluids. Archimedes' Principle. Determination of Density by Immersion

XII.    KINETICS OF FLUIDS.
The Fundamental Equation. The Flow of Liquids. The Speed of Efflux of a Liquid. Lateral Diminution of Pressure. The Siphon. The Speed of Efflux of a Gas. The Steam Injector

XIII.    PROPERTIES OF MATTER.
General Considerations. Definitions. Coefficient of Elasticity. Young's Modulus. The Bulk Modulus. Simple Rigidity. Vibration of Elastic Bodies. Viscosity. Newton's Law of Gravitation. The Gravitation Constant. Solution Pressure. Diffusion. Osmotic Pressure. Cohesion. Surface Tension. Capillarity. Boyle's Law. The Closed Manometer. Dalton's Law

MISCELLANEOUS TEST QUESTIONS IN DYNAMICS


CHAPTER IX - Energy

Definitions. - The accomplishment of motion against a resisting force is called work. It has been shown (Art. 19) that work is measured by the product of the force acting on the body moved and the resolved part of the displacement of the body in the line of action of the force. The rate of doing work, that is, the amount of work performed per second is called power. Energy has been defined (Art. 17) as stored work, or as the ability to do work. The quantity of energy possessed by a system of bodies, is the amount of work it can do against external forces in passing from its present condition to some standard condition.

The energy of a system may be due to the motion of part of it with reference to other parts. This type is called kinetic energy. Or the energy of the system may be due to stresses between different parts of the system. This type is called potential energy.

In certain cases work may be done by a body, not on ac- count of the motion or the strained condition of the body, but on account of the heat energy stored in the body. For example, a body can do work either by expanding and thereby overcoming pressure, or by expanding and thereby liberating heat, which in turn is converted into work. In the former case the work is due to a diminution of the potential energy of the strained body, while in the latter case, the work is due to a diminution of the internal energy of the body. Again, the heat absorbed when water is converted into steam is stored up as internal energy until the steam resumes the liquid form. Steam equals water plus a quantity of internal energy. According to the kinetic theory of matter, which will be considered when the subject of Heat is taken up, the internal energy of a body is a form of potential energy due to the arrangement of the ultimate parts of which a substance is considered to be composed.

Work and energy are measured in the same units. In the C. G. S. absolute system of units, the unit of work is the amount of work done when one dyne of force moves the body to which it is applied, in the direction of the force, through a distance of one centimeter. This unit is called the dyne centimeter or erg. 107 ergs is called a joule. In this system, the unit of power is the erg per second. 107 ergs per second is called a watt.

In the F. P. S. gravitational system, the unit of work is the amount of work done when a force of one pound weight moves a body in the direction of the force through a distance of one foot. This unit is called the foot pound. In this system the unit of power is the foot pound per second. 550 foot pounds per second is called a horse power.

Kinetic Energy. - Imagine that in t seconds a body of mass m is brought from rest up to a speed s by the application of a uniform force F. If, during this time the body moved through a distance x with a uniform acceleration a, the work done upon the body is

W [= Fx] = max.

Due to its motion, the body now possesses an amount of energy which can be determined by measuring the amount of work it can perform. Suppose that in overcoming a constant force F’, the body is brought to rest in t seconds; and that during this time the body traversed a distance x’ with a uniform acceleration – a’.

Kinetic energy of a body is equal to the work done upon it in bringing it from rest to its present speed y and that this equals one half the product of the mass of the body and the square of its linear speed.

Potential Energy. - Consider the particular case where the potential energy of a body is due to its position c above the surface of the earth. Imagine a body of mass m to start from C and fall to a point A. Let the initial position of the body be at a height h above the horizontal plane passing through its final position.

The potential energy of a body is measured by the work it can do in going from its present position to some standard position. In the above example, the difference between the potential energy of the body when at C and when at A equals mgh. That is, if A is the standard position from which potential energy is reckoned, the potential energy of the body when at C is mgh. Taking the earth's surface as the standard for reference, and neglecting the variation of the acceleration due to gravity at different positions on the earth's surface, the potential energy of a body due to gravitation is the- same at all points at equal heights above the surface of the earth.

In strictness, it is not correct to speak of the potential energy of a body because the energy really belongs to whatever agent is under stress. But in order to avoid suph circumlocutions as "the potential energy of the strained medium between the parts of the given system of bodies" custom has sanctioned such expressions as "the potential energy of the given body with respect to the earth." When the context shows what is taken to be the standard position from which potential energy is reckoned, it is common to speak simply of the "potential energy of the body."

The Potential Energy of a System tends to become a Minimum. - When a body moves in opposition to the resultant force upon it, positive work is done against the force, i. c, the potential energy of the system of which the body forms a part is increased. Conversely, when a body moves in the same direction as the resultant force acting upon it, the potential energy of the system is diminished. Now when a motion takes place between the parts of a system due to the action of a force between them, the direction of the displacement is in the direction of the force. Consequently, if under the action of a force between two parts of a system of bodies a motion is produced, the direction of the displacement will be such as to produce a diminution in the potential energy of the system.

If work is required to separate two bodies, the bodies will, if left to themselves, tend to approach one another. The direction of the line of approach may not be coincident with the path along which they were originally displaced: it will ailways be in that direction along which the change in the potential energy of the system, per unit distance traversed, is the greatest.

The Sum of the Kinetic and Potential Energies is Constant. - That greatest of all the generalizations of physics - the principle of the conservation of energy - has been enunciated by Maxwell in the following form: - "The total energy of any material system can neither be increased nor diminished by any action between the parts of the system, though it may be transferred into any of the forms of which energy is susceptible. Since energy can be only kinetic and potential, it follows as a direct deduction from the above principle, that, so long as no external forces act upon a system, the sum of its kinetic and potential energies must be a constant quantity. A simple example is a body falling freely under the influence of gravitation.