Theoretical mechanics - An elementary treatise

PREFACE.

In preparing the present work, which was designed to include in a single volume of moderate compass the elementary portions of Theoretical Mechanics, no formal division of the subject into Kinematics, Statics and Kinetics has been made. The topics often included under the first head it was thought best to introduce separately, each at the point where it is required for immediate application to the treatment of the motions produced by forces. For example, the expressions for radial and transverse accelerations are not introduced until required in the discussion of Central Forces.

The subject of Statics is, to be sure, to a large extent separable from the idea of motion. But, on the one hand, as has been recognized in all recent treatises, the fundamental notions of force are best presented, and the Parallelogram of Forces is best established, on the basis of the Laws of Motion. This requires what may be called a dynamical introduction to Statics. On the other hand, the subject cannot be completed without the Method of Virtual Velocities, an application of the Principle of Work. This principle, which is dynamical as involving forces acting through spaces, advantageously precedes the study of kinetics into which time enters explicitly, and prepares the student for the notion of Kinetic Energy, or work embodied in motion.

Accordingly, in the present volume, Chapter I consists of such a kinetical introduction to the whole subject as is referred to above; Chapters II~VI are purely statical; Chapter VII treats of the dynamical Principle of Work with its application to Statics and to the notion of the Potential Function ; and the remaining chapters treat of purely kinetical topics.

The chapters are further subdivided into sections followed by copious lists of graded examples aggregating over 500 in number; many of these were taken from examination papers set at the Naval Academy, and not a few were prepared expressly for this work.

In these examples, as well as in the numerical illustrations introduced in the text, gravitation units of force have for the most part been employed. These units in fact not only have the advantage of being rendered familiar to us by the common usages of every-day life, but they are actually more convenient than absolute units in mechanical problems, since in them the forces arise principally from the weights of bodies. Thus their use is forced upon even those writers who most deprecate the employment of a variable unit of force. The conception of an absolute unit of force, dependent upon mass and motion and not upon weight, is indeed essential to the gaining of correct ideas of the nature of force. Hence the introduction by Prof. James Thomson of the poundal, which serves this purpose when the English system of weights and measures is used, has been of very great value. At the same time the employment of the pound as a unit of mass as well as a unit of force has been the cause of confusion, so that a student is sometimes in doubt whether the result of the use of a formula is the number of pounds or of poundals, or, as he may phrase it, whether the formula is expressed in gravitation or in absolute units. To prevent this confusion, care has been taken in the present volume, while using gravitation units, to avoid such expressions as, for example, “a mass of 6 pounds,” and to speak instead of “a body whose weight is 6 pounds.” There is no doubt that the same body would be intended in either expression, but the former would imply that, in the formula W= mg, 6 is the  numerical value of m, and the latter that 6 is the numerical value of W, Inasmuch as the pound, though an absolute “standard” of mass, is properly called and legally styled a “unit of weight,” the latter is the more natural course. Accordingly the student is directed on page 13 to follow it and to remember that all the forces are thus expressed in local pounds. If the result is desired in poundals, neither is the formula changed nor is the result found in one unit and then changed to the other, but the number of pounds is taken as the numerical value of m.

For the same reason, we should not say that the weight of a body varies when it is taken to a place where has a different value, because the number which legally expresses its weight remains the same. The force of its gravity has indeed changed, but it is (when we use gravitation units) the unit of this force, and not its numerical measure, which has changed.

In the treatment of kinetics, the conception of the forces of inertia has been freely employed, and that without the apologies that some writers have thought necessary. It would seem that the resistance of a body in motion to acceleration in any direction is as much entitled to be regarded as a force as is the resistance of other bodies which, in the case of a body at rest, prevent motion. By including the latter as forces, we obtain the idea of a system of forces in equilibrium; so also, by including the former as forces, we extend this idea to the case of a body in motion, and D'Alembert's Principle presents itself in the form of “kinetic equilibrium,” instead of requiring for its statement a set of hypothetical “effective forces.”

The study of Mechanics is here supposed to follow an adequate course in the Differential and Integral Calculus, and to form a very important application of its principles. But, when these applications occur, the results are not merely presented in the shape of general formulae in the notation of the Calculus, leaving the student unaided in the process of evaluation. Instead of this, pains has been taken to instruct the student in the methods best adapted in various cases to obtaining numerical results. Particularly in the treatment of statical moments and of moments of inertia it is hoped that the book will be found a useful supplement to the course of instruction m the processes of integration.

Throughout, the practice of relying upon substitution in general formulae is discouraged as far as possible, and the opposite practice inculcated - namely, that of applying general principles directly to the problem in hand.

Special prominence is given to those results which it is the most important to make familiar to the student of Applied Mechanics, and to the readiest ways of recalling them when they have slipped the memory. Although preference is given to analytical processes, a not inconsiderable use is made of graphical methods. These have, however, been introduced rather as diagrammatic aids to the comprehension of general principles, and to the calculation of numerical results, than as methods of obtaining results by measurement from accurately constructed diagrams — the latter belonging rather to the province of Applied Mechanics.

 

I. Mechanics is the science which treats of the motions of material bodies^ and the causes of these motions.

A force is an action, applied to a material body or to any part of it, which when unresisted produces motion. A solid body is one which resists relative motion between its parts, so that it does not readily change its shape. When the forces under consideration can produce no change of shape, the body is said to be rigid and it moves only as a whole. If the motion of a rigid body is such that every straight line drawn in its substance remains always parallel to its original position, the motion is said to be one of translation. When this is the case, it is obvious that the motion of a single point of the body (whether it be in a straight or in a curved line) is sufficient to determine completely the motion of the body.

The whole amount of matter contained in a body is often imagined to be concentrated at a single point. In this case it is called a material particle. The motion of a body in translation is completely represented by the motion of a particle.

2. We discuss in this book only the motions of rigid bodies, and at first consider motions of translation, so that the body may be regarded as a particle, and the forces as acting at a single point. In this first chapter, we consider the general relations between forces and the motions they produce, from which is derived the mode in which they are measured and subjected to mathematical analysis.


Velocity or Speed.

3. When a body is in motion, we have to consider both the speed and the direction of the motion. The term velocity is often used to include both these notions; in such case, the velocity of a body is not said to be constant unless the direction of the motion as well as its speed is unchanged ; that is, unless the motion is rectilinear as well as uniform.

In the first section of this chapter, we shall suppose the motion to be in a single straight line, so that speed only will at present be considered.

The speed is uniform when the spaces described in any intervals of time are always proportional to the intervals. When this is the case, its measure is the number of units of space described in a unit of time. Thus, if t denotes the number (integral or fractional) of units of time in any interval, and s denotes the number of units of space described or passed over in that interval, the velocity is uniform when the ratio of s to t is the same for all corresponding values of s and s.


Variable Speed.

5. When the spaces passed over in equal intervals of time are not equal, the speed is variable, and the quotient arising from dividing the space by the time gives what may be called the average speed for the given time. But at any given instant of time the speed has a definite value of which the numerical measure is the number of units of space which would he described in a unit of time if the body moved uniformly throughout that interval with the speed which it had at the instant considered.


Acceleration and Retardation.

8. The motion of a body is said to be hastened or accelerated when the velocity is increasing, and it is said to be uniformly accelerated when the increments of velocity which take place in any two intervals of time are proportional to the intervals. Thus, the motion considered in Art. 6, namely, that of a freely falling body, is a case of uniformly accelerated motion; for the expression V = gt shows that in any one second the velocity changes from gt to g(t+1), that is, it receives the increment g; in any two seconds it receives the increment 2g; in any half-second, the increment 2g; and so on.

Under these circumstances, the increment of velocity received in a unit of time is taken as the measure of the acceleration. Thus, in the case of the falling body, the acceleration is constant and equal to g. Supposing the motion to start from rest at the beginning of the interval, so that v = 0 when t = 0, the acceleration is the same as the velocity acquired in the first second, or the quotient arising from dividing the velocity acquired in any interval by the number of units of time in that interval.


The Laws of Motion.

12. The science of Mechanics is based upon certain first principles which must be regarded as established by experience. These, having been first clearly formulated by Sir Isaac Newton in the Philosophiae Naturalis Principia Mathematical are known as Newton's Laws of Motion. We shall in the succeeding articles give the three laws in literal translation from the Latin of the Principia^ each followed by the necessary explanations. Inertia.

13. Law I - Every body keeps in its state of rest or of moving uniformly in a straight line except so far as it is compelled by forces acting on it to change its state.

This law, which is sometimes called the Law of Inertia asserts that, while some external cause which we call force is necessary to put a body in motion, no such external action is necessary to keep it in uniform rectilinear motion after it has acquired a velocity; but, on the contrary, force is then required either to deflect it from a rectilinear path, or to alter its speed. This is contrary to the notion of the ancients, who regarded the earth as at rest, and attributed the observed tendency of bodies put in motion to come to rest to an inherent property of matter which they called inertia. On the other hand, we now hold that the earth itself is in motion, but that this does not in any way disturb the relative motion of bodies with respect to it. We regard inertia as opposed to any change of motion ; so that, when bodies already in motion come to rest relatively to the earth, the fact must be attributed to external causes or forces.

We cannot completely prove the first law of motion experimentally, because it is impossible to free the body on which we experiment entirely from the action of external forces; but we can show that the nearer we approach to this condition the nearer we realize a state of uniform rectilinear motion.


The Measure of Force.

14. Law II - Change of motion is proportional to the moving force acting, and takes place in the straight line in which the force acts.

This is the most important of the three laws. We defer to the next section its application to forces and motions in various directions, and here consider only the case of a single force acting upon a freely moving body. The direction of the force is of course that of the straight line in which the body, starting from rest, begins to move under the influence of the force acting freely - that is, when no other forces are acting. This line is called the line of action of the force. If, after the body has acquired motion in this line, the force continues to act in the same direction, the body will continue to move in the same straight line; for there is no reason why it should deviate from it to one side rather than the other. In the case of the single body now under consideration, " change of motion " means change of velocity. The second law therefore asserts that the, change of velocity produced in any interval of time is proportional to the force acting during that time.


Reaction.

24. Law III - There is always a reaction opposite and equal to an action or the ctctions of two bodies upon one another are always equal and oppositely directed.

This third law of motion, which is often called the law of reaction assumes that every force acting upon a body and tending to produce motion is of the nature of a tendency in the body to approach or to recede from some other body. This tendency is called the action of the second body upon the first. The law of reaction asserts that in every case there is an equal force acting upon the second body, which is called the reaction of the first body upon it. Moreover, these actions take place in two opposite directions along the straight line joining the two bodies, which are here considered as particles.

When the mutual action takes place at a distance there is said to be an attraction or a repulsion between the bodies according as they tend to approach or to recede. For example, the weight of a body is due to an attraction between the body and the earth : an electrical action may be an attraction or a repulsion. If, in the case of action at a distance, the bodies are free to move, the equal forces acting simultaneously on the two bodies give rise to equal impulses, so that, by the second law, the momenta produced in any given interval are equal. It follows that, if the bodies start from rest, the velocities are inversely proportional to the masses.