Elementary dynamics a text book for engineers

Those who have had experience in teaching elementary dynamics to students of Engineering will agree that the majority find considerable difficulty in grasping the fundamental principles on which the subject is based. There are new physical quantities to be understood, and new principles to be accepted which can only be expressed in terms of these quantities. Unfortunately, in many cases, this subject has to be introduced at a stage of development in mathematics at which the student expects some proof of what he is told to believe. He is not so prepared to accept things without proof as he was when first told that 2 x 3 = 6. Even at that early stage, probably an attempt was made to prove to him that 2 x 3 = 6, yet, in the end, he merely accepted the fact and memorised it.

The author believes that the difficulty experienced is partly, though by no means entirely, due to the way the subject is often presented. In the student's mind it is associated with branches of pure mathematics such as trigonometry, analytical geometry, or infinitesimal calculus, and he is inclined to think that salvation lies in memorising a number of formulae which are to be used in solving problems, instead of looking upon dynamics as a fundamental branch of physical science, in which mathematics is of secondary importance and the physical ideas of primary importance. Quite commonly in elementary text-books, the second law of motion is at first summed up in the form, force = mass x acceleration, and the student is then given a number of examples to work out, most of which consist in substituting numbers in a formula. This very successfully disguises the true meaning of momentum, and the extraordinary generality of the second law of motion. The same applies to problems dealing with motion. The student is generally presented with certain formulae for motion involving a constant acceleration. These formulae, are of very little, if any, use to him later on, and have merely enabled him to get answers to certain problems without thinking.

At first sight the remedy would appear to lie in teaching dynamics experimentally, but the author's experience is that this is not so for the majority of students. The phenomena of everyday life provide innumerable qualitative experiments, and to most students quantitative laboratory experiments in dynamics are neither interesting nor convincing.

In the following pages an attempt has been made to present the principles of elementary dynamics, and to explain the meaning of the physical quantities involved, partly by definition and description, but mainly by worked examples in which formulae have been avoided as far as possible. By continually having to think of the principle and the physical quantities involved, the student gradually acquires the true meaning of them, and they become real to him.

It will be observed that the first of Newton's Laws of Motion is expressed in a somewhat different form from that in which it is usually given, and the laws are called the Laws of Momentum.

In working examples the absolute unit of force has generally been adopted, and, where applicable, the answers have been reduced to units of weight. It matters little, in the author's opinion, whether absolute or gravitation units are used, so long as mass is not defined as weight divided by the acceleration due to gravity. To say that the engineer's unit of mass is 32*2 lbs. is almost to suggest that he is rather lacking in intelligence, and cannot be expected to understand the difference between equality and proportionality. If weight is introduced in the early conception of mass, the student's conception of mass is extremely vague, and his conception of momentum as a physical quantity is even more vague or erroneous. A student who cannot understand the difference in the two units of force, and who has merely to rely on formulae expressed in one particular set of units, is not likely to get any knowledge of dynamics which will be of real use to him.

A number of graphical examples have been worked out in the text, and a number are included in the examples to be worked by the student. These frequently require more time than analytical examples, but they are more useful and instructive. This is particularly the case with the engineer, who is so frequently faced with problems which can only be solved graphically.

Probably the majority of students will be learning differential and integral calculus at the same time as dynamics, and they should be encouraged to use the calculus in working examples, although all the examples given can be worked without its use.

The examples at the ends of the chapters are arranged more or less to follow the text, and students should work them as they proceed with the reading, and not wait until they have completed the chapter. The miscellaneous examples, at the end of the book, are intended for revision, and for this reason they are not arranged either in order of difficulty or in the order of the chapters dealing with the principles involved. The answers have mostly been obtained by means of a slide rule. It is hoped that the errors in them are not numerous.

Though primarily written for engineering students the book may be useful to some others. The course covered is approximately that required for the Qualifying Examination which Cambridge students have to pass before their second year, if they wish to take an honours degree in Engineering.

The author wishes to thank Mr J. B. Peace, Fellow of Emmanuel College, for valuable suggestions and for having contributed a large number of examples, also Mr W. de L. Winter of Trinity College for very kindly reading the proofs and for useful criticism and suggestions.

The subject commonly called Dynamics is a part of the much bigger subject called Mechanics. In its broadest aspect, mechanics deals with bodies or parts of bodies which are acted upon by certain forces, and analyses and examines the effect of these forces in producing motion, or in maintaining a state of rest. For the present purpose mechanics may conveniently be divided into three branches as follows:

-  Kinematics
-  Kinetics
-  Statics

Kinematics is the branch of the subject which deals with the motions of bodies. The bodies may be of any size or shape, and at times it may be convenient to consider them indefinitely small, i.e. as points.

Kinetics deals with the causes of the motions of bodies, and attempts to find a definite relationship between these causes producing or maintaining motion, and the motions themselves.

Statics treats of bodies which are at rest and examines how this state may be maintained.

The subject of dynamics, as will be seen, does not in itself form one of the main branches of mechanics, but it may be said generally to include kinetics and a part of kinematics. It deals with motions in so far as they are required in the examination of the forces producing them, and it also deals with the general consideration of the mechanical energy possessed by bodies, either in virtue of their position or of their motion.

Now there are certain fundamental conceptions, and there are also certain principles or laws, which form the basis of the whole subject. Neither the conceptions nor the laws are numerous.


Fundamental Conceptions

These consist of the ideas of space, mass and time. It would be difficult, if not impossible, to define accurately either of these, or to explain exactly how the human mind understands their meaning. The conceptions are acquired in childhood or are born in us. Space, mass, and time are the three fundamental physical quantities, and all the other physical quantities we shall deal with may be defined in terms of them. It is easy to realise that before much practical use can be made of these, we must decide on some unit of measurement of them. We shall here only concern ourselves with two systems of units:

(1) Foot, Pound, Second system, (FPS)
(2) Centimetre, Gram, Second system, (CGS)

Space. We may say that this is what possesses length, breadth and thickness. Each of these is a length and hence space is most conveniently measured in units of length.

Mass. This may be defined as the quantity of matter or the quantity of stuff in a body. Although it is difficult to define precisely what is meant by mass, we have no difficulty in realising what we understand by the term. If, for example, we ask for half a pound of tobacco, we are not probably interested in the amount of space it occupies, or even its weight, so long as we get the correct quantity of stuff. It may be compressed in the form of a cake and occupy little volume, or it may be loose and occupy a considerable volume. We shall see later how we can compare masses. Probably our earliest conception of mass in childhood is when we try to throw things about. We find that some are more difficult to move or throw than others, and we soon discover that this does not depend upon the size. We consider that those which are more difficult to throw or move have more stuff in them.

In the FPS system the unit of mass is the pound (1 lb.). The standard pound is a particular lump of platinum deposited in the Exchequer Office.

In the CGS system, the unit of mass is the gram. Originally this was intended to be the mass of a cubic centimetre of pure water at 4 degrees centigrade, but the standard is now one of platinum like that of the pound.

In dealing with mass we might conveniently define what is meant by density. The density of a substance is the mass per unit volume.

This should not be confused with specific gravity. The specific gravity of a substance is the ratio of the mass of a given volume of the substance to the mass of an equal volume of water.

Time. The unit of time which is adopted in both systems of units is the second.


Vectors

The various physical quantities which we have to deal with can be divided into two classes:

(1) Scalar quantities.
(2) Vector quantities.

A scalar quantity is one which possesses magnitude only, for example, an interval of time, as 3 seconds. There is here no idea of direction. Again, 2 lbs. of bread. This is merely a definite quantity of stuff and has no connection with direction.

In dealing with scalar quantities we add and subtract by the ordinary rules of arithmetic. For example, in making a certain article in a workshop, work may have to be done on it in three different machines, and the lengths of time in these may be 15 minutes, 40 minutes, and 10 minutes. The total time for machining is then, (15 + 40 + 10) = 65 minutes.

A vector quantity is one which possesses both magnitude and direction for example, the weight of a body. We know that this acts vertically downwards, and that it is generally easier to push an object along than to raise it up.

Suppose we are dealing with the displacement of a body from a given position. Here we want to know, not only how far the body is from its original position, but also in what direction this distance is. Let us take an actual example.