Elements of mechanics including kinematics, kinetics and statics

1. Mechanical experiences are without doubt of great antiquity. The earliest investigations concerning mechanical principles are ascribed to Archytas of Tarentum (b.c. 400). He is said to have worked out the theory of the pulley. Later, in the writings of Archimedes of Syracuse (B.C. 287-212), are found applications of geometry to various mechanical questions, including a treatise on levers and other machines. From Archimedes to Galileo and Stevinus, a period of nearly two thousand years, no marked advance was made. It is to Galileo and Stevinus that we owe the transition from Mechanics in its original signification as the Science of Machines to Mechanics as the term is now understood - in fact they are to be regarded as the founders of the science of mechanics.

2. The qualities of natural phenomena become known to us through our senses. Certain of these qualities are assumed to be fundamental, in the sense that no one can be expressed in terms of the others. They are incapable of definition and include space, matter, and time. Closely related to these fundamental ideas are the ideas of motion and force.

But although it is not possible to define these qualities, we may consider their mutual relations. In order to investigate these relations it is necessary to compare by measurement the quantities that enter. The science which treats of the relations of matter motion, and force and of their measurement is called Mechanics.

3. The term body is applied to a limited portion of matter. Bodies are said to occupy different positions relative to neighboring bodies. We define the position of any point in a body by reference to points in some other body chosen as points of reference.

4. When a body is changing its position it is said to be in motion. The line drawn through the successive positions occupied forms the path of the moving body.

Now as we contemplate the body moving in its path, questions arise as to the influence of the body itself on the motion. We may, however, consider the motion only, apart from the body moving, and study the nature of the path traced out as the body moves from one position to another. Of course no such separation exists. It is a mere abstraction introduced to reduce questions of motion to a purely mathematical form and to serve as an introduction to the more complex problem itself.

This science which investigates motion without considering the nature of the body moved or how the motion is produced is called Phoronomics [= law of going] or, more commonly, but less properly, Kinematics.

Since in changing position a certain time is taken, the elements of a motion may be said to be distance, direction, and time. Kinematics, therefore, deals with distance, direction, and time, and may be regarded as an extension of geometry by the introduction of the idea of time. Like geometry, it is a purely abstract science resting upon certain ideal assumptions.

44. Hitherto we have considered motion in the abstract - how represented and how measured. No reference has been made to the nature of the body moving, and the problem has been as purely ideal as a problem of geometry.

We shall now consider motion with reference to the body moving and the force acting, that is, pass from kinematics to dynamics. This brings us to the region of sense and makes our results capable of verification, or such that we can test computation by observation and measurement. In order to verify results certain fundamental postulates are taken as bases of operation, which will now be stated.

45. The relations of matter, motion, and force, which constitute the science of dynamics, may be based upon three postulates known as Newton’s laws of motion. These laws were known to Galileo and other forerunners of Newton, but were first stated by Newton in concise terms. They are not axiomatic in the sense of a geometrical axiom, because when stated they are not at once assented to. They do not commend themselves to the mind either as true or as false.

In stating Newton’s laws certain rude experiments will be indicated which are sufficient to suggest the truth of the laws, but not to establish it. No direct proof is possible. The proof is indirect and is made in this way. Assume the laws true, and certain consequences follow which can he tested experimentally. This has been done in so many ways and by so many independent observers, particularly in astronomical work, that we are justified in accepting them as true. For example, the Ephemeris or Nautical Almanac is published several years beforehand, and the predictions made in it and based on these laws are always found to agree with the occurrences when observed. Such, for instance, are the predictions of the times of eclipses of the sun and moon, the positions of the planets, etc.

46. Law I. Every body [particle] continues in its state of rest or of uniform motion in a straight line except in so far as it may he compelled by external force to change that state.

The law lies beyond our experience, as we have no experience of one body not acted upon by another. Our direct experience goes, however, a certain distance in confirmation of the law. Thus, as suggested by Galileo, consider a body placed on a level surface. If at rest it will remain at rest; if in motion it will come to rest after going a distance depending upon the smoothness of the surface. The smoother the surface the farther it goes and the more nearly in a straight line. Conceive a surface perfectly smooth and the air to have no influence on the motion, and we cannot think of any reason why the body should not continue to move uniformly in a straight line.

47. From the law we learn that rest and motion are equally states of a body, the body being wholly without influence on its rest or motion. This property of matter is called inertia [the vis insita of Newton], and the law itself is often named the law of inertia.

From the law we also learn that by the term forge is meant a cause of change in motion, not in the sense of moving agent, but in the sense of antecedent. Force is thus not to be regarded as the cause of a state of motion, but of a change of state, from rest to motion, motion to rest, or to an alteration of motion - in a word, of acceleration. Whenever force acts, an acceleration of the motion of the body acted upon is produced.

48. The law guides us in finding a timekeeper. A body in motion and not acted upon by external forces would afford a means of measuring times. For the distances passed over by such a body in equal times are equal.

We know of no permanent motion that is at the same time uniform and rectilinear. The standard motion for the measurement of time is the rotation of the earth on its axis. We assume that the earth revolves uniformly or through equal angles in equal times, and find that predictions of astronomical phenomena made on this hypothesis agree closely with subsequent observation.

49. Law II. Having learned that a characteristic manifestation of force is acceleration, our next inquiry is as to the relation between force acting, body acted upon, and acceleration produced - in a word, as to how force is measured.

Now it is found that when the same body is exposed to action of the same force it has the same change of motion. Thus the same pull of a spring balance - equal pulls being measured by equal stretch of spring gives the same body the same acceleration at all times and places. The same general result is found no matter how the manner of making the experiment is varied.

If two bodies exposed to the action of the same force receive the same acceleration we say that they are of the same mass, and if the accelerations are not the same we say that the bodies are of different mass. The term mass is thus applied to that physical quality of a body that determines its acceleration. Experiment shows that mass is a definite entity altogether independent of the physical state of the body.

Newton defines "the quantity of any matter as the measure of it by its density and volume conjointly/' and states that this quantity is what he shall understand by the term mass or body.

56. Law III. In order to exert force the agent acting must meet a resistance. Thus the hand in motion does not exert force until it meets some object. The object reacts on the hand. Press the table and the table will press the hand. Force is always a mutual action: in other words, forces are never single, but act in pairs - one the action and the other the reaction. This pair of actions between two bodies or two parts of the same body is known as a stress. If it is of the nature of a push, preventing approach of the two bodies, it is called compression or pressure; if of the nature of a pull, preventing separation, it is called tension; if of the nature of a shear, preventing sliding, it is called a shearing stress or shear.

When we speak of a force acting on a body we consider only one of the two bodies between which stress exists. The force is the component of the stress on the body - the action. This was the case in discussing the preceding two laws.

But since a force cannot exist by itself, - forces being dual, - the view given in laws I and II is only partial and requires to be supplemented. This is done by the law of stress, or Newton’s third law of motion, which is:

When one body acts on another, the reacting force (reaction) is equal in magnitude and opposite in direction to the acting force {actio7i), or, as it may be expressed:

The mutual actions of two bodies are always equal and act in opposite directions.

67. In some cases the relation between the action of the agent and the reaction of the resistance is sufficiently evident. Thus if one body rests upon another it will be granted that the pressure exerted by the upper is equal to the counter-pressure exerted by the lower: if a horse hauls a canal-boat to which he is attached by a rope, the pull of the rope on the horse is equal to its pull on the boat, and so on. But when a stone falls from a height it is not evident whether the action of the earth on the stone is equal to the action of the stone on the earth. Nor is the relation evident between the actions of a magnet and a piece of iron,* nor between bodies widely separated, as the earth and the moon. But the law asserts that in all cases the acting force and reacting force are equal.

Newton points out the consequence of denying the truth of the law: "For instance, if the attraction of any part of the earth, say a mountain, upon the remainder of the earth were greater or less than that of the remainder of the earth upon the mountain, there would be a residual force acting upon the system of the earth and the mountain as a whole which would cause it to move off with an ever-increasing velocity through infinite space. This is contrary to the first law of motion, which asserts that a body does not change its state of motion unless acted upon by external force."