period, and phase, the resultant curve is drawn so that at every point its ordinate is equal to the algebraic sum of the ordinates of the component curves at that point. By suitably choosing the period and amplitude of

the component harmonic curves, it is possible, as illustrated in Figs. 45 and 46, to produce a periodic resultant curve of a type very different from a sine curve.

Fourier first showed that any periodic curve, as long as it nowhere goes to an infinite distance from the axis of X, can be built up by compounding together a finite number of harmonic curves the periods of which are commensurate. This last condition is necessary, for otherwise the resultant curve obtained by compounding the curves would never

exactly repeat itself, and would not be periodic. Hence it follows that any periodic motion can be considered as the resultant of a number of commensurate S.H.M.'s. If T is the period of the complex periodic motion, then the periods of the component S.H.M.'s will be included in the numbers 7, T2, T3, T4, &c.

As an illustration of the way in which a periodic curve of a given form may be built up by the combination of a number of S.H.M.'s, suppose the required curve to be represented by the lines ABCDEFG (Fig. 46). The thick continuous curve given in the figure is obtained by compounding the three S.H.M.'s shown dotted, of which the frequencies are in the ratio 1:35, while the amplitudes are as 1:1/31/5. It will be seen that even with three terms an approximation to the required form is produced. In Fig. 47 the result of combining 100 S.H.M.'s, having frequencies

FIG. 47.

proportional to the numbers 1, 3, 5, 7, 9, &c., and amplitudes proportional to 1, 1/3, 1/5, 1/7, 1/9, &c., is shown on a reduced scale. It will be noticed that in this case the required curve is almost perfectly reproduced.

Machines have been devised, called harmonic analysers, to determine mechanically the amplitudes of the S.H.M.'s of the periods T, 7/2, 7/3, &c., required to build up any giver curve. Other machines are capable of drawing the resultant of a certain number of S. H. M.'s of given amplitude and period.

E

PART III-DYNAMICS

CHAPTER VIII

NEWTON'S LAWS OF MOTION

56. Subdivisions of Dynamics.-Up to the present the motion of bodies has been considered quite in the abstract, and although we have assumed that the motion varied in certain ways, we have not inquired into the causes of these variations. We now pass on to consider the effects of force as shown in its action on the motion or equilibrium of material bodies. This branch of the subject of mechanics is called Dynamics. Dynamics is sometimes subdivided into two sections; in one, called Kinetics, the effect of forces on the motion of bodies is studied, while in the other, called Statics, the conditions which must exist if a body remains at rest when acted upon by a system of forces are investigated.

57. Stress. When one portion of matter acts on another portion, so as to influence its state, then the whole phenomenon of the mutual action of the two portions of matter is called in general a stress. In certain particular cases the stress has received a special name; thus we have a tension, a pressure, a torsion, an attraction, a repulsion, &c.

The term stress includes the consideration of both the mutually influencing portions of matter; it is, however, sometimes useful to concentrate our attention on one aspect of a stress, namely, the action on one of the portions of matter, so that we regard the stress as something acting on this piece of matter. From this point of view we say that the phenomena which we observe are the effect of External or Impressed Force on the portion of matter in question, and are due to the ACTION of the other portion of matter. The opposite aspect of the same stress would in this case be called the reaction on the other portion of matter. Hence Action and Reaction are simply different aspects of a stress, just as buying and selling are different aspects of one and the same transaction, according as we look at it from the point of view of one or other of the persons taking part in the transaction.

58. Newton's Laws of Motion.-The effect of external or, as it is sometimes called, impressed force on the motion of bodies is defined in three laws which are known as Newton's Laws of Motion. The first of these laws deals with the behaviour of a body when no external force

acts on it. The second tells us how the external force, when acting, may be measured. The third compares the two aspects of a stress, namely, Action and Reaction. These laws are Axioms, and do not admit of direct experimental proof; they depend, however, on convictions drawn from experiment, and their truth is universally admitted by those who have sufficient physical knowledge to thoroughly understand their purport.

59. Newton's First Law." Every body continues in its state of rest or of uniform motion in a straight line, unless it be compelled by impressed force to change that state."1

This law is also known as the law of Inertia, since it states that no body is capable of altering its state of rest or of motion without the intervention of some outside influence; and this fact we express in scientific language by saying that every body has inertia.

The law in the first place gives a definition of force, since it states that force is that action by means of which the state of rest or motion of a body is changed, and that unless a force acts no such change will occur. We may therefore define force as that which tends to produce change of motion in a body on which it acts.

In the next place the law tells us how a body will move when it is unacted upon by external forces. It says that if the body is in motion then it will continue moving uniformly in a straight line, if at rest it will continue at rest.

Indirectly the law may be taken as defining equal times. The times which a body, unacted upon by external forces, takes to pass through equal spaces are equal.

Since we are unable to obtain a body which is entirely unacted upon by external force, we cannot experimentally prove that if once set in motion it would continue to move uniformly. We find, however, that the more we reduce the magnitude of the impressed forces acting on a body, the greater is its tendency to continue moving at a uniform rate in a straight line when once it has been set in motion. Thus we know that if a stone is thrown along the surface of a road it will soon lose its motion. If thrown along the surface of smooth ice-in which case the friction, which is an impressed force tending to check the motion, is much less than in the case of the road-it will, however, continue to move very much longer.

A much more powerful argument for the validity of the law is obtained by considering that we can by its means solve problems in mechanics, and the solutions thus obtained always agree with observation, so that we conclude that our fundamental assumption is correct. Thus every one who makes use of the Nautical Almanack to discover the position of a star or the time of an eclipse, tacitly allows the correctness of Newton's law, for it is by the assumption of the correctness of the law that the numbers there given have been calculated.

1 Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur statum suum mutare.

60. Newton's Second Law.--The first law having stated that it is force alone which can produce change of motion, the second law tells us how the change of motion depends on the magnitude and direction of the force.

Before stating the law in Newton's words, we must consider some definitions which he prefixes to the laws.

(1) The Quantity of Motion, or the Momentum, of a rigid body moving without rotation is proportional to its mass and its velocity. The reasonableness of this definition will appear if we remember that the effort required to stop a body of great mass, such as a railway train, when moving with given velocity, is much greater than that required to stop a body of small mass, say a marble, when moving with the same velocity. Again, a greater effort is required to stop a bullet projected from a rifle with a high velocity than to stop a similar bullet when simply thrown by hand, and thus moving with a comparatively slow velocity. If, then, we take as the unit of momentum that of unit mass moving with unit velocity, the momentum of a mass moving with a velocity v will be mv. The dimensions of momentum are [L1M1T-1].

The change in momentum of a body is proportional to the mass of the body and the change in velocity. This follows at once, since the mass of body cannot alter; hence the only thing that can effect the magnitude of the momentum is a change in velocity. The rate of change of momentum is proportional to the mass and the acceleration (since the acceleration is the rate of change of the velocity). It must be remembered that the term velocity is used in the above in its most general sense (§ 30), and hence the momentum of a body changes when the direction of motion changes, although the speed may remain constant.

We may now state Newton's second law :-"Change of motion is proportional to the impressed force, and takes place in the direction of the straight line in which the force acts."1 By motion Newton means quantity of motion or, as it is now called, momentum, and in the same way the term impressed force includes the idea of time, for the magnitude of the change of momentum produced will depend on the time during which the force acts as well as on the magnitude of the force. The product of the magnitude of a force into its time of action is called the impulse of the force. Hence we may restate the first part of the law and say: Change of momentum is proportional to the impulse of the impressed force. It is important to notice that this law states that it is the change in momentum which is proportional to the impulse of the force, and hence it is immaterial whether the body on which the force acts is originally at rest or in motion in any direction; the change in its momentum in the direction in which the force acts is always proportional to the impulse of the force.

1 Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.