PART I.

CHAPTER I.

PERIODIC MOTION.

1. The Simple Periodic Motion. A motion which is repeated at regular intervals of time is called a periodic motion. The simplest kind of periodic motion is that in which a particle moves in a straight line, in such a way that its distance, x, from a fixed centre satisfies the equation

x = a sin w (t0)........

(1), where t is the time and a and w are constants. The equation shows that the particle continuously oscillates between two points which are at a distance a from the centre. This distance is called the amplitude. The velocity (u) of the particle which moves according to (1) is

and the acceleration (ƒ) is

u = aw cos w (t − 0)

f=-aw2 sin w (t − 0)..

The particle passes through its central position (x = 0) when

m being an integer. The velocity even, and wa when m is odd.

.(2),

·(3).

of the particle is then wa when m is Hence the velocity has its greatest value when x=0, but may be positive or negative according as the particle passes through its central position from the negative or from the positive side.

If the time t is increased by 2π/w, no change is made in the values of either a or u, so that after a time interval of 2/w the position and state of motion are the same. The period is called the "time of oscillation," the "periodic time," or simply the "period" of the motion. Its relation to the constant is expressed by the equation:

T= 2π/W.

Equations (1) and (2) may take different forms by a change in the value of 0. Thus by writing we, we, we obtain

=

When dealing with one particle only, so that the origin of time may be chosen according to convenience, we may adopt the simpler forms of either (1) or (1a), obtained by making 0 or 0, equal to zero.

Ө

I proceed to show that equations (2) and (3) are necessary consequences of (1).

In Fig. 1 consider a point P moving uniformly in a circle of radius

Fig. 1.

a = OA.

Let OM be the projection of OP on a diameter AB. If the angle POM be denoted by 4, and the distance OM by x,

x = a cos p.

If the particle passes through the position B when t=01, and takes a time to complete a whole revolution,

T

This shows that the point M moves in the simple periodic motion indicated by equations (1a) or (1) and we have the important proposition that this periodic motion may be represented as an orthogonal projection of a uniform circular motion. The periodic time is the time of revolution of the point P, the amplitude is the radius of the circle, and the constant 0, represents the smallest positive value of the time at which the particle reaches its extreme position on the positive side.

The proper expressions for the velocity and acceleration of the point M are obtained by considering that these are equal to the projections on AB of the velocity and acceleration of P.

The velocity of P is constant and equal to U.

The minus sign is explained by the fact that for positive values of , the velocity of M is from right to left, or in the negative direction. The whole circumference of the circle being described in a time t with velocity U, it follows that

The expression for the acceleration of the point M is obtained in a similar manner. The acceleration of the point P is directed radially inwards towards the centre of the circle and is equal to U2/a, and the acceleration ƒ of M is the projection of this acceleration upon the diameter AOB.

A periodic motion may be of a more complicated character than that indicated by the above equations. If we were to take e.g. the orthogonal projection of a particle moving with uniform speed in an ellipse, we should get a motion which is strictly periodic, but which could not be represented by the simple equations we have given. Even the oscillations of a simple pendulum can only be approximately represented by our equations, the approximation being the more nearly correct, the smaller the amplitude.

I shall call a "simple" or "normal" oscillation one which can be represented as the orthogonal projection of a uniform circular motion. A normal oscillation is identical with that often called "harmonic motion." I avoid this term because "harmony" means a relation between different things, and not a property of any particular thing.

The character of the motion of a particle performing normal oscillations is completely determined by the amplitude and period, but the state of motion at any time requires a third quantity for its definition. If the oscillation is considered to be the projection of a uniform circular motion, it is convenient to take the angle between the radius vector OP (Fig. 1) and some fixed radius as the quantity defining the state of motion. This angle is called the "phase" of motion, and is to a certain extent arbitrary, as the fixed radius may be drawn in any direction.

If we express the motion in the form

x = a sin ∞ (t-0)

it is usual to define zero phase as the phase at the time the particle passes through its mean position in the positive direction. The radius of reference will then be OC (Fig. 1) at right angles to AB, and w(t − A) will measure the phase.

On the other hand, if we choose the form

x = α cos w (t −02)

as the equation of motion, we may define zero phase as the phase at the time the particle reaches its extreme position on the positive side; then w(t0) will be the phase, and the radius of reference will be OB, or the positive branch of the direction on which the circular motion is projected. The absence of uniformity in the choice of the direction

which defines the zero phase, causes no inconvenience, as we are nearly always concerned with differences of phase, and this difference is perfectly determinate. Thus if in Fig. 6 two periodic motions are represented by the projections of the circular motions of two particles P and Q on the same straight line, the angle POQ will always represent the difference between the phases, whatever line is taken to be the direction of zero phase.

The difference in phase between two normal periodic motions having the same period is independent of the time.

2. Normal Oscillations under the action of forces varying as the distance. The equations for the displacement a and the acceleration ƒ of a particle which has a simple periodic motion are x = a sin w (t − 01)

f=-aw2 sin w (t − 01).

By combining these we obtain the relation:

f=- w2x......

.(4).

This is an equation of great importance, for it shows the necessary condition which must be satisfied in order that a particle shall execute normal oscillations when acted on by a force directed to a centre. This condition is, that the force tending to bring the particle back to its position of equilibrium is proportional to the distance of the particle from that position.

Consider a particle constrained to move in a straight line and attracted to a fixed centre by a force F, which is proportional to the displacement. If m is the mass of the particle and F=-n2x

N2

X. m

This agrees with (4) if w2 is equal to n2/m, and hence the time of oscillation is obtained in terms of m and n, for

All forces of nature diminish with increasing distance, and the particular law of force which produces normal oscillations may not at

first sight seem therefore to have any practical importance. As a matter of fact, this law, however, holds in almost all cases when the displacements are small, for when the particle is kept at rest under the action of opposing forces, the resultant of these forces will always, if the displacements are sufficiently small, increase proportionally to the distance of the particle from its position of equilibrium. As this is an important fact, it is well to give a few examples.

Example 1. The Simple Pendulum. A heavy particle is suspended from a fixed point by a thin string of length 7 and is set in motion. Let (Fig. 2) be the angular deviation of the string from the vertical at any instant. The only forces acting

F

S

2

Fig. 2.

mg.

on the particle are its weight and the tension of the string. The particle is constrained to move in a circle, and the force which tends to draw back the particle to its position of equilibrium is found by resolving these forces along the tangent to the arc.

If m is the mass of the particle, its weight is mg. The tension of the string has no component in the direction of the tangent to the arc, and therefore the resultant force acting on the particle is mg sin 0.

If is so small that we can neglect 02 compared to unity, we may replace sin by the angle 0, so that the force F acting on the particle is :

where s is the displacement of the particle along the arc corresponding to the angular displacement 0.

This equation shows that the particle moves along the arc, as if it were subject to a restoring force which is proportional to the distance of the particle from the lowest point of the arc. Therefore the particle will describe normal oscillations about this point. The acceleration of the particle at any distance s is F/m or gs/l. By comparing this with (4) it follows that w2=g/l or that the period is determined by

This is the well known equation for the time of oscillation of a simple pendulum.