Page:EB1911 - Volume 17.djvu/995

be accepted as a representation of the initial stages of the forced oscillation. To obtain the complete solution of (11) we must of course superpose the free vibration (6) with its arbitrary constants in order to obtain a complete representation of the most general motion consequent on arbitrary initial conditions.

A simple mechanical illustration is afforded by the pendulum. If the point of suspension have an imposed simple vibration ξ = a cos σt in a horizontal line, the equation of small motion of the bob is

mẍ = −mg	x − ξ	,
	l

This is the same as if the point of suspension were fixed, and a horizontal disturbing force mgξ/l were to act on the bob. The difference of phase of the forced vibration in the two cases is illustrated and explained in the annexed fig. 63, where the pendulum virtually oscillates about C as a fixed point of suspension. This illustration was given by T. Young in connexion with the kinetic theory of the tides, where the same point arises.

Fig. 63.

We may notice also the case of an attractive force varying inversely as the square of the distance from the origin. If μ be the acceleration at unit distance, we have

u	du	= −	μ
	dx		x2

(15)

In the case of a particle falling directly towards the earth from rest at a very great distance we have C = 0 and, by Newton’s Law of Gravitation, μ/a² = g, where a is the earth’s radius. The deviation of the earth’s figure from sphericity, and the variation of g with latitude, are here ignored. We find that the velocity with which the particle would arrive at the earth’s surface (x = a) is √(2ga). If we take as rough values a = 21 × 10⁶ feet, g = 32 foot-second units, we get a velocity of 36,500 feet, or about seven miles, per second. If the particles start from rest at a finite distance c, we have in (16), C = − 2μ/c, and therefore

dx	= u = − √{	2μ (c − x)	},
dt		cx

(17)

the minus sign indicating motion towards the origin. If we put x = c cos² 1/2φ, we find

t =	c3/2	(φ + sin φ),
	√(8μ)

(18)

no additive constant being necessary if t be reckoned from the instant of starting, when φ = 0. The time t of reaching the origin (φ = π) is

t1 =	π c3/2	.
	√(8μ)

(19)

This may be compared with the period of revolution in a circular orbit of radius c about the same centre of force, viz. 2πc^3/2 / √μ (§ 14). We learn that if the orbital motion of a planet, or a satellite, were arrested, the body would fall into the sun, or into its primary, in the fraction 0.1768 of its actual periodic time. Thus the moon would reach the earth in about five days. It may be noticed that if the scales of x and t be properly adjusted, the curve of positions in the present problem is the portion of a cycloid extending from a vertex to a cusp.

In any case of rectilinear motion, if we integrate both sides of the equation

mudu/dx = X,

(20)

which is equivalent to (1), with respect to x between the limits x₀, x₁, we obtain

1/2 mu12 − 1/2 mu02 = $\int _{x_{0}}^{x_{1}}$ Xdx.

(21)

We recognize the right-hand member as the work done by the force X on the particle as the latter moves from the position x₀ to the position x₁. If we construct a curve with x as abscissa and X as ordinate, this work is represented, as in J. Watt’s “indicator-diagram,” by the area cut off by the ordinates x = x₀, x = x₁. The product 1/2mu² is called the kinetic energy of the particle, and the equation (21) is therefore equivalent to the statement that the increment of the kinetic energy is equal to the work done on the particle. If the force X be always the same in the same position, the particle may be regarded as moving in a certain invariable “field of force.” The work which would have to be supplied by other forces, extraneous to the field, in order to bring the particle from rest in some standard position P₀ to rest in any assigned position P, will depend only on the position of P; it is called the statical or potential energy of the particle with respect to the field, in the position P. Denoting this by V, we have δV − Xδx = 0, whence

X = dV/dx.

(22)

The equation (21) may now be written

1/2 mu12 + V1 = 1/2 mu02 + V0,

(23)

which asserts that when no extraneous forces act the sum of the kinetic and potential energies is constant. Thus in the case of a weight hanging by a spiral spring the work required to increase the length by x is V = ∫x0 Kxdx = 1/2Kx², whence 1/2Mu² + 1/2Kx² = const., as is easily verified from preceding results. It is easily seen that the effect of extraneous forces will be to increase the sum of the kinetic and potential energies by an amount equal to the work done by them. If this amount be negative the sum in question is diminished by a corresponding amount. It appears then that this sum is a measure of the total capacity for doing work against extraneous resistances which the particle possesses in virtue of its motion and its position; this is in fact the origin of the term “energy.” The product mv² had been called by G. W. Leibnitz the “vis viva”; the name “energy” was substituted by T. Young; finally the name “actual energy” was appropriated to the expression 1/2mv² by W. J. M. Rankine.

The laws which regulate the resistance of a medium such as air to the motion of bodies through it are only imperfectly known. We may briefly notice the case of resistance varying as the square of the velocity, which is mathematically simple. If the positive direction of x be downwards, the equation of motion of a falling particle will be of the form

du	= g − ku2;
dt

(24)

this shows that the velocity u will send asymptotically to a certain limit V (called the terminal velocity) such that kV² = g. The solution is

u = V tanh	gt	,   x =	V2	log cosh	gt	,
	V		g		V

(25)

if the particle start from rest in the position x = 0 at the instant t = 0. In the case of a particle projected vertically upwards we have

du	= −g − ku2,
dt

(26)

the positive direction being now upwards. This leads to

tan−1	u	= tan−1	u0	−	gt	,   x =	V2	log	V2 + u02	,
	V		V		V		2g		V2 + u2

(27)

where u₀ is the velocity of projection. The particle comes to rest when

t =	V	tan−1	u0	,   x =	V2	log ( 1 +	u02	).
	g		V		2g		V2

(28)

For small velocities the resistance of the air is more nearly proportional to the first power of the velocity. The effect of forces of this type on small vibratory motions may be investigated as follows. The equation (5) when modified by the introduction of a frictional term becomes

ẍ = −μx − kẋ.

(29)

If k² < 4μ the solution is

x = ae^−t/τ cos (σt + ε),

(30)

where

τ = 2/k,   σ = √(μ − 1/4k²),

(31)

and the constants a, ε are arbitrary. This may be described as a simple harmonic oscillation whose amplitude diminishes asymptotically to zero according to the law e−t/τ. The constant τ is called the modulus of decay of the oscillations; if it is large compared with 2π/σ the effect of friction on the period is of the second order of small quantities and may in general be ignored. We have seen that