momentum is equal to the impulse is (it maybe remarked) equivalent
to Newton’s own formulation of his Second Law. The form
(1) is deduced from it by putting t ′ − t = δt, and taking δt to be
infinitely small. In problems of impact we have to deal with
cases of practically instantaneous impulse, where a very great
and rapidly varying force produces an appreciable change of
momentum in an exceedingly minute interval of time.
In the case of a constant force, the acceleration u̇ or ẍ is, according to (1), constant, and we have
| d 2xdt 2 = α, | (3) |
say, the general solution of which is
| x = 12 αt 2 + At + B. | (4) |
The “arbitrary constants” A, B enable us to represent the circumstances of any particular case; thus if the velocity ẋ and the position x be given for any one value of t, we have two conditions to determine A, B. The curve of positions corresponding to (4) is a parabola, and that of velocities is a straight line. We may take it as an experimental result, although the best evidence is indirect, that a particle falling freely under gravity experiences a constant acceleration which at the same place is the same for all bodies. This acceleration is denoted by g; its value at Greenwich is about 981 centimetre-second units, or 32.2 feet per second. It increases somewhat with the latitude, the extreme variation from the equator to the pole being about 12%. We infer that on our reckoning the force of gravity on a mass m is to be measured by mg, the momentum produced per second when this force acts alone. Since this is proportional to the mass, the relative masses to be attributed to various bodies can be determined practically by means of the balance. We learn also that on account of the variation of g with the locality a gravitational system of force-measurement is inapplicable when more than a moderate degree of accuracy is desired.
We take next the case of a particle attracted towards a fixed point O in the line of motion with a force varying as the distance from that point. If μ be the acceleration at unit distance, the equation of motion becomes
| d 2xdt 2 = −μx, | (5) |
the solution of which may be written in either of the forms
| x = A cos σt + B sin σt, x = a cos (σt + ε), | (6) |
 |
| Fig. 61. |
where σ= √μ, and the two constants A, B or a, ε are arbitrary. The particle oscillates between the two positions x = ±a, and the same point is passed through in the same direction with the same velocity at equal intervals of time 2π/σ. The type of motion represented by (6) is of fundamental importance in the theory of vibrations (§ 23); it is called a simple-harmonic or (shortly) a simple vibration. If we imagine a point Q to describe a circle of radius a with the angular velocity σ, its orthogonal projection P on a fixed diameter AA′ will execute a vibration of this character. The angle σt + ε (or AOQ) is called the phase; the arbitrary elements a, ε are called the amplitude and epoch (or initial phase), respectively. In the case of very rapid vibrations it is usual to specify, not the period (2π/σ), but its reciprocal the frequency, i.e. the number of complete vibrations per unit time. Fig. 62 shows the curves of position and velocity; they both have the form of the “curve of sines.” The numbers correspond to an amplitude of 10 centimetres and a period of two seconds.
The vertical oscillations of a weight which hangs from a fixed point by a spiral spring come under this case. If M be the mass, and x the vertical displacement from the position of equilibrium, the equation of motion is of the form
| M | d 2x | = − Kx, |
| dt 2 |
provided the inertia of the spring itself be neglected. This becomes identical with (5) if we put μ = K/M; and the period is therefore 2π√(M/K), the same for all amplitudes. The period is increased by an increase of the mass M, and diminished by an increase in the stiffness (K) of the spring. If c be the statical increase of length which is produced by the gravity of the mass M, we have Kc = Mg, and the period is 2π√(c/g).
 |
| Fig. 62. |
The small oscillations of a simple pendulum in a vertical plane also come under equation (5). According to the principles of § 13, the horizontal motion of the bob is affected only by the horizontal component of the force acting upon it. If the inclination of the string to the vertical does not exceed a few degrees, the vertical displacement of the particle is of the second order, so that the vertical acceleration may be neglected, and the tension of the string may be equated to the gravity mg of the particle. Hence if l be the length of the string, and x the horizontal displacement of the bob from the equilibrium position, the horizontal component of gravity is mgx/l, whence
| d 2x | = − | gx | , |
| dt 2 | l |
The motion is therefore simple-harmonic, of period τ = 2π√(l /g). This indicates an experimental method of determining g with considerable accuracy, using the formula g = 4π2l /τ2.
In the case of a repulsive force varying as the distance from the origin, the equation of motion is of the type
| d 2x | = μx, |
| dt 2 |
the solution of which is
where n = √μ. Unless the initial conditions be adjusted so as to make A = 0 exactly, x will ultimately increase indefinitely with t. The position x = 0 is one of equilibrium, but it is unstable. This applies to the inverted pendulum, with μ = g/l, but the equation (9) is then only approximate, and the solution therefore only serves to represent the initial stages of a motion in the neighbourhood of the position of unstable equilibrium.
In acoustics we meet with the case where a body is urged towards a fixed point by a force varying as the distance, and is also acted upon by an “extraneous” or “disturbing” force which is a given function of the time. The most important case is where this function is simple-harmonic, so that the equation (5) is replaced by
| d 2x | + μx = ƒ cos (σ1t + α), |
| dt 2 |
where σ1 is prescribed. A particular solution is
| x = | ƒ | cos (σ1t + α). |
| μ − σ12 |
This represents a forced oscillation whose period 2π/σ1, coincides with that of the disturbing force; and the phase agrees with that of the force, or is opposed to it, according as σ12 < or > μ; i.e. according as the imposed period is greater or less than the natural period 2π/√μ. The solution fails when the two periods agree exactly; the formula (12) is then replaced by
| x = | ƒt | sin (σ1t + α), |
| 2σ1 |
which represents a vibration of continually increasing amplitude. Since the equation (12) is in practice generally only an approximation (as in the case of the pendulum), this solution can only
