(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 | d2x | = − Kx, |
| dt2 |
(7)
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).