| 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
| d2x | = − | gx | , |
| dt2 | l |
(8)
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
| d2x | = μx, |
| dt2 |
(9)
the solution of which is
x = Aent + Be−nt,