Question

The spherical pendulum: Consider a simple pendulum of length l. At equilibrium the string is vertical along the z-axis and the bob is at x = 0, y = 0. Find the motion of the bob for small oscillations (x and y are small).

Answer 1

Suppose A is the position of the bob at any instant of time (Fig. 2.14). From A we drop a perpendicular AB on the z-axis. We have

which shows that z is a small quantity of second order. The potential energy of the bob at A is

Force on the bob along the x-direction is

 and the force along the y-direction is

Therefore, we have two uncoupled differential equations along x-and y-directions.

These are simple harmonic motions. These equation can be solved independently:

The constants A₁, A₂, ϕ₁ and ϕ₂ are determined by the initial conditions of displacement and velocity in the x-and y-directions. The complete motion can be thought of as a superposition of the motions i  x and j  y when we neglect the motion in the z-direction. Depending on the phase relationship between ϕ₁ and ϕ₂ we get an ellipse or a straight line for the path of the bob. For the x- and y-modes of vibrations we have the same frequency ω; the two modes are then said to be ‘degenerate’.