Question

A simple pendulum of bob mass m is suspended vertically from O by a massless rigid rod of length L (Fig. 1.17 (a)). The rod is connected to a spring of spring constant k at a distance h form O. The spring has its relaxed length when the pendulum is vertical.

Write the differential equation of motion and determine the frequency for small oscillations of this pendulum.

Answer 1

Let θ be a small deflection of the pendulum from its equilibrium position. The spring is compressed by x₁ and it exerts a force F_s = kx₁ on the rod. We have

x₁ = h sin θ and x₂ = L sin θ.

Taking the sum of torques about the point O we obtain (for small deflection θ):

Since sin θ ≈ θ for small oscillations we get SHM with frequency