Question

A simple pendulum is suspended from the ceiling of a car accelerating uniformly on a horizontal road. If the acceleration is a₀ and the length of the pendulum is l, find the time period of small oscillations about the mean position.

Answer 1

The car accelerates with acceleration a. In the reference frame of car the effective value of acceleration due to gravity is

We shall work in the car frame. As it is accelerated with respect to the road, we shall have to apply a pseudo force m a₀ on the bob of mass m.

For mean position, the acceleration of the bob with respect to the car should be zero. If θ be the angle made by the string with the vertical, then tension, weight and the pseudo force will add to zero in this position. 

Suppose at some instant during oscillation, the string is further deflected by an angle α so that the displacement of the bob is x. Taking the components perpendicular to the string, component of T = 0,

component of mg mgsin =(α+θ) and component of

This is an equation of simple harmonic motion with 

As easy working rule may be found out as follows. In the mean position, the tension, the weight and the pseudo force balance. From figure, the tension is 

This plays the role of effective ‘g’. Thus the time period