Question

A simple pendulum is suspended from the ceiling
of a car accelerating uniformly on a horizontal road.
If the pendulum oscillates about its mean position,
then find out the maximum angular displacement
from vertical achieved by pendulum.​

Answer 1

Answer:

We shall work in the car frame As it is accelerated with respect to the road we shall have to apply a pseudo force ma0 on the bob of mass m

For mean position the acceleration of the bob with respect to the car should be zero If θ0 be the angle made by the string with the vertical the tension weight and the pseudo force will add to zero in this position

Hence resultant of mg and ma0 (say F = mg2+a02) has to be along the string

∴ tanθ0=mgma0=ga0 Now suppose the string is further deflected by an angle θ as

shown in figure Now restoring torque can be given by (Fsinθ) l=−m l2α

Substituting F and using θ≃θ for small θ (mg2+a02) lθ=−m l2α

or α=− lg2+a02θ      so             ω2= lg2+a02

This is an equation of simple harmonic motion with time period

T=ω2π=2π