Simple Pendulum in Non Inertial Frame

If a simple pendulum is made to oscillate in a non inertial frame the pseudo force must or torque due to pseudo force should also be taken into account. For a simple pendulum inside a frame accelerating with acceleration , with respect to an inertial frame, time period is given by

For example,

1. Time period of pendulum inside a cart moving with acceleration a on horizontal road

2. Time period of the pendulum inside an elevator accelerating in upward direction with acceleration a

Note: If acceleration of non-inertial frame with respect to inertial frame is not in vertical direction then in equilibrium position, the string of pendulum makes some angle with vertical.

Exercise 8 : Find the time period of a simple pendulum of length l suspended from the ceiling of a car moving with a speed v on a circular horizontal rod of radius r.

Exercise 9 : Find the time period of a simple pendulum of length L having a charge q on its bob when the pendulum is oscillating in a uniform electric field E directed (a) parallel to g (b) perpendicular to g

Exercise 10 : A hollow metal sphere is filled with water and a small hole is made at its bottom. It is hanging by a long thread and is made to oscillate. How will the time period change if water is allowed to flow through the hole till the sphere is empty.