Question

A simple pendulum of length l with a bob of mass M is suspended in a car. The car is moving on a circular track of radius R with a uniform speed v. If the pendulum makes small oscillations in a radial direction about its equilibrium position, what will be its time period?

Answer 1

The bob of the simple pendulum will experience acceleration due to gravity and the centripetal acceleration provided by the circular motion of the car.

Acceleration due to gravity = g

Centripetal acceleration =\( \frac{v^2}{R}\)

Where,

v is the uniform speed of the car

R is the radius of the track

Effective acceleration (a_eff) is given as: 

\(a_{eff}=\sqrt {g^2+(\frac{v^2}{R})^2}\)
Time period,  \(T=T=2π\sqrt \frac{l}{a_{eff}}\)

Where, l is the length of the pendulum

∴ Time period, \(T = 2π\sqrt{\frac{l}{g^2+\frac{v^4}{R^2}}}\)