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A simple pendulum of length $l$ and having 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 ?
Solution
The bob of the simple pendulum will experience the 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 ( $\left.a_{\text {eff }}\right)$ is given as:
$a_{ eff }=\sqrt{g^{2}+\left(\frac{v^{2}}{R}\right)^{2}}$
Time period, $T=2 \pi \sqrt{\frac{l}{a_{\text {eff }}}}$
Where, $l$ is the length of the pendulum
Time period $T=2 \pi\sqrt{\frac{1}{g+\frac{v^{2}}{R^{2}}}}$
