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  4. 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, its time period of oscillation is

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Q. 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, its time period of oscillation is

Oscillations

Solution:

The bob of the pendulum has two accelerations:
(i) Centripetal acceleration, $a_{c}=\frac{V^{2}}{R}$ It acts horizontally
(ii) Acceleration due to gravity = g. It acts vertically downwards
The effective acceleration due to gravity.
$g'=\sqrt{g^{2}+a_{c}^{2}}$
$=\sqrt{g^{2}+\left(\frac{v^{2}}{R}\right)^{2}}$
$\sqrt{g^{2}+\frac{v^{4}}{R^{2}}}$
$\therefore $ Time period, $T=2\pi \sqrt{\frac{L}{g'}}=2\pi $
$\sqrt{\frac{L}{\sqrt{g^{2}+\frac{v^{4}}{R^{2}}}}}$