Ncert Exercises & Solutions

14.17 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?

The bob of the simple pendulum will have two types of accelerations; 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 and R is the radius of the track.
Effective acceleration ( $a_{eff}$ ) is:
$a_{eff} = \sqrt{g^{2} + {(\frac{v^{2}}{R})}^{2}}$
$Time period, T = 2 π \sqrt{\frac{1}{a_{eff}}}$
$∴ Time period, T = 2 π \sqrt{\frac{l}{\sqrt{g^{2} + \frac{v^{4}}{R^{2}}}}}$

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