Time taken by man to complete one revolution on platform – Rankers Physics

Time taken by man to complete one revolution on platform

A circular platform is mounted on a frictionless vertical axle. Its radius $R = 2\text{ m}$ and its moment of inertia about the axle is $200\text{ kg m}^{2}$. It is initially at rest. A $50\text{ kg}$ man stands on the edge of the platform and begins to walk along the edge at the speed of $1\text{ ms}^{-1}$ relative to the ground. Time taken by the man to complete one revolution is: (2012 Mains)

$\pi\text{ s}$
$3\pi/2\text{ s}$
$2\pi\text{ s}$
$\pi/2\text{ s}$

Solution:

Angular velocity of man $\omega_{m} = \frac{v}{R} = 0.5\text{ rad/s}$. By conservation of angular momentum, $I_{p}\omega_{p} + I_{m}\omega_{m} = 0$, giving $\omega_{p} = -0.5\text{ rad/s}$. Relative angular velocity $\omega_{rel} = \omega_{m} - \omega_{p} = 1\text{ rad/s}$. Time $T = \frac{2\pi}{\omega_{rel}} = 2\pi\text{ s}$.

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