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(C) Since the system is initially at rest,the initial angular momentum $L_i = 0$.According to the principle of conservation of angular momentum,the fi

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$2\pi \, s$

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(C) Since the system is initially at rest,the initial angular momentum $L_i = 0$.According to the principle of conservation of angular momentum,the final angular momentum $L_f$ must also be $0$.Let $\omega$ be the angular velocity of the platform. The angular momentum of the man is $L_m = m v R$ and the angular momentum of the platform is $L_p = I \omega$.Since the total angular momentum is zero,the man and the platform move in opposite directions: $m v R - I \omega = 0$.$\omega = \frac{m v R}{I} = \frac{50 	imes 1 	imes 2}{200} = 0.5 \, rad/s$.The angular velocity of the man relative to the ground is $\omega_m = \frac{v}{R} = \frac{1}{2} = 0.5 \, rad/s$.The angular velocity of the man relative to the platform is $\omega_r = \omega_m + \omega = 0.5 + 0.5 = 1 \, rad/s$.The time taken to complete one revolution is $T = \frac{2\pi}{\omega_r} = \frac{2\pi}{1} = 2\pi \, s$.

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