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Question

(C) When the ball slides on the lawn,the friction force acts in the backward direction,which creates a torque about the contact point $O$ on the groun

Vedclass · Accepted Answer

$\frac{5}{7} v_0$

Vedclass · Answer

(C) When the ball slides on the lawn,the friction force acts in the backward direction,which creates a torque about the contact point $O$ on the ground. Since the torque due to friction about the point of contact $O$ is zero,the angular momentum of the ball about $O$ is conserved. Initial angular momentum about $O$ is $L_i = m v_0 R$. When the ball starts rolling without slipping,its velocity is $v$ and angular velocity is $\omega = v/R$. The final angular momentum about $O$ is $L_f = I_{cm} \omega + m v R = (\frac{2}{5} m R^2) (\frac{v}{R}) + m v R = \frac{2}{5} m v R + m v R = \frac{7}{5} m v R$. By conservation of angular momentum,$L_i = L_f$: $m v_0 R = \frac{7}{5} m v R$ $v_0 = \frac{7}{5} v$ $v = \frac{5}{7} v_0$.

$A$ ball is thrown on a lawn in such a way that it initially slides with a speed $v_0$ without rolling. It gradually picks up rotational motion. Find the speed of the ball at which there will be rolling without slipping.

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