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(C) Let the mass of the ball be $m$ and its radius be $R$. The moment of inertia of a solid sphere about its center is $I = \frac{2}{5} mR^2$.Initiall

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$\frac{5}{7} v_0$

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(C) Let the mass of the ball be $m$ and its radius be $R$. The moment of inertia of a solid sphere about its center is $I = \frac{2}{5} mR^2$.Initially,the ball has linear velocity $v_0$ and zero angular velocity.As the ball slides,friction acts on it,creating a torque that increases its angular velocity until it starts rolling without slipping.During this process,the angular momentum about the point of contact on the ground remains conserved because the torque due to friction about this point is zero.Initial angular momentum about the point of contact: $L_i = m v_0 R$.Final angular momentum about the point of contact when rolling without slipping occurs (where $v = \omega R$): $L_f = I_{cm} \omega + m v R = (\frac{2}{5} mR^2) (\frac{v}{R}) + m v R = \frac{2}{5} m v R + m v R = \frac{7}{5} m v R$.Equating $L_i = L_f$: $m v_0 R = \frac{7}{5} m v R$.Solving for $v$: $v = \frac{5}{7} v_0$.

$A$ 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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