A homogeneous ball of mass $M$ and radius $R$ is struck impulsively at its center, causing it to go from rest to a horizontal speed of $v_0$. Assuming a constant coefficient of friction $\mu$, find the distance traveled by the ball before it begins to roll without slipping.

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Answer

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The requirement for rolling without slipping is $\omega R = v_{x,cm}$. We need to write down the equations of motion for $x_{cm}$, $y_{cm}$, the position of the center of mass, and $\theta$, the total angle the ball has rolled, and solve. Applying all forces and torques, we get for the $y$, $x$, and $\theta$ motions

(1)Solving for $v_x$, we get

(2)Solving for $\omega$, we get

(3)Setting $\omega R = v_x$ at time $T$, we get

(4)To find the distance it traveled in this period, we must find the position of the center of mass at this time

(5)at time $T$, this is

(6)
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