Rolling While Slipping

A bowling ball released with pure slide and no spin skids because its surface is moving relative to the lane. That relative sliding means kinetic friction acts on it, backward on the center of mass since the contact point slides forward relative to the floor. This single friction force does two jobs at once: it decelerates the translational speed v through F = ma, and it torques the ball about its center through tau = I alpha, spinning it up. Both effects come from the same friction force, applied at the same instant, but they change different quantities at different rates set by the object's moment of inertia.

Translation slows linearly: v(t) = v0 - mu g t. Rotation speeds up linearly: omega(t) = alpha t, where alpha = mu g R / I for a sphere or cylinder of radius R. Rolling without slipping begins the moment v(t) = R omega(t). For a uniform solid sphere, I = (2/5)MR², and solving gives t = v0 / (3.5 mu g). Swap in a disk or hoop and the shape factor changes, since I changes, so always redo the algebra rather than memorizing the 3.5.

This topic is the bridge between pure friction-and-torque problems and the rolling-without-slipping analysis used everywhere else in the unit. Once v = R omega is reached, friction (if any) switches to static, and energy methods using rotational kinetic energy become valid again. Before that point, mechanical energy is not conserved, which connects directly to the next idea: where that lost energy goes.