Classical Mechanics (Kibble & Berkshire) · Problem 11.15

The key is recognizing that friction acts on the cylinder in two ways: it accelerates the center of mass and creates a torque that spins it. Since there's no slipping, the contact point's velocity must match the belt's velocity $V(t)$—this constraint links linear and angular motion. You'll use Newton's second law for translation and rotation simultaneously, along with the no-slip condition $v_{\text{cm}} + \omega R = V(t)$, to find how the cylinder's velocity evolves. The second part asks you to think about when rotational inertia dominates: if $k$ is large relative to the radius, most friction goes into spinning rather than accelerating the center of mass forward.

The key is recognizing that friction acts on the cylinder in two ways: it accelerates the center of mass and creates a torque that spins it. Since there's no slipping, the contact point's velocity must match the belt's velocity $V(t)$—this constraint links linear and angular motion. You'll use Newton's second law for translation and rotation simultaneously, along with the no-slip condition $v_{\text{cm}} + \omega R = V(t)$, to find how the cylinder's velocity evolves. The second part asks you to think about when rotational inertia dominates: if $k$ is large relative to the radius, most friction goes into spinning rather than accelerating the center of mass forward.