The Rolling Constraint: $v_{cm}$ = Romega. The contact point has zero instantaneous velocity. The topmost point has velocity 2$v_{cm}$.

Kinetic Energy of Rolling Body: $KE_{total}$ = \frac{1}{2}$$Mv_{cm}^2 + \frac{1}{2}$$I_{cm}*$omega^2$ = \frac{1}{2}$$Mv^2(1 + $k^2$/$R^2$)

where k is the radius of gyration (I = $Mk^2$).

For pure rolling on incline (no energy lost to friction): a = g*sin(theta) / (1 + $k^2$/$R^2$)

Friction is static (not kinetic) in pure rolling — it does zero net work but provides the torque needed for angular acceleration.

Key Result: The acceleration (and hence speed at bottom) is independent of mass and radius — it depends only on the shape ($k^2$/$R^2$ ratio).