The Rolling Constraint: v_cm = Romega. The contact point has zero instantaneous velocity. The topmost point has velocity 2v_cm.

Kinetic Energy of Rolling Body: KE_total = (1/2)Mv_cm^2 + (1/2)I_cm*omega^2 = (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).