Rolling and Torque

Moment of inertia (I) measures how hard it is to change an object's rotation, just as mass measures how hard it is to change its motion. It depends on the mass and how far it is from the axis: I = Σ m r². Mass far from the axis, as in a ring, gives a large moment of inertia.

A solid sphere wins, then a solid disc (or cylinder), then a hollow sphere, and a ring comes last. The order depends only on the shape, through I/(mr²), not on the mass or radius.

Rolling without slipping means the object rolls without skidding: the point touching the ground is momentarily at rest, and the speed of the centre is v = ωr, where ω is the angular speed. Static friction makes it spin but does no work, so no energy is lost.

Torque τ = r F sin θ, where r is the distance from the axis to where the force acts, F is the force and θ the angle between them. Torque causes angular acceleration: τ = I α, the rotational version of F = m a.

For rolling without slipping, a = g sin θ / (1 + I/(m r²)). For a solid sphere that is (5/7) g sin θ, for a disc (2/3) g sin θ and for a ring (1/2) g sin θ. An object sliding without friction has a = g sin θ.

A rolling body has both translational and rotational kinetic energy: KE = ½ m v² + ½ I ω². For a ring the two parts are equal; for a solid sphere the spinning part is 2/7 of the total.