Rotation Has Kinetic Energy Too, ½Iω²
A spinning body holds energy too — the rotational kinetic energy ½Iω². The work a torque does, W = Mθ, changes this energy, and a rolling body carries both translation ½mv² and rotation ½Iω² at once.
Rotation has kinetic energy too — KE = ½ I ω². It is the same form as ½mv² with mass m replaced by I and speed v by ω. Raise ω and the KE grows with ω squared.
Just as work was W = F d in straight-line motion, in rotation the work a torque does turning through an angle θ is W = M θ. Drag θ and the work piles up in proportion; the power is P = M ω.
A rolling body carries two kinetic energies at once — the center's translation ½mv² and the spin's rotation ½Iω². For rolling, ω = vr makes the rotational part ½βmv², so the total KE = ½mv²(1+β).
Rolling down a ramp, the potential energy mgh converts entirely to rolling kinetic energy ½mv²(1+β). But part of it goes into spin, so from the same height the finish speed is slower than just sliding. Drag h.
So even starting from the same height, different shapes finish at different speeds — v = √(2gh(1+β)). The sphere, with small β losing the least energy to spin, is fastest, and the hoop is slowest. Pick a shape to compare.