Rotational Kinetic Energy

Rotational kinetic energy: K_rot = ½Iω². Direct analog of K_trans = ½mv².

Total KE of a rolling body = K_trans + K_rot = ½Mv² + ½Iω². With v = rω (rolling condition).

For a rolling solid sphere: K_total = ½Mv² + ½·(2MR²/5)·(v/R)² = (7/10)Mv². So 71% translation, 29% rotation.

Disk rolling: K_total = (3/4)Mv². Hollow sphere: (5/6)Mv². Ring: Mv² (50/50 split).

Energy stored in a flywheel: K = ½Iω². Large I and large ω ⇒ huge storage. Used in regenerative braking.

Work done by torque: W = ∫τ·dθ = ΔK_rot.

Power delivered to rotating body: P = τ·ω.

Rotation kinetic energy is real KE — has the same units (J) and converts to/from translational KE in rolling.