5.6 Rotational Kinetic Energy

Kinetic energy of a spinning object

A rotating rigid body stores energy in its motion just like a translating one, but the formula swaps mass for moment of inertia (I) and speed for angular velocity (ω): KE_rot = ½Iω². Every point in the object moves in a circle, and summing ½mv² over all the mass elements yields this compact expression.

Rolling without slipping

An object that rolls without slipping has both translational and rotational kinetic energy: KE_total = ½mv_cm² + ½Iω², with the constraint v_cm = ωR. This is why a rolling ball and a sliding block with the same speed have different total kinetic energies — the ball has extra KE tied up in spinning.

Energy conservation with rotation

For objects rolling down an incline or swinging as pendulums, mechanical energy conservation still applies: PE_i = KE_trans + KE_rot (plus any other KE terms). Objects with larger I (relative to mR²) convert less PE into translational speed, so they accelerate down ramps more slowly — a hoop lags behind a solid sphere of the same mass and radius.

Key terms

Rotational kinetic energy formula

KE_rot = ½Iω², where I is moment of inertia and ω is angular velocity.

Total KE of a rolling object

KE_total = ½mv_cm² + ½Iω² — sum of translational and rotational kinetic energy.

Rolling without slipping condition

v_cm = ωR, linking the center-of-mass speed to the angular velocity and radius.

Effect of larger moment of inertia on rolling speed

Larger I means more PE converts to rotational KE and less to translational KE, so the object accelerates more slowly down an incline.

Energy conservation for rotating systems

Mechanical energy (PE + translational KE + rotational KE) is conserved in the absence of non-conservative forces like friction that dissipate energy (though static friction enabling rolling does no work).

Try an AP-style question

A disk (I = 0.5 kg·m²) spins at 4 rad/s. What is its rotational KE?

1. 8 J
2. 1 J
3. 2 J
4. 4 J