6.1 Rotational Kinetic Energy

Rotational Kinetic Energy Basics

A rotating rigid body has rotational kinetic energy given by KE = ½Iω², analogous to ½mv² for translation. Here I is the moment of inertia (rotational analog of mass) and ω is angular velocity. This energy comes from the sum of KE of all mass elements moving in circles about the rotation axis.

Total Kinetic Energy for Rolling Objects

An object that both translates and rotates (like a rolling ball) has total KE = ½mv_cm² + ½Iω², combining translational and rotational terms. For rolling without slipping, v_cm = ωR links the two speeds, letting you express total KE in terms of a single variable using I about the center of mass.

Energy Conservation with Rotation

In systems with no non-conservative forces (like rolling without slipping, where friction does no work), total mechanical energy is conserved: KE_i + PE_i = KE_f + PE_f. This lets you solve problems like objects rolling down inclines by comparing gravitational PE loss to gains in both translational and rotational KE.

Key terms

Rotational Kinetic Energy formula

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

Total KE of a rolling object

KE_total = ½mv_cm² + ½Iω² (translational + rotational).

Rolling without slipping condition

v_cm = ωR, linking translational and rotational speeds; static friction does no work.

Moment of Inertia (I)

Rotational analog of mass; depends on mass distribution relative to the rotation axis.

Energy conservation in rolling motion

Loss in gravitational PE equals gain in both translational and rotational KE when no energy is lost to friction.

Try an AP-style question

Two point masses of 1 kg each are at the ends of a massless 2 m rod spinning about its center at 3 rad/s. What is the rotational KE?

1. 4.5 J
2. 3 J
3. 9 J
4. 18 J