6.1 Rotational Kinetic Energy

Rotational Kinetic Energy Formula

A rotating object stores energy due to its motion, called rotational kinetic energy: KE_rot = ½Iω². Here I is the moment of inertia (rotational inertia, depends on mass distribution) and ω is angular velocity. This is the rotational analog of translational KE = ½mv².

Total Kinetic Energy of Rolling Objects

An object that rolls without slipping has both translational KE (½mv²) and rotational KE (½Iω²). Total KE = ½mv² + ½Iω². Since v = ωr for rolling without slipping, objects with larger I (like hoops) move slower down a ramp than objects with smaller I (like solid spheres) for the same mass and radius, given equal starting energy.

Moment of Inertia Depends on Shape and Axis

Moment of inertia (I) measures resistance to changes in rotational motion and depends on both mass and how that mass is distributed relative to the axis of rotation. Mass farther from the axis contributes more to I. This is why a hollow cylinder has greater I than a solid cylinder 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.

Moment of Inertia (I)

A measure of an object's resistance to rotational acceleration; depends on mass and its distribution relative to the axis of rotation.

Total KE of a rolling object

KE_total = ½mv² + ½Iω² (translational KE + rotational KE).

Why hoops roll slower than solid spheres

Hoops have larger moment of inertia, so more energy goes into rotational KE, leaving less for translational speed.

Rolling without slipping condition

v = ωr, linking translational speed and angular velocity for a rolling object.

Try an AP-style question

An object with I = 2 kg·m² spins at 3 rad/s. What is its rotational kinetic energy?

1. 6 J
2. 9 J
3. 18 J
4. 3 J