5.4 Rotational Inertia and Angular Acceleration

Rotational Inertia (Moment of Inertia)

Rotational inertia (I) measures an object's resistance to changes in rotational motion, just as mass resists changes in linear motion. It depends on both the mass and how that mass is distributed relative to the axis of rotation — mass farther from the axis contributes more to I. For a point mass, I = mr². Objects with mass concentrated far from the axis (like a hoop) have greater I than those with mass near the axis (like a solid disk) of equal mass and radius.

Newton's Second Law for Rotation

The rotational analog of F = ma is τ_net = Iα, where τ_net is net torque, I is rotational inertia, and α is angular acceleration. This means a net torque causes angular acceleration, and the same torque produces smaller α on objects with larger I. This relationship lets you predict how quickly an object will spin up or slow down given applied torques.

Distribution of Mass Matters

Two objects with identical mass and radius can have very different rotational inertias depending on shape. A solid sphere or disk has mass closer to the axis (smaller I), while a hoop or ring has mass concentrated at the rim (larger I). This is why a hoop rolling down a ramp accelerates more slowly than a solid disk of the same mass and radius — more of its mass resists angular acceleration.

Key terms

Rotational inertia (I)

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

τ_net = Iα

Newton's second law for rotation: net torque equals rotational inertia times angular acceleration.

Point mass rotational inertia formula

I = mr², where m is mass and r is distance from the rotation axis.

Why does mass distribution matter for I?

Mass farther from the axis contributes more to rotational inertia than mass close to the axis, even with equal total mass.

Hoop vs. solid disk (same mass & radius)

The hoop has greater rotational inertia because its mass is concentrated at the rim, farther from the axis than the disk's evenly spread mass.

Try an AP-style question

A net torque of 12 N·m acts on an object with I = 4 kg·m². What is α?

1. 16 rad/s²
2. 48 rad/s²
3. 0.33 rad/s²
4. 3 rad/s²