Rotational Inertia & Newton’s Second Law for Rotation

You’ll be able to

Define rotational inertia and its dependence on mass distribution
Apply the rotational form of Newton’s second law, τ_net = Iα
Compare the rotational inertia of different shapes

Rotational inertia resists angular acceleration

Rotational inertia (moment of inertia, I) is the rotational cousin of mass: it measures how hard it is to change an object’s spin. But unlike mass, it depends on where the mass sits relative to the axis. For a single point mass, I = mr². The farther the mass is from the axis, the larger I becomes — and the square means distance matters a great deal.

Newton’s second law, rotational form

Just as ΣF = ma governs straight-line motion, the net torque sets the angular acceleration: τ_net = Iα. A given torque produces less angular acceleration on an object with large rotational inertia. Because I depends on mass distribution, two objects of equal mass can respond very differently — a hoop (mass at the rim) is far harder to spin up than a solid disk (mass spread inward).

Rotational inertia and Newton’s second law

I_point = mr² · τ_net = Iα

I is measured in kg·m². Mass far from the axis raises I steeply because of the square on r.

Worked example

A net torque of 12 N·m acts on a wheel with rotational inertia 4 kg·m². Find its angular acceleration.

1.Use the rotational second law: τ_net = Iα.
2.Rearrange for α: α = τ_net ÷ I.
3.Substitute: α = 12 ÷ 4.
4.Compute: α = 3 rad/s².

Answer: α = 3 rad/s²

Watch out

Equal mass does not mean equal rotational inertia. Move that mass outward and I grows with the square of the distance. Always ask where the mass is, not just how much there is.

Checkpoint

A net torque of 20 N·m acts on a wheel with rotational inertia 5 kg·m². What is its angular acceleration?

Checkpoint

Two objects have equal mass. Object A is a hoop with its mass at the rim; object B is a solid disk with mass spread throughout. Which has the greater rotational inertia about its center?

On the exam

On the AP formula sheet you are given I for standard shapes (hoop MR², solid disk ½MR², sphere ⅖MR²). You are not expected to derive them — just choose the right one and remember the pattern: mass farther out gives a bigger coefficient.

Answer the 2 checkpoints as you read.