Rotational Inertia
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
Rotational inertia I is how hard it is to change something’s rotation, just as mass is for straight-line motion. It depends on the mass and how far that mass is from the axis: for a single point mass, I = mr².
1. Where the mass is matters
2. It depends on the axis
Because r is measured from the axis, the same object has different rotational inertias about different axes. A rod is much easier to spin about its middle than about one end.
Rotational inertia doesn’t depend on how fast something spins. It only depends on the masses and their distances from the axis.
A figure skater who pulls in her arms brings mass closer to the axis, which lowers her rotational inertia.
3. Example, then try it
Two 2.0 kg masses sit on a light rod, each 0.50 m from the axis at the center. Find I. What if each moves out to 1.0 m?
I = 2 × (2.0)(0.50)² = 1.0 kg·m².
At 1.0 m: I = 2 × (2.0)(1.0)² = 4.0 kg·m². Twice the distance, four times the rotational inertia.
A light 1.0 m rod has a 1.0 kg ball at each end. Find its rotational inertia (a) about the middle and (b) about one end.
(a) Middle: each ball is 0.50 m away: I = (1.0)(0.50)² + (1.0)(0.50)² = 0.50 kg·m².
(b) One end: one ball is on the axis (r = 0); the other is 1.0 m away.
I = (1.0)(0)² + (1.0)(1.0)² = 1.0 kg·m².
Answer: 0.50 kg·m² about the middle and 1.0 kg·m² about an end: the same object, twice as hard to spin about the end.
4. Common mistakes
A spinning figure skater pulls her arms in close to her body. What happens to her rotational inertia?
Pulling her arms in moves some of her mass closer to the axis, so Σmr² gets smaller. (In Unit 6 you’ll see why that makes her spin faster.)
Ready to practice?
5 questions on Rotational Inertia, each with a hint and an explanation for every wrong answer.