5.4 Rotational Inertia

What is rotational inertia?

Rotational inertia (moment of inertia, I) measures how hard it is to change an object's rotation rate. It depends on both mass and how that mass is distributed relative to the axis: I = Σmr² for point masses, or I = ∫r² dm for continuous bodies. Mass farther from the axis contributes more, since r is squared.

Common shapes and formulas

Standard objects have known formulas: solid sphere I = (2/5)MR², solid cylinder/disk I = (1/2)MR², thin rod about center I = (1/12)ML², thin rod about end I = (1/3)ML², thin hoop/ring I = MR². These come from integrating r² dm over the object's geometry.

Parallel axis theorem

To find I about an axis not through the center of mass, use the parallel axis theorem: I = I_cm + Md², where d is the distance between the new axis and the center-of-mass axis. This lets you reuse known I_cm formulas for shifted rotation axes without re-integrating.

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, I = ∫r² dm.

Solid cylinder/disk about central axis

I = (1/2)MR²

Thin rod about center vs. end

About center: I = (1/12)ML²; about end: I = (1/3)ML².

Parallel axis theorem

I = I_cm + Md², used to find I about an axis parallel to and a distance d from the center-of-mass axis.

Thin hoop/ring about central axis

I = MR², since all mass lies at radius R from the axis.

Try an AP-style question

A uniform rod of mass M and length L rotates about one end. What is its rotational inertia?

1. ML²/12
2. ML²/2
3. ML²/3
4. ML²