Rotational inertia

Also called Moment of inertia

Rotational inertia measures how strongly a rigid system resists a change in its rotation. It depends on the total mass and on how that mass is distributed relative to the chosen axis, so a single object has many values, one for each axis.

Mass says how hard something is to push. Rotational inertia says how hard it is to spin up, and unlike mass it is not a fixed property of the object. EK 5.4.A.1: it measures a rigid system's resistance to changes in rotation and is related to the mass of the system and the distribution of that mass relative to the axis of rotation. It appears in $\alpha_{\text{sys}} = \tau_{\text{net}}/I_{\text{sys}}$ , in $K = \frac{1}{2}I\omega^2$ and in $L = I\omega$ . Units are $\text{kg}\cdot\text{m}^2$ .

The axis is part of the answer. A meter stick spun about its center and the same stick spun about one end have different values. Quote an axis, or the number means nothing.

For a point mass a perpendicular distance $r$ from the axis, EK 5.4.A.2 gives $I = mr^2$ , and for a collection of objects the values simply add:

I = \sum m_i r_i^2

AP Physics C: Mechanics extends this to continuous bodies, and its equation sheet prints the integral form:

I = \int r^2\,dm

What AP Physics 1 expects. The Topic 5.4 boundary statement limits calculations to systems of five or fewer objects in a two-dimensional configuration, and says students do not need to know the rotational inertia of extended rigid systems because those are provided on the exam. What is required is the qualitative rule: mass farther from the axis counts for more, which is why a hoop has more rotational inertia than a solid disk of the same mass and radius.

Moving between axes is the job of the parallel axis theorem.

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