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Rotational Inertia

Topic 5.3 named the rotational input, torque. Topic 5.4 names the rotational resistance: rotational inertia, the rotational analog of mass. Force changes velocity, and a body's mass sets how much velocity-change a given force buys. Torque changes angular velocity, and a body's rotational inertia $I$ sets how much.

The catch: where mass is a property of an object alone, $I$ is a property of an object about an axis. The same body has different $I$ values about different axes, and the way mass is distributed matters at least as much as the total mass. The headline definition for a system of point masses, $I = \sum_i m_i r_i^2$, makes both factors plain: mass enters linearly, distance from the axis enters squared.

Fig. 5.4.1 Two objects, same mass ($M = 3$ kg) and same outer radius ($R = 2$ m), about their central axes. All the hoop's mass sits at the maximum distance from the axis, so each kilogram contributes the maximum $r^2$. On the disk, mass is spread from $r = 0$ outward, and the inner mass contributes very little. The factor-of-two gap is geometry, not mass.

Inertia Lab · Open the sandbox →

Two common traps. First, deciding which body has more $I$ by looking at one factor in isolation: heavier wins ignoring the geometry, or more spread out wins ignoring the mass. The formula has both factors and neither one decides on its own. Second, treating $I$ as a property of the body alone. The same rod about its center versus about one end has rotational inertias that differ by a factor of four. Specify the axis before you quote a rotational inertia.

The work

3 ways in · any order

Lesson

Rotational Inertia ›

Build $I$ from scratch: the definition $I = \sum m_i r_i^2$, the five standard shape formulas from the AP equation sheet, the axis-dependence of $I$, and the parallel-axis theorem. Closes with a ten-scenario applet that targets the mass-and-distribution trap and the axis-independence trap.

Skill check · 10 scenarios

Open lesson →

Diagnostic

10-item topic check ›

Ten items, mixed symbolic, factor-of-change, quantitative, and conceptual. Each wrong-answer chip is tagged with a specific misconception code; whichever traps fire seed your targeted-practice queue. Plan on about 15 minutes.

Not started · 10 items · ~15 min

Start diagnostic →

Targeted Practice

Drill a single misconception ›

Pick one of the failure modes you've missed and grind it on its own. The round is adaptive: two correct in a row clears the misconception and you move on.

Take the diagnostic to identify your misconceptions

Diagnostic first