Physics

Problem

For a rigid body of total mass

M

, the rotational inertia about an axis located a perpendicular distance

d

from an axis that passes through the body’s center of mass is given by which of the following expressions? a)

I = I_{cm} - M d^{2}

I = I_{cm} + \dfrac{1}{2} M d^{2}

I = I_{cm} + M d^{2}

I = M d^{2}

Intuitive explanation

This is the parallel axis theorem, which connects rotational inertia about two different axes. The key insight is that rotating about an off-center axis is equivalent to rotating about the center of mass plus the effect of the entire mass concentrated at that distance—so you add a term $Md^2$ to account for the "extra" rotational resistance from being farther away. Think of it as: inertia shifts outward when you move the rotation axis away from the center.

Intuitive explanation

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