The Parallel-Axis Theorem Shifts a Moment of Inertia

The Parallel-Axis Theorem Shifts a Moment of Inertia

A rigid body resists a change in rotation. That resistance is its moment of inertia. Tables usually give the value about an axis through the centre of mass. Real problems often use a different parallel axis. The parallel-axis theorem makes that shift.

The theorem is compact. If $I_{\mathrm{cm}}$Icm​ is the moment about a centre-of-mass axis, and $I$I is the moment about a parallel axis a distance $d$d away, then

$$I = I_{\mathrm{cm}} + Md^{2}$$I=Icm​+Md2

Here $M$M is the mass of the body. The extra term $Md^{2}$Md2 is never negative. Therefore the moment about any parallel axis is larger than the centre-of-mass value.

The idea is geometric. Mass farther from the chosen axis contributes more. Moving the axis off the centre of mass increases those distances on net. The formula captures that increase without a fresh integration.

The conditions matter. The two axes must be parallel. They must also be fixed relative to the body in the usual rigid-body treatment. The theorem does not apply to a random pair of skewed axes. It also does not replace the need for $I_{\mathrm{cm}}$Icm​ itself.

Students use it constantly. A rod may swing about one end, not its midpoint. A disc may rotate about a rim point. A door rotates about its hinges. In each case, start from the known centre-of-mass result. Then add $Md^{2}$Md2.

The perpendicular-axis theorem is a different tool. It relates moments about three mutually perpendicular axes in a plane lamina. Do not mix the two. The parallel-axis theorem only moves an axis sideways. It keeps the direction the same.

Limits still exist. A continuous object with an unknown $I_{\mathrm{cm}}$Icm​ still needs integration or a standard result first. Composite bodies need a sum of pieces. Each piece can use the theorem about the common axis.

The practical gain is speed. One standard value plus a distance term solves many rotation problems. Energy, torque, and angular momentum all need the correct $I$I. The parallel-axis theorem supplies that value when the axis is parallel, but not central.