A.4.7 (HL)—Point-mass moment of inertia

Point-mass model

For discrete masses rotating about an axis,

$$I=\sum mr^2$$

where each $r$ is the perpendicular distance from the axis.

Build the sum

Treat each small sphere, blade or mass element separately, calculate $mr^2$, and add the contributions. Use symmetry when identical masses have equal radii.

Worked example from local Question Bank row 128743

Two $10\,\mathrm{kg}$ point masses are $8.0\,\mathrm{m}$ apart and rotate about the midpoint. Each is $4.0\,\mathrm{m}$ from the axis:

$$I=\sum mr^2=2(10)(4.0)^2=320\,\mathrm{kg\,m^2}$$

The full $8.0\,\mathrm{m}$ separation is not the radius of either mass.

Common trap

Do not use the distance between two masses as $r$ for both; use each mass’s distance to the rotation axis.