04 / CHAPTER 04
The parallel-axis theorem
Move an inertia calculation between useful reference axes.
ANCHOR EQUATION
Parallel-axis theorem
In symbols: I = Icm + Md²
Shift an inertia about the center of mass to a parallel axis.
The parallel-axis theorem shifts a known inertia
If an axis moves a distance d away from the center-of-mass axis without changing direction, every mass element gains an additional squared distance and a cross term that cancels by the center-of-mass definition.
Expand the squared distance
Let r = r′ + d, where r′ is measured from the center of mass. Then I = ∫|r′ + d|²dm = ∫r′²dm + 2d·∫r′dm + d²∫dm. The middle term is zero because ∫r′dm = 0, leaving I = I_cm + Md².
A 3 kg uniform rod of length 1.2 m has I_cm = ML²/12 = 0.36 kg·m². About an end, d = 0.6 m, so I_end = 0.36 + 3(0.36) = 1.44 kg·m².
Only parallel axes qualify
The theorem does not rotate the axis direction or replace a full tensor calculation for arbitrary three-dimensional axes. The center-of-mass reference is special because it removes the cross term.
Work it: Verify the rod result by integrating I = ∫x² dm from x = 0 to L. Then compare the end-axis inertia with the point-mass estimate ML².