Moments of Inertia Fun with math § 9.5–9.6
Parallel-Axis Theorem The moment of inertia of an object of mass M whose center of mass is distance d from the rotation axis is I = Icm + Md2 where Icm is its moment of inertia about the parallel axis passing through the object’s center of mass
Parallel-Axis Theorem Prove 1/2 (Icm + Md2)w2 = K = 1/2 Icmw2 + 1/2 M(dw)2 = 1/2 Icmw2 + 1/2 Mvcm2 1/2 Icmw2 = Krot in com frame 1/2 Mvcm2 = “K of com”
More General Result K = K' + 1/2 Mvcm2 K' = S [1/2 mi (vi – vcm)2] Parallel-axis theorem is a corollary
I by Integration I = ∫r2dm Examples Thin bar about middle Thin bar about end directly with parallel-axis theorem Cylinder about principal axis