1. Moments of Inertia
Fun with math
§ 9.5–9.6

2. 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

3. 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”

4. More General Result K = K' + 1/2 Mvcm2 K' = S [1/2 mi (vi – vcm)2]
Parallel-axis theorem is a corollary

5. I by Integration I = ∫r2dm Examples Thin bar about middle
Thin bar about end
directly
with parallel-axis theorem
Cylinder about principal axis