Moments of Inertia

1
Moments of Inertia

• Fun with math

9.59.6
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 objects 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

• 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