Introduction to Classical Mechanics · Problem 9.35

Angular momentum, Part II (General L) — Morin

Physics — Mechanics · tilted-moments

Problem

((a)) Consider a planar object in the

x

y

plane. If the

x

and

y

axes are principal axes, use the rotation matrix in Eq. (9.94) to show that the moment of inertia around the

x'

axis, which makes an angle

\theta

with the

x

axis, is

I_{x'} = I_x \cos^2\theta + I_y \sin^2\theta

. ((b)) Consider a general three-dimensional object whose principal axes are the

x

y

, and

z

axes. Consider another axis that points along the unit vector

(\alpha, \beta, \gamma)

. Show that the moment of inertia around this axis is

\alpha^2 I_x + \beta^2 I_y + \gamma^2 I_z

. Hint: The cross product, discussed in Appendix B, provides a nice method of calculating the distance from a point to a line.