- #1
B3NR4Y
Gold Member
- 170
- 8
Homework Statement
Consider an arbitrary rigid body with an axis of rotational symmetry, which we'll call ## \hat z ##
a.) Prove that the axis of symmetry is a principal axis. (b) Prove that any two directions ##\hat x## and ##\hat y ## perpendicular to ##\hat z ## and each other are also principal axes. (c) Prove that the principal moments corresponding to these two axes are equal: λ1=λ2
Homework Equations
The moment of inertia tensor for a rigid body is ## I_{ij} = m_\alpha (r_\alpha \delta_{ij} -r_{\alpha, \, i} r_{\alpha, \, j} )##
For a continuous mass distribution ## I_{ij} = \int dV \rho(\vec r) (r^2 -r_i r_j ) ##
If an axis is a principal axis (eigenvalue of the inertia tensor), then the inertia tensor about the principal axes is diagonally λ1 λ2 λ3, and elsewhere zero.
The Attempt at a Solution
I'm not sure where to start proving that an axis of rotational symmetry is a principal axis. I'd imagine I'd have to work out the arbitrary λ for each axis, but this seems like it would be really ugly and I think there should be a slicker way to do it but I can't find it.