Inertia Problem Solution - Finding Torque on a Spinning Block

[ehrenfest]

[SOLVED] inertia problem

Homework Statement

A uniform mass m and dimensions a by 2a by 3a spins about a long diagonal with angular velocity \omega. Use a coodinate system with origin at the center of the block. Find the magnitude of the torque that must be exerted on the block if the angular velocity is constant in magnitude and direction.

Homework Equations

The Attempt at a Solution

I found the principal moments of inertia. They are:
\left(a^2\frac{m}{12}13, a^2\frac{m}{12}10, a^2\frac{m}{12}5\right)
I let \vec{\omega} be \omega(1,2,3)/\sqrt{14}.
Now I believe the angular velocity is:
\vec{L} = \frac{m a^3 \omega}{12 \sqrt{14}} (13,20,15)
Those are the components of that vector in the body system, right? I am confused about what to do next. If I differentiate w.r.t time, it seems like I get zero. But I guess I need to differentiate the basis vectors as well and that I am rather unsure of how to do... Anyone have any ideas? Am I right so far?

 

[pam]

\frac{d{\vec L}}{dt}={\vec\omega}\times{\vec L}.