$3$ Problem $3(25$ points$)$

A flat rectangular plate has mass $\backslash ($ m $\backslash )$ and dimensions $\backslash (2$ a $\backslash$times a $\backslash ).$ There is an axle passing through bearings located at diagonal corners. It is set into rotation with constant angular velocity $\backslash (\backslash$omega $\backslash ).$

$1.$ The principal axes are obvious. What are the three moments of inertia?

$2.$ Using Euler's equations calculate the torque supplied by the bearings.

$3.$ From the previous results calculate the force on the bearings.