Principle Axes and Euler's Equation

AI Thread Summary
To find the force on each bearing of a flat rectangular plate rotating about an axle through its diagonal corners, one must calculate the moments of inertia using the three principal axes. The angular momentum L can be expressed as L = I * ω, where I is the inertia tensor and ω is the angular velocity. The change in angular momentum, dL/dt, corresponds to the external torque generated by the forces acting on the bearings. The expected result for the force is F = (M * ω^2) / (10 * √5). Understanding the relationship between angular velocity, moment of inertia, and torque is crucial for solving this problem.
Ed Quanta
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A flat rectangular plate of Mass M and sides a and 2a rotates with angular velocity w about an axle through two diagonal corners. The bearings supporting the plate are mounted just at the corners. Find the force on each bearing.

I am not sure how to find force using Euler's equations since they just relate angular velocities and moments of inertia. The answer is supposed to be F=maw^2/10*sqrt5. Anyone know how I use the 3 principle axes to solve this?
 
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Yes, you relate angular velocity and moment of inertia, but that's the point! Because now you can find the m. of in. as a function of time, L(t). From that, you compute dL/dt which is equal to the exterior torque produced by the two forces on the bearings.

Bruno
 
Sorry, I've made a mess with moments of inertia/angular momentum. Anyway, my answer remains more or less valid: Find the moments of inertia in the system of the three princple axes to get the tensor of inertia Î, then L=Î*omega, and... <look above>
 
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