Find the principal moments of inertia of a flat rectangular plate

[Anro]

Hello everyone; here is the problem that I’m currently working on:
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a- Find the principal moments of inertia of a flat rectangular plate (mass = 30g, a = 80 mm, b = 60 mm) that rotates about a diagonal with velocity ω = 15 rad/s.

b- What are the components of angular momentum parallel to the edges?

c- Given that the plate is mounted on an axis of length 120 mm, which is held vertical by bearings at its ends, what is the horizontal component of the force on each bearing?
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For part (a) I will set up a coordinate frame attached to the plate with the origin at the center of mass (assuming the plate is uniform), the x-axis parallel to b = 60 mm, the y-axis along the normal, and the z-axis parallel to a = 80 mm. Thus my x, y, and z axes are my principal axes, and then I can find the principal moments of inertia by finding I(xx), I(yy), and I(zz). Here is where I’m stuck on this part:
When setting up the integrals they will go like this:
ρ int dx int dy int dz
where ρ is the density = M/V, my problem with finding V is that I’m not given a thickness “t;” just height and width. So how do I go about that?
Also, if I draw a figure I can see that my limits of integration for the dx integral are from zero to b/2, the dz integral will be from zero to a/2, but I’m not sure how to go about the limits of integration for the dy integral.
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For part (b) all I need to do is to set up L = Iω in matrix form, where ω is ω(x) = 15 rad/s, ω(y) = 0, and ω(z) = 15 rad/s. Am I correct in this assumption?
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For part (c) I’m not sure how to proceed other than finding the torque, which is dL/dt, but even then I don’t know how to find it. Also, I’m not sure if there is another force that I have to find, and if so, then how to find the horizontal component of this force (totally confused here).
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Thanks for any help, and apologies for the long post.

 

[Anro]

Update:
I was able to get the inertia tensor by finding its individual components separately for I(xx), I(yy), and then using the perpendicular axes theorem to find I(zz); it turned out that I was wrong about my original approach for part (a). So now I did parts (a) and (b), but still looking for a solution for part (c).

 

[Korryn]

I think Dr. Lath expects you to use an area density, like sigma = 6.25 kg/m^2. Then the double integral of (sigma)dA would yield the plate mass, and no dz would be required.