Thin Plate, Moment of Inertia

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Homework Statement


Consider a thin homogeneous plate with a principal momenta of inertia. I1 along the principal axis x1, I2 > I1 along the principal axis x2. I3 = I1 + I2.

Let the origins of the xi x'i systems coincide and be located at the center of mass O about an axis inclined at an angle a from the plane of the plate and perpendicular to the x2 axis. If I1 / I2 = cos(2a), show that at time t the angular velocity about the x2 axis is

w2 = omega*cos(a)*tanh(omega*t*sin(a))


Homework Equations





The Attempt at a Solution



I am having a hard time starting this problem.
So we know that angular momentum and energy should be conserved, but that doesn't appear to help me at all.

I'm thinking that Euler's equations should probably be used (force free)

(I2 - I3)w2w3 - I1w'1 = 0

(I3 - I1)w3w1 - I2 w'2 = 0

(I1 - I2)w1w2 - I3 w'3 = 0

But plugging in I1 = I2*cos(2a) doesn't seem to yield anything..

Where does that tanh come from?

Anyone have any hints?
 
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Answers and Replies

  • #2
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Two things. First, this really needs a diagram. Second, is this the whole problem, or does it perhaps follow on from something else? The premise only talks about moments of inertia and orientations. There is nothing to suggest motion. Then the question asks about the velocity over time. Something is missing.
 
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