# Rigid Body Motion Proof

1. Dec 1, 2006

### ohsoconfused

1. The problem statement, all variables and given/known data
Consider a thin homogeneous plate with principal momenta of inertia
I1 along axis x1,
I2>I1 along x2,
I3 = I1 + I2 along x3

Let the origins of the x and x' systems coincide at the center of mass of the plate. At time t=0, the plate is set rotatint in a force-free manner with angular velocity Q about an axis inclined at an angle of a from the plane of the plate and perpendicular to the x2-axis. If I1/I2 = cos2a, show at time t the angular velocity about the x2-axis is:

w2(t) = Qcosa*tanh(Qtsina).

3. The attempt at a solution
I know that the kinetic energy and the square of the angular momentum are constant, but I'm not positive how to calculate them with the initial conditions given. Past that, the Euler equations simplify somewhat...

(for simplicity's sake, let [x] = the first time derivative of x)

[w2] = w3w1
[w1] = -w2w3
[w3] = w1w2*(I1-I2)/(I1+I2) = w1w2*(cos2a-1)/(cos2q+1)

I'm having trouble seeing how such a bizarre function can even arise, but I think my first problem is calculating KE and L squared in terms of the initial conditions, which I'm blank on.

Anybody help?

2. Dec 2, 2006

### vanesch

Staff Emeritus
The things I put in red give you a complete geometrical description of the rotation vector at t=0, expressed in the frame (x1,x2,x3) fixed to the rotating body (which is exactly what you need for Euler's equations).
Draw it or something: you'll find the initial values for the 3 components (for instance, you already know that the second component is 0, given that the rotation vector is perpendicular to it). Once you know those, you can fill it in in the expressions for E and M^2.

Suggestion:
If I understand the problem statement well, they GIVE you the solution and you simply have to show that it is correct - you don't need to derive it. So use the solution they give you, to show that you obtain a complete solution for w1 and w3 (derivable from the conservation of M2 and E and given w2), which satisfy the Euler equation.