Euler's Equations, a freely rotating lamina

[Oijl]

Homework Statement

Consider a lamina rotating freely (no torques) about a point O of the lamina. Use Euler's equations to show that the component of $$\omega$$ in the plane of the lamina has constant magnitude.

[Hint: Use the reults of Problems 10.23 and 10.30. According to Problem 10.30, if you choose the direction e3 normal to the plane of the lamina, e3 points along a principal axis. Then what you have to prove is that the time derivative of $$\omega$$$$_{1}$$$$^{2}$$ + $$\omega$$$$_{2}$$$$^{2}$$ is zero.]

Homework Equations

The result of 10.23 is that Izz = Ixx + Iyy
The result of 10.30 is that for a lamina rotating about a point O in the body, the axis through O and perpendicular to the plane is a principal axis.

Euler's equations, as given in my book, are
y1W1 - (y2 - y1)w2w3 = N1
y2W2 - (y3 - y1)w3w1 = N2
y3W3 - (y1 - y2)w1w2 = N3
where y is lambda, the eigenvalue, w is omega, W is omega dot, and N is the torque.

The Attempt at a Solution

If the lamina is rotating freely, the N1=N2=N3=0.
If I choose e3 to be normal to the plane of the lamina, then e3 points along a principal axis, and that means that

w3 = constant,
so
W3 = 0.

Those last two statements I'm not sure about.

However, I still know that I would like to prove that the time derivative of w1^2 + w2^2 = 0. That is,

2w1W1 + 2w2W2 = 0

And solving Euler's equations for W1 and W2, with N1,2,3 = 0, you have

W1 = ((y2-y3)/y1)w2w3
W2 = ((y3-y1)/y2)w3w1

Plugging these into 2w1W1 + 2w2W2 = 0 and doing algebra, I found that, if 2w1W1 + 2w2W2 = 0 is true, then

y2^2 - y2y3 + y1y3 - y1^2 = 0.

Which doesn't seem to necessarily be true.

Help? Thank you!

 

[brainpushups]

I don't believe that you can assume that w3 is constant. The problem does not state that the lamina is rotating about a principal axis. Instead, use the result you quoted ($\lambda_{3}$=$\lambda_{1}$+$\lambda_{2}$) and use Euler's equations to show that

$\omega_{1}$$\omega_{1}^{.}$+$\omega_{2}$$\omega_{2}^{.}$=0

Also, if the lamina WERE rotating about its principal axis that is perpendicular to the surface then the angular velocity would always point along this direction in the absence of torques and$\omega_{1}$ and $\omega_{2}$ would be zero.

 

[brainpushups]

Oops. Those dots were supposed to be above omega one and omega two. I'm sure you get the idea though.