# Euler equations

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1. Nov 9, 2014

### PeteSampras

"A rigid lamina (i.e. a two dimensional object) has principal moments of inertia about the centre of mass given by $I_1=u^2-1$, $I_2=u^2+1$, $I_3=2u^2$

Choose the initial angular velocity to be $ω = µN \hat{e_1} + N \hat{e_2}$. Define tan α = ω2/ω1,
which is the angle the component of ω in the plane of the lamina makes with e1. Show that it satisfies:
$\ddot{α}+ N^2 \cos α \sin α = 0$"

(the problem does not says what is N). The problem is on http://www.damtp.cam.ac.uk/user/tong/dynamics/mf3.pdf

I tried used the Euler equation, considering that $N_1=N_2=0$

My Euler equation are:

$\dot{\omega_1}+ \omega_2 \omega_3=0$
$\dot{\omega_2}- \omega_1 \omega_3=0$
$2 \mu^2 \dot{\omega_3}+2 \omega_1 \omega_3 =N_3$

I tried of several ways of combining this equations , using that tan α = ω2/ω1, but i do not get to $\ddot{α} + N^2 \cos α \sin α = 0$.

Some Hint to do this exercice this????...

2. Nov 14, 2014

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Nov 15, 2014

### Staff: Mentor

I don't know whether this helps, but the first two equations imply that

x2=C sinα

x1=C cosα

where C is a constant.

Also,

x3=α'

Chet