Classical Dynamics of Particles and Systems · Problem 11-31

Coupled Oscillations — Marion & Thornton

Physics — Mechanics · thin-plate-force-free-rotation

Problem

Consider a thin homogeneous plate with principal momenta of inertia

\begin{aligned} &I_1 \quad \text{along the principal axis } x_1\\ &I_2 > I_1 \quad \text{along the principal axis } x_2\\ &I_3 = I_1 + I_2 \quad \text{along the principal axis } x_3 \end{aligned}

Let the origins of the

x_i

and

x_i'

systems coincide and be located at the center of mass

O

of the plate. At time

t = 0

, the plate is set rotating in a force-free manner with an angular velocity

\Omega

about an axis inclined at an angle

\alpha

from the plane of the plate and perpendicular to the

x_2

-axis. If

I_1/I_2 \equiv \cos 2\alpha

, show that at time

t

the angular velocity about the

x_2

-axis is

\omega_2(t) = \Omega\cos\alpha\tanh(\Omega\, t\sin\alpha)