Question 3

A space station consists of a uniform circular disc. Initially it rotates with angular velocity $\omega_3$ about the $x_3$-axis, perpendicular to the plane of the disc and passing through its centre. At $t=0$booster rockets fire, applying a constant torque $\tau$ about the $x_2$-axis (in the plane of the disc).

Show that, in the body-fixed system, the angular velocity of the station varies as

$$\omega_1 = -{\tau\over I_1\Omega}(1-\cos\Omega t)$$

$$\omega_2 = {\tau\over I_1\Omega}\sin\Omega t$$

$$\omega_3 = \hbox{constant}$$

and find an expression for $\Omega$.

[21 marks]

Calculate the angular momentum of the station at $t=\pi/\Omega$.

[4 marks]

[You may use the fact that Euler's equations for a rotating body have the forms

$$I_1\dot\omega_1+(I_3-I_2)\omega_2\omega_3 = \tau_1$$

$$I_2\dot\omega_2+(I_1-I_3)\omega_3\omega_1 = \tau_2$$

$$I_3\dot\omega_3+(I_2-I_1)\omega_1\omega_2 = \tau_3$$

where $I_1$, $I_2$ and $I_3$ are the principal moments of inertia and $\hbox{\boldmath$\tau$ }$ is the torque on the body.]

[Hints can be found here, but do not follow this link until you have attempted the question on your own.]