Question

A spacecraft is in empty space. It carries on board a gyroscope with a moment of inertia of Ig=20 kg*m^2 about the axis of the gyroscope. The moment of inertia of the spacecraft around the same axis is Is= 5*10^5 kg*m^2. Neither the spacecraft nor the gyroscope is originally rotating. They gyroscope can be powered up in an negligible period of time to an angular speed of 100 rad/s. If the orientation of the spacecraft is to be changed by 30 degrees, for what time interval should the gyroscope should be operated?.,

Answer 1

ws = wg*[Ig/Is] = 0.0048 rad/sec 

t = Θ/ws = (π/6)/.0048 = 109.1 sec

Answer 2

The time interval to operate gyroscope is  131 seconds.

Given:

Moment of inertia of gyroscope $\bold{= I_g = 20 \ kg.m^2}$

Moment of inertia of the spacecraft $\bold{= I_s = 5 \times 10^5 \ kg.m^2}$

Angular speed of gyroscope $\bold{= \omega_g = 100 \ rad/s}$

Orientation of the spacecraft $\bold{= \theta_s = 30 \ degrees = 0.523 \ radian}$

To find:

Time interval to operate gyroscope = ?

Formula used:

$\bold{I_s \omega_s = I_g \omega_g}$

Where,

$\bold{I_s}$ = Moment of inertia of spacecraft

$\bold{I_g}$ = Moment of inertia of gyroscope

$\bold{\omega_g}$ = Angular speed of gyroscope

$\bold{\omega_s}$ = Angular speed of spacecraft (Not Given)

$\bold{\omega_s = \frac{\theta_s}{t}}$

Where,

$\bold{\theta_s}$ = Orientation of the spacecraft

t = Time interval

Solution:

$\bold{I_s \omega_s = I_g \omega_g}$

$\bold{I_s (\frac{\theta_s}{t}) = I_g \omega_g}$

$\bold{t = \frac{ I_s \times \theta_s }{ I_g \times \omega_g }}$

$\bold{t = \frac{5 \times 10^5 \times 0.523}{20 \times 100}}$

$\bold{\therefore t = 131 \ s}$