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?.

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?.

fretje · Answer

ok so: the angular momentum in the gyroscope after power up is
L = Ig . omega = 20 kg*m^2 . 100 rad/s = 2*10^3 kgm^2/s
this is the same as the angular momentum of the spacecraft after powering up the gyroscope, but inversed in sign.
so, we get the angular speed of the spacecraft
omega_spacecraft = L/Is = 2*10^3kgm^2/s / 5*10^5 kg*m^2 
is = 0.004 rad/s
time needed is (30 degrees is pi/6 rad) 
t = pi/6 / omega_spacecraft = pi/6 / 0.004 rad/s = 130,9 seconds