Question

Rockets can be spin-stabilized; that is, they can fly straighter through space if they rotate around their long axis. Generally, 1 rotation per second is a good angular speed for spin-stabilization.

a. Assume the rocket acts like a solid cylinder. The rocket has a mass of 30000 kg (almost all of it fuel) and a radius of 1.00 m. Calculate the angular momentum of the rocket as it spins. Pay attention to units.

b. A problem arises because fuels for larger rockets are liquid. Due to the centripetal effect, the fuel is pushed to the outer skin of the rocket. So now the rocket acts more like a thin-walled hollow cylinder with a mass of 30000 kg and a radius of 1.00 m. Calculate the new angular velocity of this rocket, assuming that it is the same rocket as in part a. Describe what happened to the rocket’s spin.

Answer 1

a.Moment of inertia of solid cylinder about its longitudinal axis is given by mr²/2 where m is mass of the cylinder and r is its radius.

Mass of rocket m=30000 kg , radius of rocket r=1 m

So,moment of inertia of rocket=30000*1*1/2=15000 kg-m²

Also,angular velocity of rocket =w=1 rotation per second =2 rad/s

Angular momentum is given by:L=I*w,where L is angular momentum, I is moment of inertia, w is angular velocity.

So, angular momentum of given rocket=I*w=15000*2=94247.78 kg-m²/s

b..Moment of inertia of hollow cylinder about its longitudinal axis is given by mr² where m is mass of the cylinder and r is its radius.

Mass of rocket m=30000 kg , radius of rocket r=1 m

So,moment of inertia of rocket=30000*1*1=30000 kg-m²

Let angular velocity of the rocket be w_f

So,angular momentum=I*w=30000*w_f

Since, there are no external torques on the rocket, angular momentum is conserved.

Initial angular momentum=final angular momentum

=>94247.78=30000*w_f=>w_f=94247.78/30000=3.14 rad/sec= rad/sec=half rotation per second

So,Angular velocity is decreased as a result of increased moment of inertia.

Spin would not remain stable because angular velocity is half rotation per second, while for stable spin angular velocity of 1 rotation per second is required.