Question

2. A communications satellite is spinning out of control and you are in charge of making it better again! It has Not mass Msat, radius R, length L, and can be modeled as a spinning solid cylinder. Fig. 2 Fig. 1 The satellite should be spinning with period Tspin, but it is actually taking time Tspin/4 to complete one revolution. It's spinning too fast! Spins Mar together Your “good-on-paper” plan to slow the satellite down (so that its period is Tspin) is this: You will launch two identical non-rotating solid spheres of radius R (same KR radius as the satellite) toward the satellite. The spheres Not are not rotating initially. The spheres will gently collide spinning with and then grab on to the satellite, attaching permanently. This will slow the satellite down. As a result, the spheres (which were not spinning initially) will now be spinning at the same final rate as the satellite. After the collision, all objects will rotate as a unit with period Tspin. ART What mass Msphere must each sphere have? Simplify your answer as much as possible; the final result should not depend on the quantities Tspin or R or L, though your intermediate results might depend on some of them. (Hint for dealing with the combined object: Moments of inertia just add.)

Answer 1

In the first picture, the angular momentum is

27 Lsat = Istwo? = Moat RP (Dyna) - "Thin 32 Myat 7² R2

The final momentum is

Lfina = ('1sat + 2 sphere)o? = (3Nosat + Msphere) R+ x (.)

This is equal to the initial momentum.

Therefore

(fara tusenere) #® (2) - 3201 Isnin

or

$4\left(\frac{1}{2}M_{sat}+\frac{4}{5}M_{sphere} \right )=32 M_{sat}$

or

(16/5) Msphere = 30M sat

or

Msphere = Msat