A space station is constructed in the shape of a hollow ring of mass 5.45 times

Question

A space station is constructed in the shape of a hollow ring of mass 5.45 times 10^4 kg. Members of the crew walk on a deck formed by the inner surface of the outer cylindrical wall of the ring, with radius 110 m. At rest when constructed, the ring is set rotating about its axis so that the people inside experience an effective free-fall acceleration equal to g. (See figure below.) The rotation is achieved by firing two small rockets attached tangentially to opposite points on the rim of the ring. (a) What angular momentum does the space station acquire? kg middot m^2/s (b) For what time interval must the rockets be fired if each exerts a thrust of 140 N? s

Explanation / Answer

PART A:

The Moment of Inertia should be I = m*r^2 (circular hoop)
I =5. 45*10^4kg * 110^2 m^2 = 6.59*10^8 kg.m^2

Centripetal acceleration must be 9.8 m/s^2
so v^2 / r = r * ^2 = 9.8
so the required angular velocity is
= sqrt(9.8 / 110) rad/s = 0.298 rad/s

So the angular momentum is

L= I * = 6.59*10^8 kg.m^2*0.298 rad/s = 1.96*10^8 kg.m^2/s

PART B:
The applied torque is T = 2*140N * 110m = 30800 N.m
So the angular acceleration is
= T / I = 30800/6.59*10^8 = 4.67*10^-5 rad/s

= * t
so the required time is
t = / = 0.298/4.67*10^-5 = 6381 s (3 s.f.)

(about 1hr 46min 35s)

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