A space station is to provide artificial gravity to support long-term stay of as

Question

A space station is to provide artificial gravity to support long-term stay of astronauts and cosmonauts. It is designed as a large wheel, with all the compartments in the rim, which is to rotate at a speed that will provide an acceleration similar to that of terrestrial gravity for the astronauts (their feet will be on the inside of the outer wall of the space station and their heads will be pointing toward the hub). After the space station is assembled in orbit, its rotation will be started by the firing of a rocket motor fixed to the outer rim, which fires tangentially to the rim. The radius of the space station is R = 54.1 m, and the mass is M = 4.67 · 105 kg. If the thrust of the rocket motor is F = 141 N, how long should the motor fire?

Explanation / Answer

here,
Mass = m = 4.67 * 10^5 kg
Radius = r = 54.1 m
Thrust = F = 141 N

from second law of motion
F = mg
m*w^2* r = m * g

Solving for angulr velocity w,
w = sqrt(g/r)
w = sqrt(9.8/54.1)
w = 0.4256 rad/s

Moment of inertia of station:
I = 0.5 * M * r^2
I = 0.5 * 4.67 * 10^5 * 54.1^2
I = 683410135 kg.m^2

From work Energy Theoram we have :
Work done = rotational Kinetic Energy
F * R * t = 0.5 * I * w^2

solving for time t,
t = (0.5 * I * w^2) /(F*r)
t = (0.5*683410135*0.4256^2)/(141*54.1)
t = 8114.061 s or 2.254 hr

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