A space station shaped like a giant wheel has a radius of 110 m and a moment of

Question

A space station shaped like a giant wheel has a radius of 110 m and a moment of inertia of 4.82 108 kg · m2. A crew of 150 lives on the rim, and the station is rotating so that the crew experiences an apparent acceleration of 1g. When 100 people move to the center of the station for a union meeting, the angular speed changes. What apparent acceleration is experienced by the managers remaining at the rim? Assume that the average mass of each inhabitant is 65.0 kg.

Explanation / Answer

You need to use conservation of angular momentum to solve this problem. To have 1g force experienced by the people at the rim, you need a centripetal acceleration of 1g is needed. This means that the angular velocity is given by

ac = R w^2

w = sqrt (ac/R)

The angular momentum of the station is

L = I w

as the people move to the center of the station, the moment of inertia will change and, thus, the angular velocity will change to keep the angular moment constant, that is

L = I1 w1 = I2 w2

thus

w2 = I1 w1 / w2

I will assume that the space station looks like a hoop. The moment of inertia of a hoop is given by

I = mR^2 = (mp+ms)R^2

where mp is the mass of the people and ms is the mass of the station. Using the given moment of inertia, the mass of the station is given by

ms = I/R^2 - mp = 4.82E8 / 110^2 - 150*65 = 30,219 kg

thus the moment of inertia after 100 people move to the center is

I2 = (30219 +50*65)*100^2 = 3.3469E8 kg m^2

this the new angular velocity is

w2 = 4.82E8*sqrt(9.8/110) / 3.3469E8 = 0.429 rads/s

and the new acceleration is

ac2 = 110*(0.429)^2 = 20.244 m/s2 = 2.02 g

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