A space station shaped like a giant wheel has a radius 99 m and a moment of iner

Question

A space station shaped like a giant wheel has a radius 99 m and a moment of inertia of 4.93 ? 108 kg

A space station shaped like a giant wheel has a radius 99 m and a moment of inertia of 4.93 ? 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 acceleration is experienced by the managers remaining at the rim? Assume that the average mass of each inhabitant is 65.0 kg.

Explanation / Answer

Centripetal Acceleration

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

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 = 4.93*10^8 / 99^2 - 150*65 = 40550.98 kg

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

I2 = (40550.98+50*65)*99^2 = 4.2*10^8 kg m^2

this the new angular velocity is

w2 = 4.9*10^8*sqrt(9.8/99) / 4.2*10^8 = 0.367 rads/s

and the new acceleration is

ac2 = 99*(0.367)^2 = 13.33 m/s^2

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