Why is the following situation impossible? A space station shaped like a giant w

Question

Why is the following situation impossible? A space station shaped like a giant wheel has a radius of r = 100 m and a moment of inertia of 5.00 108 kg · m2. A crew of 150 people of average mass 65.0 kg is living on the rim, and the station's rotation causes the crew to experience an apparent free-fall acceleration of g (see figure below). A research technician is assigned to perform an experiment in which a ball is dropped at the rim of the station every 15 minutes and the time interval for the ball to drop a given distance is measured as a test to make sure the apparent value of g is correctly maintained. One evening, 100 average people move to the center of the station for a union meeting. The research technician, who has already been performing his experiment for an hour before the meeting, is disappointed that he cannot attend the meeting, and his mood sours even further by his boring experiment in which every time interval for the dropped ball is identical for the entire evening.

Explanation / Answer

The total moment of inertia = I_n = 5*10^8 + 150*65*100^2 = 5.975*10^8 kg·m^2

To experience the acceleration equal to g, the value of ^2r must be equal to 9.81. i.e. ^2* 100 = 9.8m/s^2

Hence = 0.313 rad/s.

When only 50 people remain on the rim,

The new moment of inertia of the system will bw:

I_mtg = 5*10^8 + 50*65*100^2 = 5.325*10^8 kg·m²

But here the momentum will be conserved since there is no external iimpulse.Hence

(I * )_mtg = (I * )_n

5.325 * = 5.975*0.313

or = 0.351 rad/s

So the acceleration will be a = ^2 r = 0.351^2 * 100 = 12.34 m/s^2

This value is greater that the acceleration g = 9.81 which is about 26% higher.

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