A space station shaped like a giant wheel is located far from anything else in s

Question

A space station shaped like a giant wheel is located far from anything else in space, and has a radius of 100.0m. When unoccupied, it has a moment of inertia I = 5.000 × 108 kg m2. A crew of 150 is living on the rim, and the space station’s rotation causes the crew to experience an apparent “gravity” (in reality, their centripetal acceleration, as seen by an inertial observer) of 1 g. When 100 people move to the center of the station for a union meeting, the angular speed changes. Assume the average inertia of each inhabitant is 65kg. It is also very safe to assume that the average person is very short, compared to 100m. What apparent “gravity” (centripetal acceleration) is experienced by the managers remaining at the rim?

Explanation / Answer

Moment of Inertia of the empty wheel

    I = 5.0x108 kg-m^2 = 540 kg-m^2

Inertia of each person = 65 kg

Radious r = 100 m

Number of persons = 150

Moement of inertia of the wheel with occupants at the rim

                 I1 = 540 + 150*65*(100)^2 = 975.0054x10^5

Moment of inertia with 100 people moving to the center

               I2 = 540 +50*65*(100)^2 = 325.0054x10^5

Moment of inertia of 100 people at the center =0 as r=0

Let initial Angular speed =

Centripetal force F = m ^2 r = 1g

Let 1 be the new angular speed

As there is no external force the rotational KE will remain same

0.5*I1* ^2 = 0.5*I2* 1^2

New angular speed

1^2 = I1 ^2/I2

new apparent gravity Fn = m *1^2*r = (mr)* I1 ^2/I2

                                               = 1g *I2/I1,   ( ^2 = 1g/mr)

                                                 = 1g*(325.0054x10^5)/(975.0054x10^5)

                                                 = 0.334g

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