A cylindrically shaped space station is rotating about the axis of the cylinder

Question

A cylindrically shaped space station is rotating about the axis of the cylinder to create artificial gravity. The radius of the cylinder is 79.8 m. The moment of inertia of the station without people is 2.70 times 10^9 kg Middle Dot m^2. Suppose that 505 people, with an average mass of 67.9 kg each, live on this station. As they move radially from the outer surface of the cylinder toward the axis, the angular speed of the station changes. What is the maximum possible percentage change in the station's angular speed due to the radial movement of the people?

Explanation / Answer

Treating each person as a point mass [with M.o.I = mR²)].. their total M.o.I at the rim is ..
505 x [67.9kg x (79.8m)² ] = 2.18 x 10^8 kg.m²

Total M.o.I of station + people at the rim = (2.7 x 10^9 + 2.18 x 10^8) = 2.9 x 10^9 kg.m²

M.o.I of people at centre = 0 (as R = 0) .. Total M.o.I = 2.7 x 10^9kg.m² (station only)

By conservation of angular momentum ..
Mom.with people at rim (Ir x r) = mom. with people at centre (Ic x c)
Ir x r = Ic x c
c / r = Ir / Ic .. 2.9 x 10^9 / 2.7 x 10^9 .. .. c/r = 1.0740

(Max) fractional change in as people go to centre ..
= (c - r) / r = (c/r) - 1 .. .. 1.0740 - 1 = 0.074 .. .. .. = 7.4% (increase)

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