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 173 m. The moment of inertia of the station without people is 5.08 x 109 kg·m2. Suppose 332 people, with an average mass of 70.0 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 relative change (/) max in the station's angular speed due to the radial movement of the people?

I went through the steps three time. The final step of subtracting. If I subtract the number I get a percent over 100% if I subtract 2 I get a 13.6909% and that is still the wrong answer.

Explanation / Answer

as there is no external torque, total angular momentum is conserved.

angular momentum=moment of inertia*angular velocity

let initial angular velocity be w1 and final angular velocity be w2.

initial moment of inertia when all the people are on outer surface=moment of inertia of the station alone + 332*moment of inertia of each people

=5.08*10^9+332*mass*radius^2

=5.08*10^9+332*70*173^2=5.77554996*10^9 kg.m^2

final moment of inertia (when all the people are on axis, their contribution towards moment of inertia will be zero)=5.08*10^9 kg.m^2

hence using the principle of conservation of angular momentum:

5.7755*10^9*w1=5.08*10^9*w2

==>w2=1.13692*w1

then maximum change in angular speed=(w2-w1)/w1=0.13692=13.692%

hence maximum relative change is 13.692%.

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