Question

A space station in the shape of a uniform disk (mass 4.45x105 kg, radius 262 m) rotates with period 86.1 seconds. There are also 734 astronauts (whom you can treat as point particles) working inside the space station, each of mass 155 kg, and all standing on the outside rim and rotating with the station. Now, all the astronauts move to a conference room at the very center of the space station. Find the new period of the rotation of the space station.

Answer 1

Mass of the space station is

Radius of the space station is R=262m

Moment of inertia of the space station is equal to the moment of inertia of a disk of mass M and radius R

Astronauts can be considered as point masses. The moment of inertia of a point mass m at a distance r from the axis of rotation is

There are 734 astronauts, the moment of inertia of the astronauts is

Initially, astronauts are at r=R

Finally, they are at the center r=0

Total initial moment of inertia of the space station+astronauts system is

Total final moment of inertia is

Initial time period of rotation of the space station is T_i=86.1s

The initial angular speed of the space station is

Similarly, final angular speed of the space station is

Initial angular momentum of the system is

Final angular moment of the system is

Applying conservation of momentum