Question

Is the following situation possible?

A space station shaped like a giant wheel has a radius of r = 100 m and a moment of inertia of 5e8 kg m². A crew of 150 people of average mass 60 kg is living on the rim, and the station's rotation causes the crew to experience an apparent free-fall acceleration of g. 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.

Prove whether the ball drops at the same time period for the entire evening.

Answer 1

The space station can be considered as a ring with a moment of inertia about the central axis as

Subtituting r = 100 we get mass of the station as m = 10⁵kg

For the sake of simplicity, the people are considered as a point mass. Since the 150 people each of mass 60kg are evenly distributed around the rim, the ring can be considered as a ring of mass 10⁵kg + (150 * 60) = 1.09 x 10⁵kg

The corresponding moment of inertia about the central axis is

I₁ = (1/2) * (1.09 x 10⁵) (100)² = 0.545 x 10⁹ kg m²

The centrepetal force experiencce by the ball under this condition is g

where m is the mass of the ball

As 100 people moves towards the center of the ring for the meeting, the ring can be considered as a ring of mass 10⁵kg + (50 * 60) = 1.03 x 10⁵kg.

The corresponding moment of inertia about the central axis is

I₂= (1/2) * (1.03 x 10⁵) (100)² = 0.515 x 10⁹ kg m²

Since the torque acting on the system hasn't changed angular momentum must be conserved.

So, Substituting values we get

1.06

The corresponding centrepetal force is

So during the meeting the acceleration increases and correspondingly the time period reduces.

Using S = 0.5 at². we can conclude that the new time period reduces by a factor = 0.94

the new time period of the ball during the meeting is 94% of the time period before the meeting started.