A space station shaped like a giant wheel has a radius 105 m and a moment of inertia of 5.09 108 kg · m2. A crew of 150 lives on the rim, and the station is rotating so that the crew experiences an apparent acceleration of 1g. When 100 people move to the center of the station for a union meeting, the angular speed changes. What acceleration is experienced by the managers remaining at the rim? Assume that the average mass of each inhabitant is 65.0 kg.

To find the acceleration experienced by the managers remaining at the rim, we need to use the principle of conservation of angular momentum.

The moment of inertia of a rotating object is given by the formula:

I = m * r^2

Where:
I = moment of inertia,
m = mass of the rotating object, and
r = radius of rotation.

Given that the moment of inertia is 5.09 * 10^8 kg · m^2 and the radius is 105 m, we can rearrange the formula to find the mass:

m = I / r^2

m = 5.09 * 10^8 kg · m^2 / (105 m)^2
m = 5.09 * 10^8 kg · m^2 / 11025 m^2
m ≈ 46177.7 kg

Since the average mass of each inhabitant is 65.0 kg, we can calculate the initial total mass of the crew:

initial total mass = 150 inhabitants * 65.0 kg/inhabitant
initial total mass ≈ 9750 kg

When 100 people move to the center of the station, the total mass changes to:

new total mass = 50 inhabitants * 65.0 kg/inhabitant
new total mass ≈ 3250 kg

Now, we can calculate the change in total mass:

change in mass = initial total mass - new total mass
change in mass = 9750 kg - 3250 kg
change in mass = 6500 kg

Next, we need to find the change in angular velocity. The principle of conservation of angular momentum states that the initial angular momentum equals the final angular momentum.

The initial angular momentum is given by:

initial angular momentum = initial moment of inertia * initial angular velocity

The final angular momentum is given by:

final angular momentum = final moment of inertia * final angular velocity

Since angular momentum is conserved, we can equate the initial and final angular momentum:

initial moment of inertia * initial angular velocity = final moment of inertia * final angular velocity

Solving for the final angular velocity:

final angular velocity = (initial moment of inertia * initial angular velocity) / final moment of inertia

To find the initial angular velocity, we use the apparent acceleration of 1g. The formula relating apparent acceleration (a) to angular velocity (ω) is:

a = ω^2 * r

Solving for ω:

ω = √(a / r)

Substituting the values:

ω = √(9.8 m/s^2 / 105 m)
ω ≈ 0.3109 rad/s

Now we have all the values we need to calculate the final angular velocity:

final angular velocity = (5.09 * 10^8 kg · m^2 * 0.3109 rad/s) / 5.09 * 10^8 kg · m^2
final angular velocity = 0.3109 rad/s

To find the angular acceleration, we can use the formula:

angular acceleration = (final angular velocity - initial angular velocity) / time

Since we are given no information about time, we can assume it to be 1 second for simplicity:

angular acceleration = (0.3109 rad/s - 0.3109 rad/s) / 1 s
angular acceleration = 0 rad/s^2

Therefore, the acceleration experienced by the managers remaining at the rim is 0 rad/s^2.