A spinning space station exhibits a change in moment of inertia.

[MacRowan]

Homework Statement

A space station shaped like a giant wheel has a radius 95.0 m and a moment of inertia of 5.03✕ 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.

So,

r = 95.0

I = 5.03*10⁸

and

m_ave = 65.0 kg

Δm = 150*m_ave - 100*m_ave, or m_f - m_i

Homework Equations

L_i = L_f ---> (Iω)_i = (Iω)_f

a_c = rω², and therefor, w = √(a_cr)

I = mr²

The Attempt at a Solution

A reminder; I'm looking for the new centripetal acceleration felt by the remaining 50 New Earth colonizers.

I also want to clarify that I chose to determine the total initial moment of inertia by adding the I of the people to that of the provided I for the space station, because I reasoned I couldn't really find the difference without being provided the mass of the space-station, so I'm already unsure of my path here.

There is a change in moment of inertia of the system, so;

L_i = L_f

I_i = (I_150people) + (5.03*10⁸) = (5.910*10⁸

I_f = (I_50people) + (5.03*10⁸) = (5.350*10⁸

but to find centripetal acceleration I need to relate a_c and ω, and then put them into the conservation equation like so;

(5.910*10^8)*(√9.81*95.0) = (5.350*10^8)*(√a_c*95.0)

When I isolated acceleration I got a value of 1.859, which my shitty electronic feedback java program says is more than 10% off. At least it's a smaller value, as it should be.

So, obviously I don't have the right answer, but can someone either point out my small error or point out how I have approached the problem entirely wrong?

 

[ehild]

Welcome to PF!

The Attempt at a Solution

A reminder; I'm looking for the new centripetal acceleration felt by the remaining 50 New Earth colonizers.

I also want to clarify that I chose to determine the total initial moment of inertia by adding the I of the people to that of the provided I for the space station, because I reasoned I couldn't really find the difference without being provided the mass of the space-station, so I'm already unsure of my path here.

There is a change in moment of inertia of the system, so;

L_i = L_f

I_i = (I_150people) + (5.03*10⁸) = (5.910*10⁸

I_f = (I_50people) + (5.03*10⁸) = (5.350*10⁸

but to find centripetal acceleration I need to relate a_c and ω, and then put them into the conservation equation like so;

(5.910*10^8)*(√9.81*95.0) = (5.350*10^8)*(√a_c*95.0)

First, a_c=w²R, so your last equation is not correct. But you can simplify by R, so it does not really count.
I just wonder, how did you get 1.859 m/s² for a_c, or was it 1.859 g?. You miscalculated something.

 

[PhysX]

You're on the right track but w= sqrt(a_cr) doesn't seem to be correct... which is affecting your last equation.
Your I_f seems a little off but should be fine for the purpose of this question.
I just got an answer using your values, only correcting the formula for w, and web assign accepted it as a correct answer.