# Centripetal Acceleration must equal Gravity in a Space Station

1. Jul 28, 2012

### jamesbiomed

1. The problem statement, all variables and given/known data

A space station is to provide artificial gravity to support long-term stay of astronauts and cosmonauts. It is designed as a large wheel, with all the compartments in the rim, which is to rotate at a speed that will provide an acceleration similar to that of terrestrial gravity for the astronauts (their feet will be on the inside of the outer wall of the space station and their heads will be pointing toward the hub). After the space station is assembled in orbit, its rotation will be started by the firing of a rocket motor fixed to the outer rim, which fires tangentially to the rim. The radius of the space station is
R = 44 m, and the mass is M = 2.5 10^5 kg. If the thrust of the rocket motor is F = 1.5 10^2 N, how long should the motor fire?

2. Relevant equations

F=ma
w=w0+at
w=v/t
angular accleration=w/t => w= angular acceleration* time
gravity= 9.81
I (wheel)=1/2 m (r1^2-r2^2)

3. The attempt at a solution

Used F=ma, divided (f=150 N) by ( M=250000)= tangential acceleration (6 * 10^-4)
To get angular acceleration multiplied tangential acceleration times the radius. Got .0264
Since I'm looking for the time to make the centripetal acceleration equal to 9.81, I used
ac=w^2r. I already know w=angular accel *time, and that gave me ac=(α^2)(t^2)R
Then I solved for t and got 19.2 seconds.

The correct answer is 34600 s

Can someone check my work? It's also likely that moment of inertia is involved here, but since there is no potential energy (we're in space) and no initial kinetic energy, I couldn't think of a way to work it in. Angular momentum seems possible but I can't see how. Any help would be great! thanks.

2. Jul 28, 2012

### Villyer

I think you have to use torque in order to find the radial acceleration, which is where you would incorporate moment of inertia.

3. Jul 28, 2012

### jamesbiomed

Ok, I'll try that and see if it helps. Thanks

4. Jul 28, 2012

### pgardn

A net torque (rocket firing) applied over a period of time causes a change in angular momentum.

And you know the angular velocity you must reach to achieve an centripetal acceleration of 9.8
V^2/r = ( ω*r)^2/r = 9.8

I think this might be the quickest route.

And you are definitely going to use I in the above. Your are trying to rotate an object with a large moment of inertia... Also since they only gave you one radius that I see, pretend the space station is a thin hoop. So thats I = Mr^2

see what ya get...

5. Jul 28, 2012

### jamesbiomed

pgardn, that worked perfectly. Thanks for the explanation as well, it made sense.

Many thanks!