# Homework Help: Rotational Speed of a Space Station

1. Apr 8, 2009

### guitarman

1. The problem statement, all variables and given/known data
Space Station
A space station has the form of a hoop of radius R, with mass M. Initially its center of mass is not moving, but it is spinning with angular speed ω0. Then a small package of mass m is thrown by a spring-loaded gun toward a nearby spacecraft as shown; the package has a speed v after launch.

(a) Calculate the center-of-mass velocity of the space station (vx and vy) and its rotational speed ω after launch. Do not worry about italics. For example, if a variable R is used in the question, just type R. To specify the angle θ simply use the word theta. Likewise, for ω0 use the word omega0.
vx =

vy =

ω =

2. Relevant equations
vx = (m/M)(-v)cos(theta)

vy = (m/M)(-v)sin(theta)

3. The attempt at a solution

I know that I must use the angular momentum principle, and that the component is out of the page (+z)
I*ωi + R*m*v1(=0)*cos(theta) = I*ωf + R*m*v2*cos(theta)
I = MR^2
so ωf = ωi - R*m*v2*cos(theta)/(MR^2)
Can somebody please let me know where I am going wrong in my derivation of ωf because apparently this is wrong, yet my book does not have any solutions.

2. Apr 9, 2009

### guitarguy1

I'm having trouble on this same problem. Now I'm not 100% sure, but for the initial angular momentum, aren't you supposed to calculate the angular momentum of the package as a separate particle rotating about the same axis.

you have:
I*ωi + R*m*v1(=0)*cos(theta) = I*ωf + R*m*v2*cos(theta)

Do you think that it should something like this:
(M*R^2)*omega0 + (m*R^2)*omega0=(M*R^2)*omegaf + R*m*v(are you sure that you multiply this times cos(theta)?)

I don't think this is completely right, but it might be a start.

3. Apr 9, 2009

### guitarman

Don't you need to use cos(theta) to take into account for the angular speed of the space station?