Rotational Speed of a Space Station

[guitarman]

Homework Statement

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 =

Homework Equations

vx = (m/M)(-v)cos(theta)

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

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.

 

[guitarguy1]

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.

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.

 

[guitarman]

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