Is My Circular Hoop Rotation Problem Solution Correct?

[awvvu]

Homework Statement

http://img410.imageshack.us/img410/6864/1975m2lm2.png

The Attempt at a Solution

Could someone see if my solution is correct?

Part a:
I = M R^2 for a circular hoop.

L = \vec{r} \times \vec{p} = I \omega

m_0 v_0 R \sin(\theta) = M R^2 \omega

\omega = \frac{m_0 v_0 \sin(\theta)}{M R}

Part b:
Using conservation of momentum to find the velocity v of the dart+wheel system:
m_0 v_0 = (m_0 + M) v
v = \frac{m_0 v_0}{m_0 + M}

K_i = \frac{1}{2} m_0 v_0^2

K_f = K_{translational} + K_{rotation} = \frac{1}{2}(M + m_0) v^2 + \frac{1}{2} (M + m_0) R^2 \omega^2

And then just plug v and \omega in from above and calculate the ratio of final to initial? So, after a bunch of algebra:

\frac{K_f}{K_i} = m_0 \left(\frac{\sin^2(\theta)}{M}+\frac{\sin^2(\theta) m_0}{M^2}+\frac{1}{M+m_0}\right)

 

[kamerling]

for part a you should have used M + m_0 instead of M as the mass of the complete system. the dart still has some angular momentum after it sticks to the now rotating wheel.

for part b I think the axle doesn't move, so K_{translational} = 0

 

[awvvu]

Yeah, you're absolutely right for both of them. Thanks.

The final answer for any future googlers is (oh wait, the problem text was in an image):

\frac{m_0 \sin^2(\theta)}{m_0 + M}