Circular hoop Rotation problem

[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):

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