 #1
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Homework Statement:

Two bodies ##m_1## and ##m_2## are connected by an ideal rope, which passes through a whole of an horizontal table free of friction; ##m_1## moves along a circular trajectory of radius ##r_0##, ##m_2## hangs below the table:
1) Find an expression for ##v_{0_1}## that ##m_1## must have so that ##m_2## is stationary.
2) Suposse that as ##m_1## moves with ##v_{0_1}##, ##m_2## mass duplicates. Find the angular momentum of ##m_1##. Is it constant?
Relevant Equations:

##\vec a= (\ddot rr{\dot{\theta}}^2)\hat r +(r \ddot{\theta} + 2\dot r \dot{\theta}) \hat{\theta}##
##\vec L= \vec r \times \vec p##
1) the motion equations for ##m_2## are: $$Tm_2 g=0 \rightarrow T=m_2 g$$
##m_1##: $$T=m_1\frac{v^2}{r_0} \rightarrow \vec {v_0}=\sqrt{\frac{r_0 g m_2}{m_1}}\hat{\theta}$$
2) This is where I am stuck, first I wrote ##m_2## motion equation just like before, but in polar coordinates: $$T+m_2 g= m_2 a_r \rightarrow a_r= \frac{T}{m_2}+g$$
##m_1##: $$\hat r: T=m_1 a_r=m_1(\frac{T}{m_2}+g) \rightarrow T=g \frac{m_1 m_2}{m_1m_2}$$
and at last I have a kind of system of equations:
$$\hat{\theta}:r \ddot{\theta}+2\dot r \dot{\theta}=0$$
$$\hat r: g \frac{m_2}{m_1m_2}=\ddot rr {\dot{\theta}}^2$$
This is all I could do, I think I might be missing some assumption or condition, but honestly I am not sure. Any help would be appreciated.
##m_1##: $$T=m_1\frac{v^2}{r_0} \rightarrow \vec {v_0}=\sqrt{\frac{r_0 g m_2}{m_1}}\hat{\theta}$$
2) This is where I am stuck, first I wrote ##m_2## motion equation just like before, but in polar coordinates: $$T+m_2 g= m_2 a_r \rightarrow a_r= \frac{T}{m_2}+g$$
##m_1##: $$\hat r: T=m_1 a_r=m_1(\frac{T}{m_2}+g) \rightarrow T=g \frac{m_1 m_2}{m_1m_2}$$
and at last I have a kind of system of equations:
$$\hat{\theta}:r \ddot{\theta}+2\dot r \dot{\theta}=0$$
$$\hat r: g \frac{m_2}{m_1m_2}=\ddot rr {\dot{\theta}}^2$$
This is all I could do, I think I might be missing some assumption or condition, but honestly I am not sure. Any help would be appreciated.