Draw a pulley system

In order to compute the tangents it is advantageous to make the circles nodes and to use tangent cs:, which comes with calc. And the angles can be conveniently drawn with the angles library, where quotes makes it somewhat simpler to add \alpha and \beta.

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{patterns,calc,angles,quotes}
\begin{document}
\begin{tikzpicture}
\fill[pattern=north east lines] (-3,0) rectangle (3,.3);
 \begin{scope}[thick]
 \draw(-3,0)--(3,0);

 \path (-2.5,-2.4) node[circle,draw,inner sep=.3cm] (L){}
 (2.5,-1) node[circle,draw,inner sep=.3cm] (R){}
 (0,-3) node[circle,draw,inner sep=.153cm,label=below:$m$] (M){};
 \draw (L.north) -- (L.north|-0,0) (R.north) -- (R.north|-0,0);
 \draw[dashed] (M) -- (0,-1)coordinate (M1);
  \draw (M)  -- (tangent cs:node=L,point={(M.center)},solution=1) coordinate (L1)
  let \p1=($(L1)-(L.center)$),\n1={atan2(\y1,\x1)},\n2={veclen(\y1,\x1)} in
  arc(\n1:180:\n2) -- ++(0,-1.5) node[below,draw]{$m_1$};
  \draw (M)  -- (tangent cs:node=R,point={(M.center)},solution=2) coordinate (R1)
  let \p1=($(R1)-(R.center)$),\n1={atan2(\y1,\x1)},\n2={veclen(\y1,\x1)} in
  arc(\n1:00:\n2) -- ++(0,-1.5) node[below,draw]{$m_2$};
 \end{scope}
 \path pic [draw,angle radius=0.5cm,"$\alpha$",angle eccentricity=1.5] {angle = M1--M--L1} 
   pic [draw,angle radius=0.7cm,"$\beta$",angle eccentricity=1.5] {angle = R1--M--M1} ;
\end{tikzpicture}
\end{document}