In the arrangement shown in figure a weight A possesses mass m, a pulley B possesses mass M. Also known are the moment of inertia I of the pulley relative to its axis and the radii of the pulley R and `2R`. The mass of the threads is negligible. Find the acceleration of the weight A after the system is set free.

Explanation

Let us depict the forces acting on the pulley and weight A, and indicate positive direction for x and `varphi` as shown in the figure. For the cylinder from the equation `F_x=m underset(cx)w` and `N_(cz)=I_cbeta_z`, we get
`Mg+T_A-2T=Mw_c` (1)
and `2TR+T_A(2R)=Ibeta=(Iw_c)/(R)` (2)
For the weight A from the equation
`F_x=mw_x`
`mg-T_A=mw_A` (3)
As there is no slipping of the threads on the pulleys.
`w_A=w_c+2betaR=w_c+2w_c=3w_c` (4)
Simultaneous solutions of above four equations gives:
`w_A=(3(M+3m)g)/((M+9m+(I)/(R^2)))`