The pulley shown in figure has a moment of inertia I about it saxis and its radius is R. Find the magnitude of the acceleration of the two blocks. Assume that the string is light and does not slip on the pulley.

Explanation

Suppose the tension in the left string is `T_1` and that in the right string is `T_2`. Suppose the block of mass M goes down with an acceleration a and the other block moves up with the same acceleration. This is also the tangential acceleration of the rim of the wheel as the string does not slip over the rim.
The angular acceleration of the wheel `alpha = a/R`. The equations of motion for the mass M, the mass m and the pulley are as follows ,

`Mg - T_1 = Ma" ...(1)"`
` T_2 - Mg = ma" ...(2)"`
`T_2 R - T_2 R = I alpha = (la)/R " ...(3)"`
Substituting for `T_1` and `T_2` from equations (i) and (ii) in equation (iii)
`[ M (g-a)-m(g+a)]R = (la)/R`
Solving, we get
`a = ((M-m)gR^2)/(I+(M+m)R^2)`