The pulley shown in figure has a moment of inertia I about it's axis 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 sleft string is `T_1 and `that in the right string is `T_2`. Suppose the block of mas M goes down with an acceleration a and the other block moves up wit 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 is theerfore `alpha=a/R`. The equation of motiooin for the mass M the mass m and the pulley are as follows:
Mg-T_1=Ma`..........i
T_2-mg=ma`......ii
`T_1R-T_2R=Ialpha=Ia/R` ............iii
putting `T_1 and T_2` from i and ii and iii
`[M(g-a)-m(g+a)]RIa/R`
which gives `a=((M-m)gR^2)/(I+(M+m)R^2`