Question

Figure shows two blocks of masses $m$ and $M$ connected by a string passing over a pulley. The horizontal table over which the mass $m$ slides is smooth. The pulley has a radius $r$ and moment of inertia $I$ about its axis and it can freely rotate about this axis. Find the acceleration of the mass $M$ assuming that the string does not slip on the pulley.

Answer 1

So, According to the equation

$Mg-T_1=Ma$ $........\left(1\right)$

$T_2=ma$ $.....\left(2\right)$

$\Rightarrow \left(T_1-T_2\right)=\frac{Ia}{r^2}$ $......\left(3\right)$ $(because,a=ra)$

Now add the equations $1$ and $2nd$

Then, we have have,

$Mg+T_2-T_1=Ma+ma$ $........\left(4\right)$

$\begin{array}{c}\Rightarrow Mg-I\frac{a}{r^2}=Ma+ma\\ \Rightarrow \left(M+m+\frac{I}{r^2}\right)a=Mg\\ \Rightarrow a=\frac{Mg}{M+m+\frac{I}{r^2}}\end{array}$