In Fig. mass $m_1$ slides without friction on the horizontal surface, the frictionless pulley is in the form of a cylinder of mass $M$ and radius $R$, and string turns the pulley without slipping. Find the acceleration of each mass.

Positive direction of angular acceleration $a$ is taken along the corresponding acceleration $a_1$ which is taken positive in the direction of net force. Also due to inertia of the pulley the tension in two parts of the string will be different. Let it is $T_1$ and $T_2$. The equation of motion for masses $m_1$ and $m_2$ are

$$\begin{gathered} T_1=m_{1}a.......\left(i\right) \\ {m}_{2}g-T_2=m_{2}a........\left(ii\right) \end{gathered}$$

For the pulley , $$\begin{gathered} T_{2}R-T_{1}R =\left(\frac{1}{2}M{R}^2\right)\left(\frac{a}{R}\right) \\ \Rightarrow T_2-T_1=\frac{1}{2}Ma .............\left(iii\right) \end{gathered}$$

Solving ,a = $\frac{m_{2}g}{m_1+m_2+{\displaystyle \frac{M}{2}}}$