Two blocks of masses $m_1$ and $m_2$ are connected by a light string passing over a pulley of mass $M$ and radius $R$. The pulley is a uniform disk. If the system is released from rest, what is the acceleration of the blocks?

Solution

1. For the two blocks, the equations of motion are $T_2-m_{2}g=-m_{2}a$ and $T_1-m_{1}g=m_{1}a$, assuming $m_2>m_1$.
2. For the pulley, the net torque equation is $(T_2-T_1)R=I\alpha$, where $I=\frac{1}{2}M{R}^2$ and $\alpha =\frac{a}{R}$.
3. Substituting the tensions into the torque equation yields $(T_2-T_1)=\frac{1}{2}Ma$.
4. Adding the equations $T_2-m_{2}g=-m_{2}a$ and $T_1-m_{1}g=m_{1}a$ with the torque result eliminates the tensions to give $(m_2-m_1)g=(m_1+m_2+M/2)a$.
5. Hence the answer is (A).