For hanging masses $m_1$ and $m_2$, where $m_1$ accelerates upwards and $m_2$ downwards, the equations are $T-m_{1}g=m_{1}a$ and $\boxed{m_{2}g-T=m_{2}a}$.

Its size, shape and rotational effects.

Friction.

For hanging mass $m_1$ moving downwards and mass $m_2$ moving towards the pulley on a smooth plane, the equations are $\boxed{m_{1}g-T=m_{1}a}$ and $T=m_{2}a$.