When the pulley has mass, tensions on opposite sides of the string are NOT equal. The net torque ($T_1$ - $T_2$)R = Ialpha drives the pulley's rotation.

For an Atwood machine with masses $m_1$, $m_2$ and a disc pulley of mass M:

• a = ($m_1$ - $m_2$)*g / ($m_1$ + $m_2$ + M/2)
• $T_1$ = $m_1$(g - a), $T_2$ = $m_2$(g + a)

For a single mass hanging from a string around a pulley:

• a = m*g / (m + I/$R^2$)

Key insight: the pulley's rotational inertia contributes I/$R^2$ as "equivalent mass" to the translational dynamics. This is why a heavier pulley reduces the system's acceleration.