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.