Massive Atwood Pulleys

In a standard Atwood machine with a massless pulley, the rope tensions are equal on both sides. When the pulley has mass, its rotational inertia resists angular acceleration. This means the tensions must differ to produce the net torque needed to accelerate the pulley.

Consider masses $m_1$ and $m_2$ (with $m_1 > m_2$) connected by a rope over a pulley of radius $R$ and rotational inertia $I$. We need three equations: Newton's second law for each mass, and $\tau_{\text{net}} = I\alpha$ for the pulley.

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