Pulley Calculator (Atwood Machine)

For an ideal Atwood machine, the acceleration is $$a = \frac{(m_1 - m_2)g}{m_1 + m_2}$$. This treats the two masses as a single system where the net driving force is the weight difference and the total inertia is the sum of both masses.

The heavier mass must accelerate downward, which requires a net downward force. Therefore $$m_1 g - T = m_1 a > 0$$, meaning $$T < m_1 g$$. Similarly, the lighter mass accelerates upward, so $$T > m_2 g$$. The tension sits between the two weights.

When $$m_1 = m_2$$, the net force is zero, so $$a = 0$$ and the system remains in equilibrium. The tension equals the weight of either mass: $$T = mg$$. Any small perturbation will not cause acceleration.

In this ideal model, the pulley is assumed massless and frictionless. In reality, a pulley with moment of inertia $$I$$ and radius $$R$$ adds an effective mass $$I/R^2$$ to the denominator, reducing the acceleration slightly.

This tool models the standard two-mass Atwood machine. For compound pulley systems with three or more masses, additional constraint equations are needed. However, the same principles of Newton's second law and string constraints apply to more complex configurations.