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Atwood with real pulley
Intermediate
Include rotational inertia of the pulley in the equations.
Write ΣF = ma for each mass and τ = Iα for the pulley, then use a = αR (no slipping). Solve for a and the two tensions.
m₁ = 3, m₂ = 1, I/R² = 0.5 kg: a = 2g/4.5 ≈ 4.4 m/s².
The tensions differ by Ia/R².
In more detail
An Atwood machine with a real pulley has different tensions on each side. Including the pulley's moment of inertia reduces the acceleration compared with the ideal case.
Real-world uses
- Engineers account for real pulleys in hoist design.
- Experiments with Atwood machines measure the pulley's moment of inertia.
- A physics lab measuring acceleration with a real pulley must include the pulley's moment of inertia to match the observed motion.
Open the interactive lesson, with simulation and quiz →
Frequently asked questions
Why are the tensions different?
Because the pulley must be accelerated rotationally, which requires a net torque.
What if the pulley is massless?
Then the tensions are equal and the ideal Atwood formula applies.
How do I include the pulley in the equations?
Add a torque term for the tensions on each side, and the moment of inertia of the pulley, and solve for the acceleration.
Related topics in Rotational Motion
Principal axes · Torque on a dipole · Pulley with mass · Yo-yo and unwinding cylinder · Angular impulse · Static friction in rolling