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Pulley with mass
Intermediate
Pulley inertia reduces the acceleration of attached masses.
a = (m₁−m₂)g/(m₁+m₂+I/R²)A real pulley has moment of inertia I, so tensions on both sides differ: (T₁ − T₂)R = Iα with α = a/R. The acceleration becomes a = (m₁ − m₂)g/(m₁ + m₂ + I/R²).
Heavy pulleys slow the system.
Don't assume equal tension.
In more detail
A pulley with mass, rotating as the rope moves, does not have equal tension on both sides. The difference in tensions provides the torque that gives the pulley its angular acceleration.
Real-world uses
- Lift systems use real pulleys with mass, which must be included in the design.
- Physics labs use pulleys with known moments of inertia to test rotational dynamics.
- A cable pulling a load over a heavy pulley must account for the pulley's own inertia, which slows the acceleration.
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Frequently asked questions
Can I ignore the pulley's mass?
Only if it is very light compared with the other masses.
How do I include it?
Use τ = Iα for the pulley and combine with F = ma for the masses.
When is the pulley's mass ignored?
When it is light and the rope does not slip, so the tension is the same on both sides and the pulley's inertia can be neglected.
Related topics in Rotational Motion
Inertia tensor · Principal axes · Torque on a dipole · Atwood with real pulley · Yo-yo and unwinding cylinder · Angular impulse