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# Statics: Massless and frictionless pulley

### 1. Massless pulley

The role of the pulley is a system is to change the direction of the tension acting on the pulley. We pass then over a pulley strings or ropes. The presence of friction and inertia in the pulley modifies the transmitted tension. Therefore, to make things simple, we often use the massless and frictionless pulley approximation.

If the pulley has a mass "m", then It has a moment of inertia I which is functiom of mass m.
Therefore, for the pulley:
Σ τ = τnet =
TSonA - TBons = I α
where α is the angular acceleration of the pulley.

If the pulley is massless (m = 0), then its moment of inertia I is zero. Therefore, for the pulley:
Σ τ = τnet = TSonA - TBons = 0

Hence:
TSonA = TBons

### 2. Frictionless pulley

Now if the pulley has friction ƒ with contact with the axle or with the string,this friction will provide torque τfriction on the pulley. This torque is:
τfriction = ƒ . R . Where R is the radius of the pulley. Theefore, Newton's second law of rotation gives for the pulley gives:
Στ = τnet = τfriction = ƒ R

Hence:
- TBons R + TSonA R - ƒ R = I α
Where I is the moment of inertia and α is the angular acceleration of the pulley.

If the pulley is frictionless, then ƒ = 0 and
- TBons R + TSonA R = I α

If, in addition the pulley is massless, then I = 0 and
- TBons R + TSonA R = 0.

Therefore:
TBons = TSonA

If the pulley is massless and frictionless, it can be omitted, and:

TSonA = TBonS

The massless frictionless pulley does not modify the transmitted tension.

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