Statics
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© The scientific sentence. 2010
| 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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