# How to solve statically indeterminate pulley?

I am working on a wire rope pulley at rest, shown below. The wire rope wraps once around the pulley (180°) and has tension T. The pulley axle is on a sliding track that is up against its hard stop, thus a reaction force R balances the free body diagram. If I introduce a force F that pulls on the axle to the left, I believe the free body diagram becomes statically indeterminate? When F is 0, R=2T. But when 0<F<2T then F+R=2T. Does the reaction diminish whilst the tension remains constant? Or does the reaction remain constant whilst the wire rope tension increases? Or something in-between? I know there are methods to solve indeterminate beams, but not sure how to apply here...

• Not indeterminate by any means. If your pulley is structurally sound, reaction decreases since the stop has no means to follow the pulley. Pretend every material has the capacity to deform and the answers will be clearer. When you exceed F=2T you will enter dynamics.
– Abel
Commented Oct 14, 2022 at 2:12

This really depends on how the tension is applied at the end of the rope. More specifically, there are two (basic) cases:

• rope is inextensible
• rope is extensible (acts like a spring).

In most cases, if there is no translation of the pulley (ie. the reaction is not zero), then the tension of the rope remains the same.

# rope as inextensible

In this assumption, (and also assuming there is no spring element at the end of the sides of the rope) then in order to have a tension T you'd need some sort of dead weight (that is the simplest way to think about it).

In the case described above, when you apply the force F then there is no change in rope tension (it remains T).

(Of course is the tension is adjusted/applied with a spring like element this changes).

# rope is extensible

If the rope is extensible then you can have two different situations depending on how the tension is applied:

1. the tension is applied (again) by a dead weight on one end of the rope and fixed on the other end. In this case, in the quasi static case the situation is the same as above (i.e. no change in the rope).

2. The tension is applied by fixing both ends of the rope to the wall, and adjusting the tension with a turn-buckle (See below)

In that case the wire tension will change, but only when the reaction R reaches 0. How much it will change, it will depend on the "spring constant" of the rope.

## to sum up

As a conclusion, if you know the force F, and the initial force T is also known, then your only unknown is the reaction. So the system is not statically indeterminate.

• My wire rope is initially tensioned with a turnbuckle. I guess I was confused about what a reaction force is. Thanks. Commented Oct 14, 2022 at 21:35