Static Equilibrium: Why Does Changing Pulley Position Affect Relative Angles?

[Kevin de Oliveira]

I have a 3 pulley system statically balanced hanging weights at a determined relative angle (we are taking into account friction). If I change the position of one pulley, angles will remain the same. However, if I have a 4 pulley system, at the same conditions, changing one's position will affect the relatives angles between them all.

I would like to know why that happen. Why, with 3 pulleys, changing one's position will not affect their relatives angles and not with 4 pulleys?

 

[Chestermiller]

I have a 3 pulley system statically balanced hanging weights at a determined relative angle (we are taking into account friction). If I change the position of one pulley, angles will remain the same. However, if I have a 4 pulley system, at the same conditions, changing one's position will affect the relatives angles between them all.

I would like to know why that happen. Why, with 3 pulleys, changing one's position will not affect their relatives angles and not with 4 pulleys?

Can you please present a specific example?

 

[Kevin de Oliveira]

Here an example attached.
Just a correction, instead of 3 it's 2 and 4 it's 3.
In that example, there are only 2 pulleys. If I apply the same conditions in 3 pulleys, angles will change.

Thank you for you reply

 

[Chestermiller]

Sorry. I still don't get what you are asking.

 

[CWatters]

I think I understand what he's asking...

Forget about the position of the pulleys for the moment. What matters is the magnitude and direction/angles of the forces acting on the central ring (see solution 1). Since it's a statics problem the vertical and horizontal components must sum to zero. The forces are fixed so in general (but not always) there will only be one solution for the angles.

In the two pulley case: If one pulley is moved the ring is also free to move horizontally and vertically so the forces acting on it stay at the angles required for the static solution.

In the three (or more) pulley case it's not always possible for the ring to move to maintain the required angles. However it _is_ possible to move a pulley in such a way that it preserves the angles.

 

[CWatters]

For example in this set up the top pulley can be moved from position A to position B without changing any of the angles. If it's moved in any other direction the angles change and a new static solution will have to be found..

 

[Kevin de Oliveira]

Thank you dor your reply. But can we mathematically prove it?

And just to add, I suppose that it's because there is one force applied on each side. In another word, if I apply an odd quantity of forces on the horizontal axis, the angles will no longer be the same. not sure if this physically makes sense

 

[CWatters]

As I see it.. In the three pulley case you are asking us to prove that the angles change if you change the angles.

 

[Kevin de Oliveira]

Yes