Transfer of forces from a pulley to a beam

[MartinLoland]

Homework Statement

I am trying to understand how forces transfer from a wheel / pulley onto a beam, and then calculating the resulting moment. The image describe the problem.

The rope is in tension with force "S", the wheel/pulley can rotate freely without any friction.

Homework Equations

[/B]
M = force*distance

The Attempt at a Solution

Method 1: If we look at the global system it should be ok to use the force*distance to find the resulting moment M.

Method 2: If we first calculate the forces acting between the pulley and the beam and then isolate the beam we should be able to find the moment.

I thought method two would be right because the pulley don't transfer moment, but then again we should also be able to look at it as a global system. Which one is correct and why doesn't the other one work? Thanks :)

 

[BvU]

Have you shown that the two methods are really different ?
Do you let the two S act at the correct point in method 2 ?

 

[MartinLoland]

Have you shown that the two methods are really different ?
Do you let the two S act at the correct point in method 2 ?

In theory the two methods shouldn't produce any differences. But if we look at the pulley as a isolated system and find the x and y forces acting in the bearing we are only left with M, but since the two S are producing the same moment with equal distance r my understanding is that M=0 in the bearing. Since the beam and pulley are connected in the frictionless bearing a can only see that the x and y component described in the picture are transferred to the beam.

When you say

Do you let the two S act at the correct point in method 2 ?

Hmm, I can't see how I should set the forces differently

 

[BvU]

Hmm, I can't see how I should set the forces differently

Simple case $\theta = \pi/2$. You let the horizontal S act at the height of the beam instead of (idem + r) and the vertical S at the end of the beam instead of (idem + r) -- but I do see the additional moments cancel, so perhaps that's what you mean ?

 

[MartinLoland]

but I do see the additional moments cancel, so perhaps that's what you mean ?

Yes, if you use method 2 the moment cancels out in the pulley so you are left with a smaller M than what you get with method 1. Is it wrong to do a global analysis as in method 1 when you have freely rotating wheels in that system?

Another way of stating the problem: Is the reaction moment force M at the end of the beam dependent on the radius of the pulley r?

 

[BvU]

Is the reaction moment force M at the end of the beam dependent on the radius of the pulley r?

So my reasoning yields answer: no.

Now we want to show that method 1 gives the same moment as method 2 ...

 

[MartinLoland]

So my reasoning yields answer: no.

Now we want to show that method 1 gives the same moment as method 2 ...

Hmm, I have no idea why my earlier results yielded different numbers. Some times all you need is someone looking over your shoulder, thank you for your patience!

 

[haruspex]

Hmm, I have no idea why my earlier results yielded different numbers

Different numbers or different-looking equations?
Your two equations involve l, r, c and theta, but from the geometry there is a relationship between those variables, so it is not apparent whether they are actually the same. Figure out the relationship and substitute for c in method 1.

 

[MartinLoland]

Different numbers or different-looking equations?
Your two equations involve l, r, c and theta, but from the geometry there is a relationship between those variables, so it is not apparent whether they are actually the same. Figure out the relationship and substitute for c in method 1.

Yes thanks. When I worked through it earlier I got different numbers. But working it out once more I found that the r cancels out when using method 1.
So it all makes sense now :)