How Much Weight Can a Walkway Hold Before Balance Is Lost?

[Emmanems08]

This problem is trying to find out how much weight can be placed in the middle of a system before the weights on the outside go over the pulleys.

A diagram of the problem can be found here http://www.physics.umn.edu/classes/...ds/208321-1201Lab3_P4_Equilibrium_Walkway.pdf

We are trying to find d (the distance of the sag) in terms of known quantities. We will know m (the mass of each counterweight), L (the length), and M the mass of the center weight.

So far I have...

∑F=Fc+Fa-Fb=0
ƩFx=Fc cosθ+Fa cosθ=0
∑Fy=Fc sinθ+Fa sinθ-Fb=0

All from the middle point P, where Fb is the force down, Fa is the force up to the left, and Fc is the force up to the right.

sinθ=d/L1 =d/√(d²+(L/2)²)

That check mark is supposed to be a squareroot if that wasn't clear.

I am not sure where to go from here to solve for d.

 

[haruspex]

The linked page is not public, so the diagram is unknown. Taking the algebra on trust to the final equation, and assuming theta is known, it is just a matter of squaring both sides, multiplying out to remove the fraction, and collecting up like terms, to find a simple expression for d².