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

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SUMMARY

This discussion focuses on calculating the maximum weight that a walkway can support before losing balance, specifically analyzing the forces acting on a system with counterweights. The key variables include the mass of the center weight (M), the mass of each counterweight (m), and the length of the walkway (L). The equations of equilibrium are established using the sum of forces in both the x and y directions, leading to the relationship involving the sag distance (d) and the angle (θ). The next steps involve algebraic manipulation to derive a clear expression for d² based on known quantities.

PREREQUISITES
  • Understanding of static equilibrium principles
  • Familiarity with trigonometric functions and their applications in physics
  • Basic algebraic manipulation skills
  • Knowledge of forces and tension in mechanical systems
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  • Study the principles of static equilibrium in mechanical systems
  • Learn about trigonometric relationships in physics, particularly in force analysis
  • Explore algebraic techniques for solving equations involving multiple variables
  • Investigate the applications of counterweights in engineering design
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Physics students, mechanical engineers, and anyone involved in structural analysis or design of balance systems will benefit from this discussion.

Emmanems08
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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.
 
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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.
 

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