How Do Counterweights Affect the Sag of a Rope Bridge?

[maroonandgold]

Homework Statement

Alright, here we go. In the problem, there is a bridge. Basically it's a bridge in the amazon over the trees to make it easier for the explorers to walk and also to protect the forest floor. The bridge is rope bridge, and it's also a pulley system. There are are three masses involved in keeping the bridge horizontal. Each end of the rope supporting the walkway goes over a branch and then is attached to a large weight hanging down. When the team member is at the center of the walkway between two trees, you need to determine how the sag of the walkway is related to the mass of the counterweights and the total mass of the team member with the equipment. Alright, that's the situation, and what I actually have to do is find an equation for the vertical displacement of the large object in the middle in terms of the mass of the large object, the mass of the weights on the other sides of the branches, and the horizontal distance between the two pulleys. And the final equation shouldn't depend on pulleys.

Homework Equations

Hm, I don't know. We just started force this week so I'm not familiar with forces yet. How about f=ma?

The Attempt at a Solution

I have been staring blankly at the problem for an hour. And the book has no problem similar to it. I know it involves trigonometry, but that's about all I know.

TIA!

 

[tiny-tim]

Welcome to PF!

Hi maroonandgold! Welcome to PF!

Hm, I don't know. We just started force this week so I'm not familiar with forces yet. How about f=ma?

a, the acceleration, is zero, so this is just the vector equation F_total = 0.

Concentrate on the bit of rope at the centre …

it has three forces acting on it, the two tension forces and the weight of the man …

so call the tension T, and the angle θ, and put that into F_total = 0 …

what do you get? ​

 

[tomcue]

Thank you for providing the problem and your thoughts on it. I can offer some guidance on how to approach this problem.

Firstly, it is important to understand the concept of a pulley system. A pulley is a simple machine that changes the direction of a force, allowing us to lift or move objects with less effort. In this case, the pulley system is used to support the bridge and keep it horizontal.

To solve this problem, we can use the principles of equilibrium. This means that the forces acting on the bridge must be balanced in order for it to remain horizontal. The three masses involved are the team member, the counterweights, and the equipment. Each of these masses exerts a force on the bridge, and we can use the equation F=ma to calculate these forces.

Next, we need to consider the forces acting on the pulley system. The tension in the rope is equal on both sides of the pulley, and this tension is what keeps the bridge in equilibrium. We can use trigonometry to determine the relationship between the tension in the rope and the angle of the rope as it passes over the branch.

Finally, we can use the fact that the bridge is in equilibrium to set up an equation that relates the vertical displacement of the large object in the middle to the masses and the horizontal distance between the pulleys. This equation should not depend on the pulleys, as they are just a means to change the direction of the forces.

I understand that this may seem overwhelming at first, but with practice and a solid understanding of the principles involved, you will be able to solve this problem and many others like it. I encourage you to continue studying forces and pulleys, and don't hesitate to seek help from your teacher or classmates if needed. Good luck!