Solving a Pulley Problem with Unknown Variables

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In summary, to solve this problem, you can use the fact that the total force on the pulley is 0, meaning the sum of the tensions due to the blue rope must be in the same direction as the green rope. You can also use the equations 20 = 2sinx + Lsin2x and 20 = 2cosx + Lcos2x to find the values of L and x, which can then be used to find the tension in the blue rope. Additionally, you can use the fact that the weight is at rest to find the tension in the green rope. Calculus is not necessary for this problem.
  • #1
jprg
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Hi,

I would like to solve this problem.
Pulley is attached to the green rope that is 2' long.
The weight is 1 unit.
a and b are unknown.
The green rope's point of attachment is 20' higher than that of the blue rope.
The distance between two walls is 20'.
I would like to solve this for the forces in the green rope and the angle the green rope is making with the wall.
I am not sure how to proceed.

Thanks.

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  • #2
I can write a computer program and use numerical approximation for this. But I would really like to solve this analytically so as to understand the mechanics of it.

By the way, the pulley diameter can be disregarded.

Do I need to get into calculus to solve this?
 
  • #3
No, calculus is unnecessary :smile:
Since total force on the pulley = 0, the sum of 2 tensions due to the blue rope must be in the same direction of the green rope (assume that friction is negligible). Denote x the angle needed, then the angle between the 2 sections of the blue rope is 2x.
We have: tanx = 1/(b/a+1) and tan2x = a/b, so: [tex]\frac{1}{tan2x}+1 =\frac{1}{tanx}[/tex]
Moreover: [tex]tan2x=\frac{2tanx}{1-tan^2x}[/tex]
From here, you may solve for x. Then the tension in the green rope can be easily found.
 
  • #4
Thanks,

Can you clarify how you got tanx = 1/(b/a+1)
 
  • #5
Oops, sorry, I was wrong :biggrin: It should be tanx = a/(b+1). My mistake led me to think that the lengths were redundant.
By the way, what's dimension of a and b? At first I thought you used them to illustrate the ratios. But if you write b+1, "1" here should mean 1 meter or 1 cm, or so. With the lengths, I think it's solvable.
 
  • #6
a and b just serve to denote force components. Their units are the same as the weight.
Since the weight is 1 unit acting straight down, and the vertical (downward) component on the blue rope's first section is denoted as b, then the green rope must counteract these too forces and have an upward vertical component equal to b+1. (b plus one unit).

Thanks.
 
  • #7
It still doesn't make sense to me, as the calculations do not take the actual dimensional layout into consideration. But the forces and angles will change as the points of attachment or the distance between the walls change.
 
  • #8
Let's do it all over.

Denote T tension in the blue rope. Because the weight is at rest: T = W = 1 (unit of force).
As the pulley is at rest as well, we have: To = 2Tcosx = 2cosx.

Denote L the length of the left section of the blue rope. We have:
_ For the distance between the walls: 20 = 2sinx + Lsin2x
_ For the distance between the attachment positions: 20 = 2cosx + Lcos2x
Technically we can solve the 2 above equations for L and x. Then plug x to the first equation for To. But I'm too lazy to work it out :biggrin: Try to give it a shot :smile:

This way is easier I think, as we don't have to deal with a and b to find To.
 

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What is a pulley problem with unknown variables?

A pulley problem with unknown variables is a type of problem in physics or mechanics that involves determining the values of unknown variables in a system of pulleys. This can include variables such as tension, force, and acceleration.

What are the key steps in solving a pulley problem with unknown variables?

The key steps in solving a pulley problem with unknown variables include identifying all known and unknown variables, drawing a free body diagram, applying Newton's laws of motion, and setting up and solving equations to determine the unknown variables.

How do I draw a free body diagram for a pulley problem with unknown variables?

To draw a free body diagram for a pulley problem with unknown variables, start by identifying all the objects and forces involved in the system. Then, draw a simplified diagram of the system, showing the direction and magnitude of each force acting on each object.

What equations can be used to solve a pulley problem with unknown variables?

The equations that can be used to solve a pulley problem with unknown variables include Newton's second law (F=ma), the equation for gravitational force (F=mg), and the equations for tension in a string or rope (T=mg or T=ma).

What are some common mistakes to avoid when solving a pulley problem with unknown variables?

Some common mistakes to avoid when solving a pulley problem with unknown variables include forgetting to include all forces in the free body diagram, using incorrect signs for forces, and setting up incorrect equations. It is also important to double check units and ensure they are consistent throughout the problem.

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