Statics problem: man lowering himself from tree w/ friction

In summary, the 180-lb tree surgeon lowers himself with the rope over a horizontal limb of the tree. If the coefficient of friction between the rope and the limb is 0.60, compute the force (P) which the man must exert on the rope to let himself down slowly.
  • #1
bkw2694
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Homework Statement


The 180-lb tree surgeon lowers himself with the rope over a horizontal limb of the tree. If the coefficient of friction between the rope and the limb is 0.60, compute the force (P) which the man must exert on the rope to let himself down slowly.

23ve102.jpg


Homework Equations


T1/T2 = e^(βμ)

Where β = the angle of the rope on the log, μ = 0.60, T1 = the bigger tension (180 lbs), and T2 = the smaller tension (P)

The Attempt at a Solution



So I looked at the picture and I believe β = 180° which = π rads.

Then I plugged those into my equation and got:

180/P = e^(.6π), this gave me P = 27.33 lbs, which is incorrect.

The correct answer should be 23.7 lbs, but I have no idea what I'm doing wrong. I'm guessing my β is the wrong number, but I'm not sure how to find the correct number if that's the case.
 
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  • #2
Your mistake is in assuming the greater tension equals the man's weight. Draw a FBD for the man.
 
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  • #3
haruspex said:
Your mistake is in assuming the greater tension equals the man's weight. Draw a FBD for the man.

Sorry, I'm a little confused so I don't know if my FBD is right. I made a diagram of just the guy and the two sides of the rope, with a downward force of 180 lbs due to his weight, a downward force of P on the piece of rope he's holding (since he's pulling down) and an upward force of 180 + P since that satisfies equilibrium. So then when I make a diagram of the branch with the two pieces of rope, I have (180 + P) going downward and a force of P going upward. Am I on the right track here?
 
  • #4
bkw2694 said:
I made a diagram of just the guy and the two sides of the rope, with a downward force of 180 lbs due to his weight, a downward force of P on the piece of rope he's holding
The rope he's holding is pushing down on him with force P??
 
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  • #5
haruspex said:
The rope he's holding is pushing down on him with force P??

Ah, so I was thinking of it as if he were pulling down, but it should be that the tension of P is upward, which would make the tension on the other side: T1 = (180 - P). Thanks so much!

And for anyone else who might find this problem, I'll finish the solution: (180-P)/P = e^(.6π) ⇒ P = 23.7 lbs
 

Related to Statics problem: man lowering himself from tree w/ friction

What is the definition of statics?

Statics is a branch of mechanics that deals with the study of objects at rest or in constant motion. It is concerned with analyzing and predicting the forces acting on objects and how they affect their equilibrium.

What is friction?

Friction is a force that opposes the motion of an object when it comes into contact with another surface. It is caused by the microscopic irregularities of the two surfaces and can either be helpful or harmful depending on the situation.

How does friction affect a man lowering himself from a tree?

When a man is lowering himself from a tree, friction between his body and the tree trunk helps in slowing down his descent and preventing him from falling too quickly. However, too much friction can also cause blisters and burns on the skin.

What are the factors that affect friction?

The factors that affect friction include the nature of the surfaces in contact, the force pushing the surfaces together, and the roughness of the surfaces. The type of motion and the speed of the objects also influence the amount of friction.

How can friction be reduced?

Friction can be reduced by using lubricants, such as oil or grease, between the surfaces in contact. Smoother surfaces and reducing the force pushing the surfaces together can also help in reducing friction. Additionally, using wheels or rollers can help reduce friction by converting sliding friction into rolling friction.

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