Rope Sliding off a Peg problem

In summary, The problem involves a limp rope with a mass of 2.5 kg and a length of 1.2 m hung on a frictionless peg. The rope is hung with 0.8 m hanging off the longer end and 0.4 m off the lower end. The question asks for the vertical velocity of the rope just as the end slides off the peg. The solution involves using conservation of energy, specifically the equation PE=mgh and KE=1/2mv^2. After an incorrect initial attempt, the correct solution involves considering the counterweight applied by the shorter end of the rope and using the ratio of mass at any given end and gravity to solve for the kinetic energy.
  • #1
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Homework Statement



A limp rope with a mass of 2.5 kg and a length of 1.2 m is hung, initially at rest, on a frictionless peg that has a negligible radius. The rope is hung such that 0.8m hangs off the longer end, and 0.4m off the lower end. What is the vertical velocity of the rope just as the end slides off the peg?

Homework Equations



PE = mgh
KE = 1/2mv^2

The Attempt at a Solution



Because the kinematics of this system seemed incredibly complicated, I figured it best to use conservation of energy in the system. Knowing that the center of mass will fall 0.4 meters, I assumed:

mgΔH = 1/2mv^2
2.5*9.81*0.4 = 1/2*2.5*v^2
v = 2.8 m/s

Unfortunately, this seems to be incorrect. I assume this has something to do with the counterweight applied by the short end of the rope, but I'm not sure how to account for this.
 
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  • #2
Aha, got it! I knew the initial problem had to do with my initial potential energy, so I realized I could set it equal to:

PE = mh*g(0.8/1.2)

As the initial acceleration is dependent upon the ratio of mass at any given end and gravity. Using this, I could go straight to kinetic energy and solve.
 
  • #3
The second attempt has the correct initial C. M.
 

1. What is the "Rope Sliding off a Peg problem"?

The "Rope Sliding off a Peg problem" is a classic physics problem that involves a rope wrapped around a vertical peg. The goal is to determine the minimum coefficient of friction needed between the rope and the peg in order for the rope to slide off the peg without slipping.

2. What are the variables involved in this problem?

The variables in this problem include the mass of the rope, the angle of the rope with respect to the peg, the coefficient of friction between the rope and the peg, and the gravitational acceleration.

3. How can this problem be solved?

This problem can be solved using principles of statics and friction. By setting up equations for the forces acting on the rope and using the condition for no slipping, the coefficient of friction can be determined.

4. What are some real-life applications of this problem?

This problem has real-life applications in situations where ropes are used to support or lift objects, such as in rock climbing or construction. It can also be used in industrial settings where ropes and pulleys are used for lifting heavy loads.

5. Are there any simplifications or assumptions made in this problem?

Yes, there are a few simplifications and assumptions made in this problem. Some of these include assuming a uniform rope with no knots or kinks, neglecting the weight of the rope, and assuming a frictionless pulley at the top of the peg.

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