"Solving Rope Through a Hole Physics Problem

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In summary, the conversation discusses the problem of a rope with mass M and length ##l## hanging over a frictionless table, with a short portion ##l_0## hanging through a hole. The solution for x(t), the length of rope through the hole, is given as ##x=Ae^{\gamma t}+Be^{-\gamma t}##, where ##\gamma^2=g/l##. The constants A and B are then evaluated to satisfy the initial conditions of ##x(0)=l_0## and ##x'(0)=0##, giving the final solution of ##x(t)=\frac{1}{2}\left(l_0e^{\gamma t}+l_0e^{-
  • #1
Saitama
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Homework Statement


A rope of mass M and length ##l## lies on a friction less table, with a short portion, ##l_0## hanging through a hole. Initially the rope is at rest.

a. Find a general solution for x(t), the length of rope through the hole.

(Ans: ##x=Ae^{\gamma t}+Be^{-\gamma t}##, where ##\gamma^2=g/l##)

b. Evaluate the constants A and B so that the initial conditions are satisfied.


Homework Equations





The Attempt at a Solution


The forces acting on the rope are weight and tension (T) due to the part of rope on the table. If x is the length of rope hanging, l-x is the length of rope on the table. Let ##\lambda## be the mass per unit length of rope.
Newton's second law for hanging part,
$$\lambda xg-T=\lambda xa$$
Newton's second law for rope on table,
$$T=\lambda (l-x)a$$
From the two equations,
$$a=\frac{gx}{l+2x}$$
I can substitute a=d^2x/dt^2 but Wolfram Alpha gives no solution for this. :confused:

Any help is appreciated. Thanks!
 
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  • #2
Pranav-Arora said:

Homework Statement


A rope of mass M and length ##l## lies on a friction less table, with a short portion, ##l_0## hanging through a hole. Initially the rope is at rest.

a. Find a general solution for x(t), the length of rope through the hole.

(Ans: ##x=Ae^{\gamma t}+Be^{-\gamma t}##, where ##\gamma^2=g/l##)

b. Evaluate the constants A and B so that the initial conditions are satisfied.


Homework Equations


Hi Pranav-Arora. Check your algebra. You made a mistake. It should be la=xg.



The Attempt at a Solution


The forces acting on the rope are weight and tension (T) due to the part of rope on the table. If x is the length of rope hanging, l-x is the length of rope on the table. Let ##\lambda## be the mass per unit length of rope.
Newton's second law for hanging part,
$$\lambda xg-T=\lambda xa$$
Newton's second law for rope on table,
$$T=\lambda (l-x)a$$
From the two equations,
$$a=\frac{gx}{l+2x}$$
I can substitute a=d^2x/dt^2 but Wolfram Alpha gives no solution for this. :confused:

Any help is appreciated. Thanks!
Hi Pranav-Arora. Your formulation is correct, but check your algebra. It should be la=xg.
 
  • #3
Chestermiller said:
Hi Pranav-Arora. Your formulation is correct, but check your algebra. It should be la=xg.

Oh yes, sorry about that. Thanks a lot! :smile:

At t=0, ##x(0)=l_0##, x'(0)=0
##x(0)=A+B=l_0##

Since ##x'(t)=A\gamma e^{\gamma t}-B\gamma e^{-\gamma t}\Rightarrow x'(0)=0=A-B##
Solving the two equations, ##A=B=l_0/2##.
Hence,
$$x(t)=\frac{1}{2}\left(l_0e^{\gamma t}+l_0e^{-\gamma t}\right)$$
Looks good?
 
  • #4
It can be solved this way too, by Newton's second law :

[tex]F=Ma=\rho g x A[/tex]

[tex]M\frac{d^2x}{dt^2}=\rho g x A[/tex]

[tex]M\frac{d^2x}{dt^2}=\frac{M}{l^3} g x l^2[/tex]

[tex]\frac{d^2x}{dt^2}=\frac{gx}{l} [/tex]

and the solution of this DE is

[tex]x(t)=x=A\cdot exp(\sqrt{\frac{g}{l}}t)+B\cdot exp(-\sqrt{\frac{g}{l}}t)[/tex]

same as yours...
 
Last edited:
  • #5
@janhaa: What are ##\rho## and ##A##? :confused:
 
  • #6
Pranav-Arora said:
@janhaa: What are ##\rho## and ##A##? :confused:
[tex]\rho[/tex] is density
and
[tex]A: area = l^2[/tex]
 

1. What is the "Solving Rope Through a Hole Physics Problem"?

The "Solving Rope Through a Hole Physics Problem" is a classic physics problem that involves a rope passing through a hole or a pulley. It is often used to demonstrate the principles of tension, force, and friction.

2. How do you solve the "Solving Rope Through a Hole Physics Problem"?

The problem can be solved using the principles of Newton's laws of motion and the equation for calculating tension. It involves setting up an equilibrium equation and solving for the unknown variables.

3. What are the key concepts involved in solving the "Solving Rope Through a Hole Physics Problem"?

The key concepts involved in solving the problem include tension, force, friction, and equilibrium. It is important to understand the relationship between these concepts and how they affect the movement of the rope through the hole.

4. What are some common mistakes made when solving the "Solving Rope Through a Hole Physics Problem"?

One common mistake is forgetting to account for the friction between the rope and the hole or pulley. Another mistake is not properly setting up the equilibrium equation, which can lead to incorrect solutions. It is also important to pay attention to the direction of the forces and ensure they are correctly accounted for in the calculations.

5. How is the "Solving Rope Through a Hole Physics Problem" relevant in real-life situations?

The problem can be applied to real-life situations such as lifting objects using a pulley system or understanding the mechanics of a rock climbing harness. It also helps to develop problem-solving skills and critical thinking, which are valuable in many scientific and engineering fields.

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