# Solving the Problem of an Infinite Chain Slipping Down a Table

• johnson12
In summary, the problem at hand is determining the position of the tip of a chain with constant density and infinite length as it slips down from a table without friction. There are a few approaches to solving this problem, such as using Newton's or Lagrange's equations. It is suggested to use the Euler-Lagrange equations and find expressions for the kinetic and potential energy. The Lagrange function can then be used to model the problem and Lagrange's equations can be applied to get a second order differential equation. The Lagrange function for this problem is T - U, where T is the kinetic energy and U is the potential energy. However, there is some confusion about the mass and movement of the chain due to its infinite length.
johnson12
Hello all, I'm having trouble with the following problem:

Pb: A chain with constant density and infinite length is slipping down from the table without friction. Determine the position of the tip of the chain at time t.

I know there are a few ways to approaching this problem, namely from Newtons equations, or lagranges equations, but I am quite rusty with this, so any suggestions would help a lot, thanks.

Hi,

do you know the Euler–Lagrange equations ?
You have to find an expression for the potential and kinetic energy,
the difference is the Lagrange-function. Put this function in the Langrange equations
and you get a second order diff.-equation.

kind regards

Hi johnson12!

Infinite length? … presumably only in one direction?

Use conservation of energy.

are you referring to this equation:

$$\frac{\partial L}{\partial x_{i}}$$ - $$\frac{d}{dt}$$$$\frac{\partial L}{\partial \dot{x_{i}}}$$ = 0, i=1,2,3

where L = T - U is the lagrange function,

how can I use this to model my problem?
(ps. I am apologize if my physics is wrong, unfortunately I am a math major).

johnson12 said:
are you referring to this equation:

$$\frac{\partial L}{\partial x_{i}}$$ - $$\frac{d}{dt}$$$$\frac{\partial L}{\partial \dot{x_{i}}}$$ = 0, i=1,2,3

where L = T - U is the lagrange function,

how can I use this to model my problem?
(ps. I am apologize if my physics is wrong, unfortunately I am a math major).

Yes, but we need only one variable x_1 = y for example. Now you have to find an expression
for the kinetic energy T which is simple and for the potential energy U which is simple
as well. U depends of course at your point of reference. Make a sketch.

I get that $$T(\dot{x}) = \frac{1}{2}m \dot{x}^{2}$$
$$U(x) = mgx$$,

$$\frac{\partial L}{\partial x}= - m g$$
$$\frac{d}{dt} \frac{\partial L}{\partial \dot{x}}= m\ddot{x}$$
Lagranges equation implies $$m\ddot{x} + mg = 0$$

but I'm a little confused, if the chain is of infinite length, would it then have infinite mass?

so how can the chain move?

## 1. How does an infinite chain slip down a table?

An infinite chain is a hypothetical construct that is often used in physics problems to simplify calculations. In this scenario, the chain is placed on a frictionless table and its weight causes it to slide down the table due to the force of gravity.

## 2. What is the main challenge in solving this problem?

The main challenge is accurately representing the chain's motion and accounting for the various forces acting on it. This includes the force of gravity, tension in the chain, and any other external forces that may be present.

## 3. How can the problem be approached mathematically?

The problem can be solved using Newton's laws of motion and principles of dynamics. This involves setting up equations of motion and solving for the chain's position and velocity at any given time.

## 4. Are there any real-life applications of this problem?

While an infinite chain slipping down a table may seem like a purely theoretical problem, it has real-life applications in industries such as robotics and manufacturing. Understanding the principles of motion and forces involved can help engineers design more efficient and reliable systems.

## 5. What are some potential factors that may affect the chain's motion in this problem?

Some factors that may affect the chain's motion include the angle at which it is placed on the table, the length and weight of the chain, and the presence of any external forces such as air resistance or friction on the table surface.

Replies
1
Views
2K
Replies
7
Views
2K
Replies
1
Views
2K
Replies
3
Views
4K
• Introductory Physics Homework Help
Replies
6
Views
2K
Replies
14
Views
3K
Replies
2
Views
2K
• STEM Educators and Teaching
Replies
22
Views
3K
• Mechanics
Replies
6
Views
2K