# Chain Sliding on a Pulley: Acceleration and Force Analysis

• Eugen
In summary, on a pulley with negligible inertia and friction, a chain of mass m and length l starts sliding from its equilibrium position. The acceleration of the chain can be expressed as a function of x: a = gx/l. The force exerted by the chain on the pulley can be represented as F = gm(1 - x2/l2).
Eugen

## Homework Statement

On a pulley with a very small radius and negligible inertia, that rotates without friction around its fixed horizontal axis, there is a chain of mass m and length l. The chain starts sliding from its equilibrium position. Let x be the distance between the ends of the chain. Express as a function of x: a) the acceleration of the chain. b) the force exerted by the chain on the pulley.

F = ma

## The Attempt at a Solution

a) The chain is composed of two parts: the part that starts sliding and keeps getting longer, of mass m1 and the part that keeps getting shorter, of mass m2. I think that on m1 act these forces:
- m1g, its weight, positive
- m2g, negative
So the acceleration of m1 should be g(m1 - m2)/m1. This should also be the acceleration of m2. But I don't know how to express this acceleration in terms of x and l.

As for the force acting on m2, it appears to me that it should also be m1g - m2g, which must be wrong, since the two bodies have the same acceleration and different masses.

Eugen said:
the acceleration of m1 should be g(m1 - m2)/m1
No. That overlooks the additional force needed to accelerate m2.
Consider the tension at the top of the chain, each side of the pulley. Consider all the forces acting on each portion of chain. Write out the ΣF=ma equation for each of the two portions of chain.

I think the forces acting on the two parts of the chain are weight and tension:

T - m2g = m2a
m1g - T = m1a
a = g(m1 - m2)/m

It can be proven that (m1 - m2)/m = x/l, so that a = gx/l.
As for point b), I'm still clueless.

Last edited:
Eugen said:
As for point b), I'm still clueless
What forces act on the pulley?

haruspex said:
What forces act on the pulley?
I think the tension acts on both left and right. The total force should be 2T. Not quite sure, though.

Eugen said:
The total force should be 2T
yes.

Well then. The force acting on the pulley is F.

F = m2a + m2g + m1g - m1a
F = g(m2 + m1) + a(m2 - m1)
F = gm + a(m2 - m1)

From the equation m2a + m2g = m1g - m1a
we find that m1 - m2 = xm/l, so m2 - m1 = - xm/l
Substituting, we find that
F = gm(1 - x2/l2)

Thank you, haruspex.

## 1. What is a chain sliding on a pulley?

A chain sliding on a pulley is a mechanical system in which a chain wraps around a pulley and slides along its surface as it is pulled or pushed by an external force. This system is commonly used to transfer motion or power from one point to another.

## 2. How does a chain sliding on a pulley work?

As the chain slides on the pulley, the pulley rotates in response to the force applied to the chain. This rotation can then be used to move other parts of the system, such as a conveyor belt or a bicycle chain. The friction between the chain and pulley is what allows the transfer of motion or power.

## 3. What are the advantages of using a chain sliding on a pulley?

One advantage of using a chain sliding on a pulley is that it allows for the transfer of motion or power over a distance without the need for direct contact between the two points. This can be useful in situations where the two points are not physically connected or when the distance is too great for a direct connection.

## 4. What are the potential problems with a chain sliding on a pulley?

The main potential problem with a chain sliding on a pulley is the potential for wear and tear on the chain and pulley. Friction between the two can cause the chain to stretch or wear down over time, which can affect the accuracy and efficiency of the system. Regular maintenance and proper lubrication can help prevent these issues.

## 5. Are there different types of chains and pulleys used in this system?

Yes, there are various types of chains and pulleys that can be used in a chain sliding on a pulley system. The type of chain and pulley chosen will depend on factors such as the amount of load and the speed of the system. Some common types of chains used in this system include roller chains and leaf chains, while common types of pulleys include V-belt pulleys and timing pulleys.

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