Find the Tension in a Flexible Chain Resting on a Cone

In summary, the problem involves a loop of flexible chain resting on a smooth, frictionless right circular cone. The goal is to find the tension in the chain. Both the original poster's solution using Newton's laws and the suggestion to use the principle of virtual work result in the same answer. However, virtual work may not be the most useful approach in this case as the problem is static and does not involve movement or potential movement.
  • #1
tobix10
7
1

Homework Statement


A loop of flexible chain, of total weight W, rests on a smooth, frictionless right circular cone of base radius r and height h. The chain rests in a horizontal circle on the cone, whose axis is vertical. Find the tension in the chain.

Homework Equations


Virtual work, but I've done it with Newton's ## F = ma ##.

The Attempt at a Solution


I considered a part of a chain that spread on an arch of angle ## \Delta \theta ## (angle is very small) The forces ## T ## on each end of arch exert horizontal force ## 2T \sin(\frac{\Delta \theta}{2}) ## which has to be equalized by horizontal component ## N_{x} ## of normal force.
Equations are:
## N_{x} = T \Delta \theta ##
## N_{y} = W \frac{\Delta \theta}{2 \pi} ##
## \frac{N_{x}}{N_{y}} = \frac{h}{r} ##
Solution is:
## T = \frac{W}{2 \pi} \cdot \frac{h}{r} ##

Is this answer correct? Any tips how to handle this problem using principle of virtual work?
 
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  • #2
Hello tobix and welcome to physicsforums.

My answer matches yours.

Why do you want to use virtual work? The problem is a static one, and virtual work typically involves movement, or at least potential movement. It might be useful if they had asked you to prove that the described configuration has the lowest potential energy. But they haven't asked you to do that.
 
  • #3
I was told to use virtual work principle, but I don't see any starting point. Nevertheless I am going to stick with static solution. Thank you for your answer.
 

Related to Find the Tension in a Flexible Chain Resting on a Cone

1. What is the tension in a flexible chain resting on a cone?

The tension in a flexible chain resting on a cone refers to the force that is pulling on both ends of the chain in opposite directions. It is a measure of the strength of the chain and is affected by factors such as the weight of the chain, the angle of the cone, and any external forces acting on the chain.

2. How do you calculate the tension in a flexible chain resting on a cone?

The tension in a flexible chain on a cone can be calculated using the formula T = (W/2) * (1 + 1/cosθ), where T is the tension, W is the weight of the chain, and θ is the angle of the cone. This formula takes into account the weight of the chain as well as the angle of the cone, which affects the tension by changing the direction of the force.

3. What factors can affect the tension in a flexible chain resting on a cone?

The tension in a flexible chain on a cone can be affected by several factors including the weight of the chain, the angle of the cone, and any external forces acting on the chain. Other factors such as the material and thickness of the chain can also play a role in determining the tension.

4. How does the angle of the cone affect the tension in a flexible chain?

The angle of the cone has a significant impact on the tension in a flexible chain. As the angle of the cone increases, the tension on the chain also increases, as the force of gravity pulling down on the chain becomes greater. This means that a steeper cone will result in a higher tension in the chain.

5. What is the relationship between the weight of the chain and the tension in a flexible chain resting on a cone?

The weight of the chain and the tension in a flexible chain resting on a cone are directly proportional. This means that as the weight of the chain increases, the tension in the chain also increases. This is because a heavier chain will exert a greater force on the cone, resulting in a higher tension in the chain.

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