Finding Equilibriums on a Cone with a Chain Loop

In summary, the system has two equilibriums: when the loop is in the horizontal plane and when it is not belonged to any plane.
  • #1

wrobel

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Imagine a right circular cone with smooth surface. The cone is stated such that its axis is parallel to the standard gravitational field g. And you have a piece a thin homogeneous chain. Then you connect the tips of the chain to obtain a loop. You put this loop on the cone:
bfc69151eab9.png

It is clear, there is a state such that whole the loop forms the circle and rests in the horizontal plane. Are there another equilibriums? How many equilibriums does this system have?

(I know the answer and the solution. It is just for those who doesn't know what to do on weekends evenings :) I enjoyed solving this problem and share it with PF :)
 
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  • #2
If we neglect friction then, I would guess that it resting in the horizontal plane is the only equilibrium point. Not wanting to get into the calculus of veriations, if I imagine just turning the gravity up to absurd levels, it seems like the chain would end up resting flat.

Also, if we consider the set of chain positions of ellipses on the cone, then the circle in the horizontal plane is where the center of mass of the chain would be lowest (of that set), and where the chain would have a minimum total potential energy.

But then, ellipses are just a small subset of possible ways to wrap a chain once around a cone. I'd be interested in knowing what the real answer is, though.
 
  • #3
Yes indeed it is a task for variational calculus. Let ##\beta\in(0,\pi)## stand for the cone's vertex angle.

Theorem. If ##\pi/3<\beta<\pi/2## then wrapped once loop of the chain has only two (up to rotational symmetry) equilibriums: 1) the trivial one when whole the chain is in the horizontal plane and 2) some another equilibrium when the loop is not belonged to any plane.
For all other values of ##\beta## there is only trivial equilibrium.
 
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1. How does the chain loop method help find equilibriums on a cone?

The chain loop method is a mathematical approach that can be used to find equilibriums on a cone. It involves looping a chain around the cone and calculating the forces acting on the chain at each point. By analyzing these forces, the location of the equilibrium point can be determined.

2. What are the limitations of using the chain loop method to find equilibriums on a cone?

One limitation of the chain loop method is that it assumes the cone is perfectly symmetrical and smooth. In reality, cones may have imperfections or irregularities that can affect the accuracy of the results. Additionally, the method is limited to finding equilibriums on cones with a circular base.

3. Can the chain loop method be applied to other shapes besides cones?

Yes, the chain loop method can be applied to other shapes as long as they have a circular base. It has been used to find equilibriums on cylinders, spheres, and even more complex 3D shapes. However, the calculations may become more complex for non-conical shapes.

4. Is the chain loop method an accurate way to find equilibriums on a cone?

The accuracy of the chain loop method depends on the assumptions made and the precision of the measurements. If the cone is symmetrical and smooth, and the measurements are accurate, the method can provide a fairly accurate estimate of the equilibrium point. However, it is always important to validate the results with other methods or experiments.

5. Are there any practical applications of finding equilibriums on a cone with a chain loop?

Yes, there are several practical applications of this method. For example, it can be used in engineering to determine the stability of structures with conical components, such as towers or chimneys. It can also be applied in physics to study the behavior of objects rolling down a cone and reaching an equilibrium point.

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