Potential in caternary problem

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In summary, the equilibrium shape of a flexible but inextensible chain suspended between two points can be found by minimizing the potential energy of the chain, which is given by the formula delta p = y(1+y'^2)^(1/2) delta x. This formula can be derived using trigonometric principles and can be found in various sources such as Wikipedia.
  • #1
thehoten
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a flexible but inextensible chain having uniform mass density is suspended between two points (of course not vertically aligned). Find the shape of the equilibrium of the chain.

The chain will settle down to a position of minimal potential energy. Let the suspending points be (a,y(a)) and (b,y(b)) where (without loss of generality) b>a and y(b)>y(a). The potential energy delta p (relative to y(a)) of a portion of chain corresponding to small delta x is gievn by delta p = y(1+y'^2)^(1/2) delta x.

I don't understand where this potential comes from
 
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  • #2
I don't recognise it either apart from the square root but its often difficult to work backwards from part of an answer- several good derivations can be found at

http://en.wikipedia.org/wiki/Catenary

Regards

Sam
 
  • #3
y' is the slope, tan(θ). Squaring, adding 1 and sqrting gives sec(θ).
Multiplying that by dx gives ds, the length of the section of chain (hypotenuse). Taking the density to be 1, that's also its mass.
The final factor, y, seems intended to be its height above the reference point, y(a), but that seems a very confusing notation.
 

1. What is the catenary problem?

The catenary problem is a mathematical problem that involves finding the shape of a curve that a hanging chain or cable will form under its own weight and the force of gravity. This curve is known as a catenary curve and is described by the hyperbolic cosine function.

2. How is potential related to the catenary problem?

In the catenary problem, potential refers to the potential energy of the hanging chain or cable. The shape of the catenary curve is determined by finding the minimum potential energy of the system, which occurs when the chain is in equilibrium.

3. What is the significance of the catenary problem?

The catenary problem has many practical applications, such as in architecture and engineering. The catenary curve is a strong and stable shape, making it useful for designing arches and bridges. It is also used in the design of suspension bridges and cables for power lines.

4. How is the catenary problem solved?

The catenary problem can be solved using calculus and the principle of minimum potential energy. The equation for the catenary curve is derived by minimizing the potential energy of the system, taking into account the weight of the chain and the force of gravity.

5. Are there any real-life examples of the catenary problem?

Yes, there are many real-life examples of the catenary problem. Some notable examples include the Gateway Arch in St. Louis, Missouri, which is shaped like a catenary curve, and the cables of suspension bridges like the Golden Gate Bridge in San Francisco. The shape of hanging power lines and telephone wires can also be described by the catenary curve.

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