Finding m_c: Tangent to Catenary Curve

In summary, the conversation discusses the use of variational calculus to minimize the surface area of a soap bubble and find its shape. It is found that the resulting function for the bubble's radius as a function of height is a catenary, with a constrained constant c. The question also mentions that the ratio of b/a must be less than a critical value, and the speaker wishes to find this value. They have observed that the tangents of the functions representing the boundary conditions appear to form a straight line with a gradient of 1.5089. This is consistent with the concept of an envelope, but the speaker is unsure how to show this analytically. They mention two restrictions but are unable to show how they lead to the same
  • #1
jimbobian
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Hi all, this question stems from a homework question but is not the homework question itself, more a discussion on something I found, hence why I have put it here.

The question involved using variational calculus to minimise the surface area of a soap bubble to find the shape it would take. The restrictions were that [itex]r=a[/itex] for [itex]z=\pm b[/itex] and I found the radius of the bubble as a function of height z to be:

[tex]r(z) = c\cosh(z/c)[/tex]

which is a catenary as expected. The constant c is constrained by the boundary conditions s.t:

[tex]a/c = \cosh(b/c)[/tex]

which the question points out only has solutions for the ratio [itex]b/a<m_c[/itex] where [itex]m_c[/itex] is some critical value. The question does not ask us to find [itex]m_c[/itex] but I wished to do so.

I plotted a selection of functions of the form [itex]a/c = \cosh(b/c)[/itex] for various [itex]c[/itex] and observed that the tangents appeared to form a straight line through the origin (which I have also added in red to help see it). This is consistent with what the question points out, but I can't seem to show why this is the case analytically (ie. that each catenary touches the same line through the origin). This line, incidentally, has a gradient of 1.5089 which I found using NR.

Could anyone point out why the catenaries form in this way, because I can't show (analytically) that they do.

Cheers,
James
 

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  • #3
Thanks for pointing that out, at least now I know what it is called and that it seems to be a general result (for some envelope). But is it possible to show that, in this case, the envelope is a straight line with gradient [itex]m_c[/itex] as above.

I find the following two restrictions:

[tex]a=c\cosh(b/c)[/tex]
[tex]c/b = \tanh(b/c)[/tex]

But no way of showing that this leads to the same gradient for each choice of c.
 

1. What is the significance of finding the tangent to the catenary curve?

The tangent to the catenary curve represents the slope of the curve at a specific point, which is important for determining the direction of motion or the force acting on an object at that point.

2. How is the tangent to the catenary curve mathematically calculated?

The tangent to the catenary curve can be calculated using the derivative of the curve's equation, which is expressed as y = a*cosh(x/a), where a is the constant of the curve and x is the independent variable representing the position along the curve.

3. What is the relationship between the catenary curve and the tangent line?

The tangent line is always perpendicular to the curve at a specific point, meaning that the slope of the tangent line is equal to the negative reciprocal of the slope of the curve at that point.

4. Can the tangent line intersect the catenary curve at more than one point?

Yes, the tangent line can intersect the catenary curve at multiple points, depending on the shape and position of the curve. However, there will always be exactly one tangent line at any given point on the curve.

5. How is the tangent to the catenary curve used in real-world applications?

The tangent to the catenary curve has many practical uses, such as in architecture and engineering for designing stable structures, in physics for calculating the trajectory of a projectile, and in calculus for finding the rate of change of a variable along the curve.

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