How to Express the Curve of a String Hung Between Two Columns

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techjumper
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1. Imagine two columns between which is hung a string. The hung string may be expressed graphically and mathematically as a parabola whose nature is exponential. If these columns are moved closer together or farther apart, the hung string becomes more loose or taut according to a trigonometric function both of the length of the string and the distance between the two columns. 1. What is this trigonometric function? 2. Create a general equation which can anticipate the mathematical expression of the parabola based on any combination of string lengths and column distances.



2. I am not sure of the relevant equations but I believe the nature of the string behaves according to a hyperbolic trigonometric function.



3. I lack the mathematical aptitude and relevant physics knowledge to provide an attempted solution. Thank you for your help and guidance. Please see my attached photo showing my attempt to visually reproduce this phenomenon.
 

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techjumper said:
2. I am not sure of the relevant equations but I believe the nature of the string behaves according to a hyperbolic trigonometric function.

Why do you believe so? I am not saying it is wrong, it would just be of help to know how you argue to come to this conclusion.

3. I lack the mathematical aptitude and relevant physics knowledge to provide an attempted solution. Thank you for your help and guidance. Please see my attached photo showing my attempt to visually reproduce this phenomenon.

It would also help to know how much physics and maths you do know. The physical principle is not very difficult - the string takes the shape that minimizes its energy. The mathematics involve variational calculus with given constraints. I assume we are considering the string to be non-elastic so that it does not stretch under its own weight (although this could be taken into account for added complexity).
 
Orodruin said:
The mathematics involve variational calculus with given constraints. I assume we are considering the string to be non-elastic so that it does not stretch under its own weight (although this could be taken into account for added complexity).
I'm not sure the solver is expected to go through that. The OP states that it is a parabola. From the further remark
according to a trigonometric function both of the length of the string and the distance between the two columns.
it would appear that the ends of the string are to be taken to be at the same height.
That should be enough to solve the problem without calculus.