Minimizing Potential Energy of a Hanging String: Calculus of Variations Approach

[brainmush]

1. A uniform string of length 2 meters hangs from two supports at the same height, 1 meter apart. by minimizing the potential energy of the string, find the equation describing the curve it forms and, in particular, find the vertical distance between the supports and the lowest point on the string.

Homework Equations

\lambdag\int y(x) dx ?

where lambda is linear density, g is acc. due to grav.

The Attempt at a Solution

Not sure how to start, thinking maybe using an equation for potential as an integral over length?

 

[gabbagabbahey]

Hi brainmush, welcome to PF!

We're not here to do your homework for you, we're here to help you learn. You must show some attempt at a solution, in order to receive help.

 

[brainmush]

i'm not asking to have my homework done for me. that's just the problem, we have 4 of us working on this problem and we've had some ideas, we're just not sure how to start the problem because everything we try ends up not working. i could post the calculations we have done that didn't work i just figured i wouldn't at first since they didn't work.

 

[lanedance]

hi brainmush

The question asks to minimise the potential energy...

might be a start to try and find the potential energy of a given string configuration... can you write this down in integral form?

 

[brainmush]

are you asking if i will as an example or if it is possible?

 

[gabbagabbahey]

are you asking if i will as an example or if it is possible?

He's hinting at the fact that you will need to first find/write down an expression for potential energy if you have any hope of minimizing it.

 

[brainmush]

well we're starting with P.E.=mgh but m is distributed as the curve between the supports, and h is given by the equation of that curve as a function of x, which is what we need to find. we're using m="lambda" int(dx) the integral under the equation section in the original post is the one we're trying to work with

 

[gabbagabbahey]

Homework Equations

\lambdag\int y(x) dx ?

where lambda is linear density, g is acc. due to grav.

I've just noticed that you added this to your post...it would have made more sense to put it in a new post where it would be noticed quicker

Okay, so this is the correct expression for the potential energy of the string; U=\lambda g \int y(x) dx...First, what are the integration limits?...second, how would you go about minimizing this integral?

 

[brainmush]

limits from 0 to 1, the minimization is what we are having trouble with if you have any suggestions

 

[gabbagabbahey]

Use the Euler Lagrange equations with f=y(x)...and keep in mind that the length of the string adds a constraint

 

[brainmush]

they are kicking us out of the lab that we are working on this in but we will all go home and see what we can come up with and when we reconvene tomorrow we will probably post again. thank you for the help.