Solving for Potential Energy of Hanging String: Integrals and Equations

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Homework Statement


This is actually just a minor part of a larger problem - I need to find the potential energy of a string of mass m and length L that is hanging over the edge of a table.


Homework Equations





The Attempt at a Solution


If we define V = 0 at the level of the talbe, then the potential energy of a mass element dm below the able is given by V = -gydm where y is the height of dm below the table. But here I blank - how can I use this to find the total potential energy o hte cord?
 
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What about thinking in terms of a mass density \delta so the potential of a string segment of length dy is -\deltaydy and integrating with respect to y?
 
I'm not sure I follow

the liear mass density would be m/L assuming that it is uniform, but how can I turn that into the integral?
 
KBriggs said:
I'm not sure I follow

the liear mass density would be m/L assuming that it is uniform, but how can I turn that into the integral?

For each segment dy of cable that is hanging over the edge a distance y it's potential is -\delta y\,dy. You have to add all these up, which you do by integrating with appropriate y limits.
 
So we get:

\int_0^y(\frac{-mg}{L}y)dy = \frac{-mg}{2L}y^2

Is that right, assuming that a length y is hanging over the edge?


The only problem is that I am not explicitly given L in the question, so I am not sure if I can use it. Is there a way to get the potential of a string of mass m hanging a distance y over the edge of a table without using the length? I can't think of anything.
 
y is the variable. You don't want it in the upper limit. If h is the length of the cable hanging over the edge your integral would go from 0 to h.
 
Alright - if you replace y by h in the above, is it correct? ^_^
 
It looks OK to me.
 
Thanks :)

Now that it's done, I see you can get the same thing without the integral by using the centre of mass of the part of the cord that is hanging over the edge.
 
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