Solving for Potential Energy of Hanging String: Integrals and Equations

[KBriggs]

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?

 

[LCKurtz]

What about thinking in terms of a mass density $\delta$ so the potential of a string segment of length dy is -$\delta$ydy and integrating with respect to y?

 

[KBriggs]

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?

 

[LCKurtz]

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.

 

[KBriggs]

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.

 

[LCKurtz]

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.

 

[KBriggs]

Alright - if you replace y by h in the above, is it correct? ^_^

 

[LCKurtz]

It looks OK to me.

 

[KBriggs]

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.