Plucked string, potential energy

[Unicorn.]

Hello, I don't understand the second question, i don't know what I have to do:

Homework Statement

A string of length L, which is clamped at both ends and has a tension
T, is pulled aside a distance h at its center and released.
(a) What is the energy of the subsequent oscillations?
(b) Find the approximative expression of this energy for small oscillations

Homework Equations

The Attempt at a Solution

In a/ I found that U=2Th²/L
For the second question I took a segment dx, so we have that
dU=T(ds-dx)
ds=dx(1+1/2(dy/dx)²)
I replaced it so that
U= T/2∫(dy/dx)²dx from 0 to L
I'm stuck here, how can I find the exact potential energy for small oscillations, I'm not even sure that I'm using the good method since in the next questions, they ask to show that the energy in b is the sum of potential energies of each mode, so I can't use y(x,t) to find dy/dx in this question, right ?
Thanks

 

[haruspex]

Looks to me that you have answered (b), not (a). I would have taken your answer to (a) as correct if it were not for the fact that it goes on to ask (b). This makes me think that (a) is asking for an exact expression, with no approximations.

 

[Unicorn.]

I don't understand, in the question (a) it's asking for the exact expression, and b for an approximation of U for small oscillations

 

[haruspex]

I don't understand, in the question (a) it's asking for the exact expression, and b for an approximation of U for small oscillations

We agree on that, but your answer to (a) is not exact, in several ways. It is only an approximation valid for small h.

 

[Unicorn.]

I'm really confused.
So the answer to (a) must be U=Tn²pi²A²/4L we use y(x,0)
And for (b) it must be U=2Th²/L
Then, for c/, they ask to show that the potential energy is the sum of potential energy of each mode by finding the same expression as b/
So I have to find U=2Th²/L, right ?
The problem now I don't know how to find b/ by using "small oscillation argument" without answering to the question c/ at the same time.

 

[haruspex]

So the answer to (a) must be U=Tn²pi²A²/4L we use y(x,0)

The initial half length is L/2. If the centre point is displaced h orthogonally to that, what is the exact extension?

And for (b) it must be U=2Th²/L

That's my interpretation.

Then, for c/, they ask to show that the potential energy is the sum of potential energy of each mode by finding the same expression as b/
So I have to find U=2Th²/L, right ?

I'm not sure what c is asking for. It sounds like they want you to do the Fourier analysis to find the energy at each harmonic, but I wouldn't know how to do that other than by assuming the total of those is what you calculated in b.

 

[Unicorn.]

I did c/ and I found U=2Th²/L
I don't understand what you're asking for a/ ..?

 

[haruspex]

I don't understand what you're asking for a/ ..?

The initial length is L. If the centre point is displaced h orthogonally to that, what is the exact length of the wire now? How much longer is that than L?

 

[Unicorn.]

It says that we neglect the extension of the wire.
Now, I don't know how to deal with the a/ question , is it a n function ?

 

[haruspex]

It says that we neglect the extension of the wire.

Not in the OP, it doesn't . Please post the question exactly as given.

 

[Unicorn.]

Sorry, I didn't notice that i forgot it. "We neglect the extension of the wire and the change of the tension."

 

[haruspex]

Sorry, I didn't notice that i forgot it. "We neglect the extension of the wire and the change of the tension."

That changes things. Now I agree with your answer for (a) but I'm mystified as to what (b) is asking. Note that you've referred to potential energy a few times, but in the question as you posed it it just says energy.