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Energy of vibrating string

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1. I'm having some trouble following the derivation for the energy of a vibrating string.

For KE:

The KE is (1/2)(m)(v)2, which is (1/2)(Ddx)(dy/dt)2 for a given segment.

Where D is the linear density and dx is the length of the segment.

y = Asin(kx-wt)

So dy/dt = -Awcos(kx-wt)

So KE for the segment is (1/2)(Ddx)(-Awcos(kx-wt))2

Integrating over the total length, L, gives (1/2)(DA2w2)(L/2)

That last term is confusing me - I don't understand how the integral of cos2(kx-wt) is L/2

For PE:

The derivation leads with PE of segment = T(dL - dx) where T is tension.

I'm pretty baffled by this statement, because in the KE derivation, L was the total length of the string, and had nothing to do with the vertical displacement - which I would expect to see in the PE.

If anyone can help explain these I'd be most grateful.
Thank you.
2. (Original post by 99wattr89)
I'm having some trouble following the derivation for the energy of a vibrating string.

For KE:

The KE is (1/2)(m)(v)2, which is (1/2)(Ddx)(dy/dt)2 for a given segment.

Where D is the linear density and dx is the length of the segment.

y = Asin(kx-wt)

So dy/dt = -Awcos(kx-wt)

So KE for the segment is (1/2)(Ddx)(-Awcos(kx-wt))2

Integrating over the total length, L, gives (1/2)(DA2w2)(L/2)

That last term is confusing me - I don't understand how the integral of cos2(kx-wt) is L/2

For PE:

The derivation leads with PE of segment = T(dL - dx) where T is tension.

I'm pretty baffled by this statement, because in the KE derivation, L was the total length of the string, and had nothing to do with the vertical displacement - which I would expect to see in the PE.

If anyone can help explain these I'd be most grateful.
Thank you.

It looks like it's been integrated with respect to x, from 0 to L, giving (1/2)(Ddx)(-Awcos(kx-wt))2, and then they've found the average KE.

Since the KE is sinusoidal, you can find the average value by looking at the average value of cos2, which is a half (try drawing the graph if you need convincing).

So yeah, check if the textbook mentions deriving the average KE, rather than the varying value.
3. (Original post by vandub)
It looks like it's been integrated with respect to x, from 0 to L, giving (1/2)(Ddx)(-Awcos(kx-wt))2, and then they've found the average KE.

Since the KE is sinusoidal, you can find the average value by looking at the average value of cos2, which is a half (try drawing the graph if you need convincing).

So yeah, check if the textbook mentions deriving the average KE, rather than the varying value.
Sorry I was slow to reply, I've just had a lot of deadlines and stuff to meet.
I think you're right about it being average KE, thanks.
4. For the potential energy I think it means the elastic potential energy - i.e. the work done in stretching a segment of the string by a length dL-dx against a tensile (restoring) force of T.

I'm puzzled why I got negged...to try and answer your question: vertical displacement doesn't appear in the expression because given a sinosiodal shape - the mass of the string below the equilibrium position is the same as the mass above it, so the overall change in GPE is 0.
5. (Original post by XiaoXiao1)
For the potential energy I think it means the elastic potential energy - i.e. the work done in stretching a segment of the string by a length dL-dx against a tensile (restoring) force of T.

I'm puzzled why I got negged...to try and answer your question: vertical displacement doesn't appear in the expression because given a sinosiodal shape - the mass of the string below the equilibrium position is the same as the mass above it, so the overall change in GPE is 0.
I see! Thank you!

I don't see why you got negged either.

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