Here is a question,

A string of mass \(m\) and length \(l\), both fixed at both ends is vibrating in its fundamental mode. The maximum amplitude is \(a\) and the tension in the string is \(T\). Find the energy of vibrations of the string in terms of \(a\), \(T\) and \(l\).

This is how I answered,

Let the equation of wave be \(y=f\left( x,t \right) =a\sin { \left( \omega t-kx \right) } \)

Differentiating w.r.t \(t\),

\(\frac { \delta y }{ \delta t } ={ v }_{ p }=a\omega \cos { \left( \omega t-kx \right) } \)

\(\therefore \) Kinetic Energy, \(K.E=\frac { 1 }{ 2 } m{ a }^{ 2 }{ \omega }^{ 2 }\cos ^{ 2 }{ \left( \omega t-kx \right) } \)

\(\therefore\)Average Kinetic Energy, \(\left< K.E \right> =\frac { \int _{ 0 }^{ T }{ \frac { 1 }{ 2 } m{ a }^{ 2 }{ \omega }^{ 2 }\cos ^{ 2 }{ \left( \omega t-kx \right) } dt } }{ \int _{ 0 }^{ T }{ dt } } =\frac { 1 }{ 4 } m{ a }^{ 2 }{ \omega }^{ 2 }\)

Potential Energy, \(P.E=\frac { 1 }{ 2 } m{ \omega }^{ 2 }{ a }^{ 2 }\sin ^{ 2 }{ \left( \omega t-kx \right) } \)

\(\therefore\) Average Potential Energy,\(\left< P.E \right> =\frac { \int _{ 0 }^{ T }{ \frac { 1 }{ 2 } m{ a }^{ 2 }{ \omega }^{ 2 }\sin ^{ 2 }{ \left( \omega t-kx \right) } dt } }{ \int _{ 0 }^{ T }{ dt } } =\frac { 1 }{ 4 } m{ a }^{ 2 }{ \omega }^{ 2 }\)

Therefore **Total energy**=\(\left< K.E \right>+\left< P.E \right>=\frac { 1 }{ 2 } m{ a }^{ 2 }{ \omega }^{ 2 }\)

Now,

\({ \omega }^{ 2 }={ v }^{ 2 }{ k }^{ 2 }={ v }^{ 2 }\frac { 4{ \pi }^{ 2 } }{ { \lambda }^{ 2 } } =\frac { T }{ \frac { m }{ L } } \frac { 4{ \pi }^{ 2 } }{ { \lambda }^{ 2 } } =\frac { 2T{ \pi }^{ 2 } }{ m\lambda } =\frac { T{ \pi }^{ 2 } }{ mL } \)

\(\therefore\) **Total Energy**\(=\boxed{\frac { T{ a }^{ 2 }{ \pi }^{ 2 } }{ 2L } }\)

But the given answer is \(\boxed{\frac { T{ a }^{ 2 }{ \pi }^{ 2 } }{ 4L} }\)

Can someone explain this?

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestAre vah yyaar maja hi aa Gaya shandaar

Log in to reply

I think you're right.

See this.

Unless, they meant only the energy due to the motion, which is the kinetic energy.

Log in to reply

Again you are there to help me! Thank you, Sir :)

Log in to reply