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?

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## Comments

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TopNewestI think it will be a standing wave and not a progressive wave, whose equation you have written. Hence, amplitude for every point of the string is different.

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I think you're right.

See this.

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

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

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Are vah yyaar maja hi aa Gaya shandaar

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