Question

A string of mass m is fixed at both ends. The fundamental tone oscillations are excited in the strong with angular frequency $\omega$ and maximum displacement amplitude A. Find the total energy curtained in the string.

• A. $\frac{1}{2}{m\omega }^2{A}^2$
• B. $\frac{1}{4}{m\omega }^2{A}^2$
• C. $\frac{1}{6}{m\omega }^2{A}^2$
• D. $\frac{1}{8}{m\omega }^2{A}^2$

Answer 1

The correct option is B $\frac{1}{4}{m\omega }^2{A}^2$

$y=A\sin kx\cos \omega t$ where $k=\frac{2\pi }{\lambda }$

$\frac{dy}{dt}=-A\omega \sin kx\sin \omega t$

Kinetic energy of element dx at position x is

$d{k}_n=\frac{1}{2}\mu dx{\left(\frac{dy}{dt}\right)}^2$

$=\frac{1}{2}\mu A^2{\omega }^2{\cos }^2\omega t\cdot {\sin }^{2}kx.dx$

$k_n={\int }_{x=0}^{x=1}d{k}_n=\frac{1}{4}M{A}^2{\omega }^2{\cos }^2\omega t$.