Question

A $200$ Hz wave with amplitude $1$ mm travels on a long string of linear mass density $6$ g/m kept under a tension of $60$ N. Find the total energy associated with the wave in a $2.0$ m long portion of the string.

Answer 1

We know that energy associated with one wavelength is given by

$U=\frac{1}{2}\mu {\omega }^2{A}^2\lambda$

where $\mu =6g/m=6\times {10}^{-3}kg/m$

$\omega =2\pi f=400\pi rad/s$

$A=1mm={10}^{-3}m$

$v=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{60}{.006}}=100m/s$

$\lambda =\frac{v}{f}=0.5m$

$⟹U=\frac{1}{2}\times 6\times {10}^{-3}\times (400\pi {)}^2\times ({10}^{-3})\times 0.5=2.37J$

Number of wavelengths $=\frac{2}{0.5}=4$

Total energy $=2.37\times 4=9.48J$