Question

Sinusoidal progressive wave of amplitude $10\text{cm}$ is to be sent along a string that has linear mass density of $4\times {10}^{–2}\text{kg/m}$ and a tension of $100\text{N}$. The source can deliver a maximum average power of $40\text{W}$. What is the maximum frequency (in Hz) at which source can operate? Take ${\pi }^2=10$

Answer 1

Maximum Average power of a wave :-
$P=\frac{1}{2}\mu v{A}^2{\omega }^2$
Here, $\omega =2\mu f$
$P=2{\pi }^2\mu v{A}^2{f}^2=2{\pi }^2\mu \sqrt{\frac{T}{\mu }}{A}^2{f}^2=2{\pi }^2\sqrt{T\mu }{A}^2{f}^2$
$f=\sqrt{\frac{P}{2{\pi }^2\sqrt{\mu T}{A}^2}}$
$=\sqrt{\frac{40}{2\times 10\sqrt{100\times 4\times {10}^{-2}}\times (10\times {10}^{-2}{)}^2}}$
$=\sqrt{\frac{40}{20\times 2\times {10}^{-2}}}=10\text{Hz}$