Question

A taut string for which linear mass density is $\mu =5\times {10}^{-2}\text{kg/m}$ is under a tension of $80\text{N}$. How much power must be supplied to the string to generate sinusoidal waves at a frequency of $60\text{Hz}$ and an amplitude of $6\text{cm}$ ?

• A. $312\text{W}$
• B. $512\text{W}$
• C. $412\text{W}$
• D. $312\text{W}$

Answer 1

The correct option is B $512\text{W}$

Given:

$f=60\text{Hz};T=80\text{N};\mu =5\times {10}^{-2}\text{kg/m}$

We know that, power of a wave,

$P=\frac{1}{2}\rho {\omega }^2{A}^{2}sv$
here $s$ is area of cross section of the string
But, $\rho s=\mu$ and $v=\sqrt{\frac{T}{\mu }}$

$\therefore \rho sv=\sqrt{\mu T}$

Hence, power$\left(P\right)$ is,

$P=\frac{1}{2}(2\pi f{)}^2{A}^2\sqrt{\mu T}=2{\pi }^2{f}^2{A}^2\sqrt{T\mu }$

$\Rightarrow P=\left(2\right)\times (\pi {)}^2\times {60}^2\times (0.06{)}^2\times \sqrt{80\times 5\times {10}^{-2}}$

$\Rightarrow P=511.64\approx 512\text{W}$

Hence, option $\left(b\right)$ is correct.