Question

A taut string for which $\mu =15\times {10}^{-2}\text{kg/m}$ is under a tension of $240\text{N}$. How much power must be supplied to the string to generate sinusoidal waves at a frequency of $18.0\text{Hz}$ and an amplitude of $18\text{cm}$?

• A. $1280\text{W}$
• B. $1300\text{W}$
• C. $1240\text{W}$
• D. $1220\text{W}$

Answer 1

The correct option is C $1240\text{W}$

Given that
$\mu =15\times {10}^{-2}\text{kg/m}$
Frequency $\left(f\right)=18\text{Hz}$
Tension $\left(T\right)=240\text{N}$
Amplitude $\left(A\right)=18\text{cm}=18\times {10}^{-2}\text{m}$
So wave velocity $\left(v\right)=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{240}{15\times {10}^{-2}}}=40\text{m/s}$
Angular frequency $\left(\omega \right)=2\pi f=2\pi \times 18.0=36\pi \text{rad/s}$

Supplied power $\left(P\right)=\frac{1}{2}\frac{{\omega }^2{A}^{2}T}{v}\text{J}$
So, $P=\frac{1}{2}\frac{{\omega }^2{A}^{2}T}{v}=\frac{1}{2}\times \frac{(36\pi {)}^2\times (18\times {10}^{-2}{)}^2\times 240}{40}$
$\Rightarrow P=1243.28\text{W}\approx 1240\text{W}$