Question

A tuning fork of frequency $440Hz$ is attached to a long string of linear mass density $0.01kg{m}^{-1}$ kept under a tension of $49N$. The fork produces transverse waves of amplitude $0.50mm$ on the string.

(a) Find the wave speed and the wavelength of the waves.

(b) Find the maximum speed and acceleration of a particle of the string.

(c) At what average rate is the tuning fork transmitting energy to the string?

Answer 1

Step 1: Given Data:

Frequency of the Tuning fork $f=440Hz$

Linear Mass density of the String $\rho =0.01kg{m}^{-1}$

Tension applied on the String $T=49N$

The amplitude of the Transverse waves $A=0.5\times {10}^{-3}m$

Step 2: Find the speed and wavelength of the wave.

If $v$ is the speed then, $v=\sqrt{\frac{T}{\rho }}$

$=\sqrt{\frac{49}{0.01}}$

$=70m{s}^{-1}$

Now, if $\lambda$is the wavelength, then $\lambda =\frac{v}{f}$

$=\frac{70}{440}$

$=0.16m$

Step 3: Find the maximum speed and acceleration of the particle:

If $y$is the displacement of the particle at time $t$, then it is given by

$y=A\sin \left(\omega t-kx\right)$

where $\omega$ is the angular velocity and $\omega =2\pi f$

$=2\times 3.14\times 440$

$=2763.2s^{-1}$

The velocity $v$ is given by $v=\frac{dy}{dt}$

$=A\omega C\mathrm{os}\left(\omega t-kx\right)$

Maximum velocity is given by $v_{max}=A\omega$

$=0.5\times {10}^{-3}\times 2763.2$

$=1.3816m{s}^{-1}$

Acceleration $a$is given by $a=\frac{dv}{dt}$

$=-A{\omega }^2\sin \left(\omega t-kx\right)$

Maximum acceleration is $a_{max}=A{\omega }^2$

$=0.5\times {10}^{-3}\times {\left(2763.2\right)}^2$

$=3.8km{s}^{-2}$

Step 4: Find the average rate:

If $r$ be the average rate, then $r=2{\pi }^{2}v{A}^2{f}^2$

$=2\times {\left(3.14\right)}^2\times 70\times {\left(0.5\times {10}^{-3}\right)}^2\times {\left(440\right)}^2$

$=0.67W$

Hence, the maximum velocity and acceleration are $\mathbf{1}\mathbf{.}\mathbf{3816}\mathbf{}\mathit{m}{\mathit{s}}^{\mathbf{-}\mathbf{1}}$ and $\mathbf{3}\mathbf{.}\mathbf{8}\mathbf{}\mathit{k}\mathit{m}{\mathit{s}}^{\mathbf{-}\mathbf{2}}$ respectively along with an average rate of energy transmittance of $\mathbf{0}\mathbf{.}\mathbf{67}\mathbf{}\mathit{W}$