Question

A tuning fork of frequency 440 Hz is attached to a long string of linear mass density 0⋅01 kg m⁻¹ kept under a tension of 49 N. The fork produces transverse waves of amplitude 0⋅50 mm 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

Given,
Frequency of the tuning fork, f = 440 Hz
Linear mass density, m = 0.01 kgm⁻¹
Applied tension, T = 49 N
Amplitude of the transverse wave produce by the fork = 0.50 mm
Let the wavelength of the wave be $\lambda$.

(a) The speed of the transverse wave is given by
$\nu =\sqrt{\left(\frac{T}{m}\right)}$
$\Rightarrow v=\sqrt{\frac{49}{0.01}}=70\mathrm{m}/\mathrm{s}$
$$\begin{gathered} \mathrm{Also}, \\ \nu =\frac{f}{\lambda } \\ \therefore \lambda =\frac{f}{v}=\frac{70}{440}=16\mathrm{cm} \end{gathered}$$

(b) Maximum speed (v_max) and maximum acceleration (a_max):
$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ y=\mathrm{A}\sin \left(\omega t-kx\right) \end{gathered}$$
$$\begin{gathered} \therefore \nu =\frac{dy}{dt}=\mathrm{A}\omega \cos \left(\omega t-kx\right) \\ \mathrm{Now}, \\ {\nu }_{\max }=\left(\frac{dy}{dt}\right)=\mathrm{A}\omega \\ =0.50\times {10}^{-3}\times 2\pi \times 440 \\ =1.3816\mathrm{m}/\mathrm{s}. \\ \mathrm{And}, \\ a=\frac{d^{2}y}{d{t}^2} \\ \Rightarrow a=-\mathrm{A}{\omega }^2\sin \left(\omega t-kx\right) \\ {a}_{\max }=-\mathrm{A}{\omega }^2 \\ =0.50\times {10}^{-3}\times 4{\pi }^2{\left(440\right)}^2 \\ =3.8\mathrm{km}/{\mathrm{s}}^2 \end{gathered}$$

(c) Average rate (p) is given by
$$\begin{gathered} p=2{\pi }^2\nu A^2{f}^2 \\ =2\times 10\times 0.01\times 70\times {\left(0.5\times {10}^{-3}\right)}^2\times {\left(440\right)}^2 \\ =0.67\mathrm{W} \end{gathered}$$