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 \[\nu = \sqrt{\left( \frac{T}{m} \right)}\] 
\[\Rightarrow v = \sqrt{\frac{49}{0 . 01}} = 70  m/s\]

\[Also,   \] 

\[\nu = \frac{f}{\lambda}\] 

\[ \therefore   \lambda = \frac{f}{v} = \frac{70}{440} = 16  cm\]
(b) Maximum speed (v_max) and maximum acceleration (a_max):

We  have:

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

\[\therefore   \nu = \frac{dy}{dt} = A\omega  \cos  \left( \omega t - kx \right)\] 

\[Now, \] 

\[ \nu_\max  = \left( \frac{dy}{dt} \right) = A\omega\] 

\[  = 0 . 50 \times  {10}^{- 3}  \times 2\pi \times 440\] 

\[ = 1 . 3816  m/s . \] 

\[And,   \] 

\[a = \frac{d^2 y}{d t^2}\] 

\[ \Rightarrow a =  - A \omega^2   \sin  \left( \omega t - kx \right)\] 

\[ a_\max  =  - A \omega^2 \] 

\[ = 0 . 50 \times  {10}^{- 3}  \times 4 \pi^2    \left( 440 \right)^2 \] 

\[= 3 . 8  km/ s^2\] 
(c) Average rate (p) is given by

\[p = 2 \pi^2 \nu A^2  f^2 \] 

\[  = 2 \times 10 \times 0 . 01 \times 70 \times  \left( 0 . 5 \times {10}^{- 3} \right)^2  \times  \left( 440 \right)^2 \] 

\[ = 0 . 67  W\]