Question

A string of length $2\text{m}$ is fixed at both ends. If the speed of transverse wave on the string be $500\text{m/s}$, then what will be the frequency of vibration of string in its fifth harmonic and corresponding wavelength of the wave?

• A. $125\text{Hz},0.5\text{m}$
• B. $625\text{Hz},0.8\text{m}$
• C. $500\text{Hz},1.25\text{m}$
• D. $250\text{Hz},0.5\text{m}$

Answer 1

The correct option is B $625\text{Hz},0.8\text{m}$

Fundamental frequency.
$f=\frac{v}{2l}$
$\Rightarrow f_n=\frac{nv}{2l}$
for fifth harmonic, n = 5
$\Rightarrow f_5=\frac{5v}{2l}$
Given $v=500\text{m/s},l=2m$
$\therefore f_5=\frac{500\times 5}{2\times 2}=625\text{Hz}$
Using $v=f\lambda$
$\therefore \lambda =\frac{500}{625}=\frac{4}{5}m=0.8\text{m}$

Alternative,
For fifth harmonic vibration, number of loops formed in strings $(n=5)$
$\Rightarrow L=n\times \frac{\lambda }{2}$
$\Rightarrow 2=5\times \frac{\lambda }{2}$
$\lambda =\frac{4}{5}=0.8\text{m}$
using $v=f\lambda$
$\therefore f=\frac{v}{\lambda }=500\times \frac{5}{4}=625\text{Hz}$