Question

A horizontal rope of $1\text{m}$ length, having a mass $100\text{g}$, is fixed at one end and is tied to a light string at the other end. Find the frequency and wavelength of the fifth overtone, if the tension in the string is $40\text{N}$.

• A. $50\text{Hz},20\text{cm}$
• B. $55\text{Hz},36\text{cm}$
• C. $110\text{Hz},36\text{cm}$
• D. $55\text{Hz},72\text{cm}$

Answer 1

The correct option is B $55\text{Hz},36\text{cm}$

Given that,
$l=1\text{m}$
$m=100\text{g}\text{or}0.1\text{kg}$
$T=40\text{N}$

Wave speed on a stretched string is

$v=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{40}{0.1}}=20\text{m/s}$

Frequency of fifth overtone for string fixed at one end

$f_5=\frac{11v}{4L}=\frac{11\times 20}{4\times 1}=55\text{Hz}$

and we know
$v=f_5{\lambda }_5$
$\Rightarrow$ wavelength ${\lambda }_5=\frac{v}{f_5}=\frac{20}{55}=0.3636\text{m}\approx 36\text{cm}$