Question

A string of length $10\text{cm}$ fixed at one end and free at another end, has its fundamental frequency half that of second overtone for another string fixed at both ends. What will be the length of string fixed at both ends?
(Assume velocity of wave in both strings is same)

• A. $10\text{cm}$
• B. $20\text{cm}$
• C. $40\text{cm}$
• D. $30\text{cm}$

Answer 1

The correct option is D $30\text{cm}$

For a string fixed at one end and free at another end, fundamental mode
$f_0=\frac{V}{4L}$
$\Rightarrow f_0=\frac{V}{4\times 0.1}=\frac{V}{0.4}$
For the $2^{nd}$ overtone of string (fixed at both ends)
$f_2=\frac{3V}{2l}$
Given $f_0=\frac{f_2}{2}\frac{V}{0.4}=\frac{3V}{2l\times 2}$
$\therefore l=0.3\text{m}=30\text{cm}$