Question

Two vibrating strings of same material but lengths $L$ and $2L$ have radii $2r$ and $r$ respectively. They are stretched under the same tension and vibrate in their fundamental modes with frequencies $f_1$ and $f_2$ respectively. The ratio of $\frac{f_1}{f_2}$ is given by (strings are fixed at both ends)

• A. 2
• B. 4
• C. 8
• D. 1

Answer 1

The correct option is D 1

Fundamental frequency for vibration is given by,
$f=\frac{v}{2L}=\frac{1}{2L}\sqrt{\frac{T}{\mu }}$
$\Rightarrow f=\frac{1}{2L}\sqrt{\frac{T}{\rho s}}$, where $\mu =\rho s$
$s\to$ cross - sectional area of string
$\because s=\pi r^2)$
$\Rightarrow \frac{f_1}{f_2}=\frac{r_2{L}_2}{r_1{L}_1}=\frac{r\times 2L}{2r\times L}$
$\therefore \frac{f_1}{f_2}=1$