Two vibrating strings of same material but lengths $l$ and $2l$ have radii $2r$ and $r$, respectively. They…
Two vibrating strings of same material but lengths $l$ and $2l$ have radii $2r$ and $r$, respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental modes, the string of length $l$ with frequency $n_1$ and the other with frequency $n_2$. The ratio of $\frac{n_1}{n_2}$ is given by
2
4
8
1
Solution
Fundamental frequency, $f = \frac{v}{2l} = \frac{\sqrt{T/\mu}}{2l}$
where, $\mu = \text{mass per unit length} = \rho s$
or $f = \frac{1}{2l}\sqrt{\frac{T}{\rho s}}$ but $s = \pi r^2$
$\therefore f \propto \frac{1}{rl}$ or $\frac{f_1}{f_2} = \frac{r_2 l_2}{r_1 l_1} = \frac{r \times 2l}{2r \times l} = 1$