Question

In order to double the frequency of the fundamental note emitted by a stretched string, the length is reduced to ${\left(\frac{3}{4}\right)}^{\text{th}}$ of the original length (mass per unit length remains same) and the tension is changed. The factor by which the tension is to be increased is

• A. $\frac{3}{8}$
• B. $\frac{2}{3}$
• C. $\frac{8}{9}$
• D. $\frac{9}{4}$

Answer 1

The correct option is D $\frac{9}{4}$

Frequency of fundamental note produced by a stretched string (fixed at both ends) is,
$f=\frac{1}{2L}\sqrt{\frac{T}{\mu }}$
$\Rightarrow f\propto \frac{\sqrt{T}}{L}$
$\Rightarrow \frac{f_2}{f_1}=\sqrt{\frac{T_2}{T_1}}\times \frac{L_1}{L_2}\text{Given:}{f}_2=2f_1,L_2=\frac{3}{4}{L}_1$
$\Rightarrow \frac{T_2}{T_1}={\left(\frac{f_2}{f_1}\right)}^2\times {\left(\frac{L_2}{L_1}\right)}^2$
$\Rightarrow \frac{T_2}{T_1}=4\times \frac{9}{16}$
$\therefore T_2=\frac{9}{4}{T}_1$
Tension has to be increased by a factor of $\left(\frac{9}{4}\right)$