Question

Spacing between two successive nodes in a standing wave on a string is $x$. If tension in the string is doubled keeping frequency constant, then find the new spacing between successive nodes.

• A. $\frac{x}{\sqrt{2}}$
• B. $\sqrt{2}x$
• C. $\frac{x}{2}$
• D. $2x$

Answer 1

The correct option is B $\sqrt{2}x$


Spacing between successive nodes is $x$.
$\Rightarrow x=\frac{{\lambda }_1}{2}$
$\Rightarrow {\lambda }_1=2x$ ..............(1)
Now, $v=f\lambda$
Here, frequency is kept fixed, so
$\lambda \propto v$
$\Rightarrow \lambda \propto \sqrt{\frac{T}{\mu }}$
$\Rightarrow \frac{{\lambda }_2}{{\lambda }_1}=\frac{\sqrt{T_2}}{\sqrt{T_1}}$
$(\because \mu$ is same for string)
$\Rightarrow \frac{{\lambda }_2}{{\lambda }_1}=\frac{\sqrt{2T}}{\sqrt{T}}\Rightarrow {\lambda }_2=\sqrt{2}{\lambda }_1$
New spacing between nodes is
$x^{\prime }=\frac{{\lambda }_2}{2}=\frac{\sqrt{2}{\lambda }_1}{2}$
$\Rightarrow x^{\prime }=\frac{{\lambda }_1}{\sqrt{2}}$
From equation (1)
$x^{\prime }=\frac{2x}{\sqrt{2}}=\sqrt{2}x$
Hence, option $\left(b\right)$ is the correct answer.