Question

A standing wave is produced on a string clamped at one end and free at the other. The length of the string

• A. may be an integral multiple of $\frac{\lambda }{2}$
  
• B. must be an integral multiple of $\frac{\lambda }{2}$
• C. must be an integral multiple of $\frac{\lambda }{4}$
• D. must be an integral multiple of $\lambda$

Answer 1

The correct option is C must be an integral multiple of $\frac{\lambda }{4}$

Draw a labelled diagram.
Find the length of the string.
As we know,
Fundamental frequency is given by,
$v=(n+\frac{1}{2})\frac{v}{2L}\dots \left(i\right)$
$v=v\lambda \dots \left(ii\right)$
From $\left(i\right)$ and $\left(ii\right)$,
$v=(n+\frac{1}{2})\frac{v\lambda }{2L}$

$L=\frac{(2n+1)\lambda }{4}$

$\therefore L=\frac{\lambda }{4},\frac{3\lambda }{4},\frac{5\lambda }{4}$

Final answer: (a)