Question

A string fixed at both ends has a standing wave mode for which the distances between adjacent nodes is $18cm$. For the next consecutive standing wave mode distances between adjacent nodes is $16cm$. The minimum possible length of the string is:

• A. $144cm$
• B. $204cm$
• C. $288cm$
• D. $72cm$

Answer 1

Answer: (A)

$144cm$

Let the vibration takes place is $nth$ mode
So, for $1st$ case, $\frac{n\lambda }{2}=L....\left(i\right)$
and for $2nd$ case, $(n+1)\frac{{\lambda }^{\prime }}{2}=L....\left(ii\right)$
From Eqs. (i) and (ii), we get
$n\frac{\lambda }{2}=(n+1)\frac{{\lambda }^{\prime }}{2}[\because \frac{\lambda }{2}=18cmand\frac{{\lambda }^{\prime }}{2}=16cm]$
$\Rightarrow 18n=(n+1)16$
$\Rightarrow n=8$
So, minimum possible length $l=\frac{n\lambda }{2}$
$\Rightarrow l=8\times 18=144cm$