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

Answer 1

Answer: (C)

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