Question

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

• A. $144\text{cm}$
• B. $204\text{cm}$
• C. $288\text{cm}$
• D. $72\text{cm}$

Answer 1

Answer: (A)

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