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 distance between adjacent nodes is $16cm.$ The minimum possible length of the string is

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

Answer 1

Answer: (A)

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