Question

A string fixed at both ends has a standing wave mode for which the distances between adjacent nodes is $18 \, cm.$ For the next consecutive standing wave mode distance between adjacent nodes is $16 \, cm.$ 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, $\frac{n \lambda}{2}=L \ldots \ldots$ (i)
And for $2^{n d}$ case, $(n+1) \frac{(\lambda)^{\prime}}{2}=L \ldots$...(ii)
From Equations. (i) and (ii), we get
$n \frac{\lambda}{2}=(n+1) \frac{\lambda^{\prime}}{2}\left[\because \frac{\lambda}{2}=18 cm\right.$ and $\left.\frac{\lambda^{\prime}}{2}=16 \,cm \right]$
$\Rightarrow 18 n=(n+1) 16$
$\Rightarrow n=8$
So, minimum possible length $l=\frac{n \lambda}{2}$
$\Rightarrow l=8 \times 18=144 \,cm$