Question

A string fixed at both ends has consecutive standing wave modes for which the distances between adjacent nodes are $18\text{cm}$ and $16\text{cm}$ respectively. The minimum possible length of the string is:

• A. $144\text{cm}$
• B. $152\text{cm}$
• C. $176\text{cm}$
• D. $200\text{cm}$

Answer 1

The correct option is A $144\text{cm}$

For a string fixed at both ends
$L=n\frac{\lambda }{2}$
the consecutive standing wave modes will occur for $n$ and $(n+1)$. So,
$L=n\frac{{\lambda }_1}{2}$...................(1)
$L=(n+1)\frac{{\lambda }_2}{2}$........(2)
From question
$\frac{{\lambda }_1}{2}=18\text{cm}$ and $\frac{{\lambda }_2}{2}=16\text{cm}$
From Eq. (1) and (2),
$n\left(18\right)=(n+1)16$
$\Rightarrow 18n=16n+16$
$\therefore n=8$
putting the value of $n$ in Eq. (1),
$L=8\times 18=144cm$
Thus, option $(a$) is the correct answer.