A string fixed at both ends has consecutive standing wave modes for which the distances between adjacent…
A string fixed at both ends has consecutive standing wave modes for which the distances between adjacent nodes are 18 cm and 16 cm respectively.
(i) What is the minimum possible length of the string?
(ii) If the tension is 10 N and the linear mass density is 4gm$^{-1}$, what is the fundamental frequency?
Solution
Sol. (i)
Let l be the length of the string. Then,
$18n = l$ ...(i)
$16(n + 1) = l$ ...(ii)
From Eqs. (i) and (ii), we get
$n = 8$ and $l = 144\ \text{cm}$
Therefore, the minimum possible length of the string can be 144 cm.
(ii) For fundamental frequency, $l = \lambda/2$
or
$\lambda = 2l = 288\ \text{cm} = 2.88\ \text{m}$
Speed of wave on the string,
$v = \sqrt{T/\alpha} = \sqrt{\dfrac{10}{4\times10^{-3}}} = 50\ \text{ms}^{-1}$
$\therefore$ Fundamental frequency, $f = \dfrac{v}{\lambda} = \dfrac{50}{2.88}$
$= 17.36\ \text{Hz}$
Answer: $\lambda = 2l = 288\ \text{cm} = 2.88\ \text{m}$