Question

In an experiment with a sonometer a tuning fork of frequency 256Hz resonates with a length of 25cm and another tuning fork resonates with a length of 16cm. Tension of the string remaining constant the frequency of the second tuning fork is.
A. $163.84Hz$
B. $400Hz$
C. $320Hz$
D. $204.8Hz$

Answer 1

Hint: when string is fixed from both ends nodes will be formed at both ends. When a string is only fixed from one end and set free at the other end node is formed at fixed end and antinode is formed at the free end. Here we match frequency of the sonometer with tuning fork frequency as they will be under resonance. We consider both ends fixed in case of a sonometer.

Formula used:
$v = f\lambda $

Complete answer:
When string is fixed at both ends initially one loop will be formed and hence wavelength of that loop be ${\lambda _1}$
For one loop distance is $\dfrac{{{\lambda _1}}}{2}$
So initially $\dfrac{{{\lambda _1}}}{2} = l$ , ${\lambda _1} = 2{l_1}$ … eq(1)
When second string is fixed at both ends initially one loop will be formed and hence wavelength of that loop be ${\lambda _2}$
For one loop distance is $\dfrac{{{\lambda _2}}}{2}$
So initially $\dfrac{{{\lambda _2}}}{2} = l$ , ${\lambda _2} = 2{l_2}$ … eq(2)
In both the cases we will consider two end fixed cases because in case of a sonometer two ends will be fixed. Neither of the ends will be set free.
It is given that tension is constant. So the velocity of the wave will be the same in both cases as the linear mass density of the string will be constant.
$v = f\lambda $
f is frequency and ‘v’ is velocity and lambda is wavelength.
From equation one and two we have
$\eqalign{
  & {v_1} = {v_2} \cr
  & \Rightarrow {f_1}{\lambda _1} = {f_2}{\lambda _2} \cr
  & \Rightarrow \left( {256} \right)\left( {25} \right) = {f_2}\left( {16} \right) \cr
  & \therefore {f_2} = 400Hz \cr} $

Hence option B is correct answer.

Note:
In the question it is given that different tuning forks are resonating with sonometer wire at different lengths. We call a phenomenon a resonance when their frequencies meet. So at resonance we had considered the frequency of tuning fork same as the frequency of the sonometer.