Question

In a resonant column method, resonance occurs at two successive levels of ${l_1} = 30.7cm$ and ${l_2} = 63.2cm$ using a tuning fork of $f = 512Hz$. What is the maximum error in measuring speed of sound using the relations $v = f\lambda $ and $\lambda = 2({l_2} - {l_1})$?
A. $256cm/\sec $
B. $92cm/\sec $
C. $128cm/\sec $
D. $102.4cm/\sec $

Answer 1

Hint: When a vibrating body applies force on another body around it and makes it vibrate with a greater frequency, it is known as resonance. In a resonant column experiment, the velocity of the sound waves in the air at room temperature is calculated. It can be done by using forks of some known frequency. In order to find the wavelength of the stationary waves produced, resonating length is obtained.

Complete step by step answer:
Step I:
Given that the two levels where the resonance occurs is
${l_1} = 30.7cm$ and ${l_2} = 63.2cm$
$f = 512Hz$
Step II:
Given $\lambda = 2({l_2} - {l_1})$---(i)
$\lambda = 2(63.2 - 30.7)$
$\lambda = 2 \times 32.5$
$ \Rightarrow \lambda = 65cm$
Step III:
Given $v = f\lambda $---(ii)
Where f is the frequency
And $\lambda$ is the wavelength of the waves
Substituting the values and solving,
$v = 512 \times 65$
$v = 33280cm/\sec $
Or $v = 332.8m/s$
Step IV:
From equation (ii) the maximum error can be calculated using
$ \Rightarrow \dfrac{{\Delta v}}{v} = \dfrac{{\Delta f}}{f} + \dfrac{{\Delta \lambda }}{\lambda }$---(iii)
The least count of the resonant column scale is $ = 0.1cm$
Total measurement of $\lambda$ is seen from equation (i) and is written as
$ \Rightarrow \lambda = 2 \times \text{Least Count}$
$ = 2 \times 0.1$
$ = 0.2cm$
Total error in the measurement of frequency is zero. Hence $\Delta f = 0$
Step V:
Substitute all the values in equation (iii) and solving for error in speed
$ \Rightarrow \dfrac{{\Delta v}}{{332.8}} = 0 + \dfrac{{0.2}}{{65}}$
$\Delta v = \dfrac{{0.2 \times 332.8}}{{65}}$
$\Delta v = 1.024m/s$
Or $ \Rightarrow \Delta v = 102.4cm/\sec $

$\therefore $The maximum error in the speed of sound waves is $ = 102.4cm/\sec $
Hence, the correct answer is option (D).

Note: It is important to remember that if the frequency and the wavelength are known, then the velocity with which the sound waves travel can be known. When the natural frequency of the sound waves becomes equal to the natural frequency of the tuning fork, then resonance occurs. As a result, a loud sound is produced in the resonance column.