Question

In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency 500
Hz is used. The length of the air column is varied by changing the level of water in the resonance tube. Two
successive resonances are heard at air columns of length 50.7 cm and 83.9 cm. Which of the following
statements is (are) true?
(A) The speed of sound determined from this experiment is 332 ms⁻¹
(B) The end correction in this experiment is 0.9 cm
(C) The wavelength of the sound wave is 66.4 cm
(D) The resonance at 50.7 cm corresponds to the fundamental harmonic

Answer 1

Explanation:

Given In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency 500 Hz is used. The length of the air column is varied by changing the level of water in the resonance tube. Two  successive resonances are heard at air columns of length 50.7 cm and 83.9 cm. Which of the following  statements is (are) true?

(A) The speed of sound determined from this experiment is 332 ms⁻¹

(B) The end correction in this experiment is 0.9 cm

(C) The wavelength of the sound wave is 66.4 cm

(D) The resonance at 50.7 cm corresponds to the fundamental harmonic

• Let the harmonic of air column of length 50.7 cm be h1 and the harmonic of air column of length  83.9 cm be h2
• The difference of harmonics will be λ/2 since both are consecutive.
• Therefore λ/2 = 83.9 – 50.7
•              = 33.2 cm
• Or λ = 33.2 x 2
•          = 66.4 cm
• Also λ/4 = 66.4 / 4 = 16.6 cm
• Since 16.6 x 3 = 49.8 and this is closer to 50.7 cm and so it is third harmonic.so the length corresponding to fundamental mode should be close to λ/4..
• Now e + 50.7 = 3λ / 4
• Or e + 50.7 = 49.8
• Or e = 49.8 – 50.7
• Or e = - 0.9 cm
• We have speed of sound v = f λ
•                  So v = 500 x 66.4  
•                          = 33200 cm / s
•                         = 332 m/s
• Therefore option A, B and C holds good.

Reference link will be

https://brainly.in/question/16193989

Answer 2

Given:

Frequency of tuning fork = 500 Hz

Length of air column at first resonance = 50.7 cm

Length of air column at second resonance = 83.9 cm

Calculation:

Consider n₁ as the harmonic frequency corresponding to 50.7 cm & n₂ as the harmonic frequency corresponding to 83.9 cm

As they are successive harmonics, their difference will be equal to λ/2

     $\frac{\lambda}{2}=83.9-50.7\\ \\ \frac{\lambda}{2}=33.2\\ \\ \lambda=66.4\ cm$

The wavelength of the sound wave is 66.4 cm

The length corresponding to fundamental frequency should be equal to λ/4. 50.7 should be closed to an odd multiple of length. Therefore 50.7 will be the third harmonic

     $e+50.7=\frac{3\lambda}{4} \\ \\e+50.7=49.8\\ \\e=49.8-50.7\\ \\e=-0.9\ cm$

The end correction in this experiment is 0.9 cm

Speed of sound is given by the formula

     $v=f\lambda \\ \\v=500\times 66.4 \\ \\v=332\ m/sec$

Speed of sound determined from this experiment is 332 m/s

Final answer:

Options A, B, C are correct