A pipe of length 0.85 m is closed at…

A pipe of length 0.85 m is closed at one end and open at the other. The speed of sound in the air in it is 340 m s^-1 at the room temperature. A…

One or more options correct

A pipe of length

0.85\ \mathrm{m}

is closed at one end and open at the other. The speed of sound in the air in it is

340\ \mathrm{m\,s^{-1}}

at the room temperature. A tuning fork of frequency

450\ \mathrm{Hz}

moves directly towards the open end at

34\ \mathrm{m\,s^{-1}}

, the air being at rest. Neglect end correction. Which of the following statements is/are correct?

A. The pipe resonates in its third overtone.
B. The pipe has exactly three displacement nodes along it (counting the closed end).
C. If the fork moved away from the pipe at $34\ \mathrm{m\,s^{-1}}$ the pipe would resonate in its fundamental mode.
D. If the air in the pipe and surroundings were warmed so that the speed of sound is $350\ \mathrm{m\,s^{-1}}$ , the pipe would still resonate with the fork approaching at $34\ \mathrm{m\,s^{-1}}$ .

Show answer and solution

Answer: B

The pipe's natural frequencies are

f_n=(2k+1)\frac{v}{4L}=100,300,500,\dots

Hz. The frequency reaching the pipe is

f'=450\times\frac{340}{340-34}=500

Hz, which is the fifth harmonic, i.e. the second overtone, so (A) is false. For the fifth harmonic

L=5\lambda/4

and nodes occur at the closed end and then every

\lambda/2

: three nodes, so (B) is true. Receding,

f'=450\times\frac{340}{374}\approx409

Hz, which is none of

100,300,500

Hz, so (C) is false. At

v=350

the natural fifth harmonic is

514.7

Hz while

f'=450\times\frac{350}{316}\approx498.4

Hz, so resonance fails and (D) is false.

Original question written in the JEE Advanced 2020 Paper 1 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake