JEE Advanced 2018 Paper 1 · Physics · Waves and Sound · hard

A taut string of linear mass density 5.0×10^-3 kg m^-1 and length 1.2 m is fixed at both ends and held at tension 72 N. A closed organ pipe of length…

One or more options correct

A taut string of linear mass density

5.0\times10^{-3}\ \mathrm{kg\,m^{-1}}

and length

1.2\ \mathrm{m}

is fixed at both ends and held at tension

72\ \mathrm{N}

. A closed organ pipe of length

0.55\ \mathrm{m}

stands near it (speed of sound in air

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

, end correction ignored). Which of the following is/are correct?

A. The fundamental frequency of the string is $50\ \mathrm{Hz}$ .
B. The third harmonic of the string is in resonance with the fundamental mode of the pipe.
C. The second harmonic of the string is in resonance with a mode of the pipe.
D. If the tension is raised by $4\%$ , the string's third harmonic and the pipe's fundamental produce about $3$ beats per second.

Show answer and solution

Answer: A, B, D

Wave speed on the string is

v=\sqrt{T/\mu}=\sqrt{72/0.005}=120\ \mathrm{m\,s^{-1}}

, so

f_1=v/2L=50\ \mathrm{Hz}

and (A) is true. The closed pipe resonates at

f=(2k-1)\dfrac{330}{4\times0.55}=150,450,750\ \mathrm{Hz}

, odd multiples of

150\ \mathrm{Hz}

. The string's third harmonic is

150\ \mathrm{Hz}

, so (B) is true. The second harmonic is

100\ \mathrm{Hz}

, which is not in the pipe's list, so (C) is false. Since

f\propto\sqrt T

, a

4\%

rise in tension changes

150\ \mathrm{Hz}

150\sqrt{1.04}\approx152.97\ \mathrm{Hz}

, giving about

3

beats per second against the pipe, so (D) is true.

Original question written in the JEE Advanced 2018 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