JEE Advanced 2020 Paper 2 · Physics · Waves and Sound · medium

A wire of length 0.80 m, mass per unit length 0.010 kg m^-1, is fixed at both ends under tension 64 N. A closed-at-one-end air column of length 1.70…

One or more options correct

A wire of length

0.80\ \mathrm{m}

, mass per unit length

0.010\ \mathrm{kg\,m^{-1}}

, is fixed at both ends under tension

64\ \mathrm{N}

. A closed-at-one-end air column of length

1.70\ \mathrm{m}

is placed beside it (speed of sound

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

). Choose the correct statement(s).

A. The third harmonic of the wire is one of the resonant frequencies of the air column.
B. The second harmonic of the wire resonates with the air column.
C. If the tension is increased by $2\%$ , the beat frequency between the wire's fundamental and the air column's fundamental is about $1.0\ \mathrm{Hz}$ .
D. If the temperature of the air rises, the fundamental frequency of the air column decreases.

Show answer and solution

Answer: A

Wire:

v=\sqrt{T/\mu}=\sqrt{64/0.01}=80\ \mathrm{m\,s^{-1}}

, so

f_n=nv/2L=50n\ \mathrm{Hz}

. Closed column:

f=(2k-1)\,340/(4\times1.70)=50,\,150,\,250,\dots\ \mathrm{Hz}

, odd multiples only. (A) is true because the wire's

150\ \mathrm{Hz}

equals the column's third mode. (B) is false because

100\ \mathrm{Hz}

is an even multiple and is absent in a closed column. (C) is false:

f\propto\sqrt T

, so

f=50\sqrt{1.02}=50.50\ \mathrm{Hz}

and the beat frequency is about

0.50\ \mathrm{Hz}

, not

1.0\ \mathrm{Hz}

(that would need a

4\%

rise). (D) is false because the speed of sound grows with temperature, so

f=v/4L

increases.

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