JEE Main 2024 · Physics · Waves and Sound · medium

Two identical organ pipes, each closed at one end and 0.85 m long, are sounded in their first overtones, one at 27^circC and the other at 51^circC.…

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Two identical organ pipes, each closed at one end and

0.85\ \mathrm{m}

long, are sounded in their first overtones, one at

27^\circ\mathrm{C}

and the other at

51^\circ\mathrm{C}

. At

27^\circ\mathrm{C}

the speed of sound is

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

and it varies as

\sqrt{T}

. The number of beats per second heard is nearly (neglect end corrections and thermal expansion of the pipes)

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Answer: (D) $12$

A closed pipe's first overtone is the third harmonic:

f=\dfrac{3v}{4L}

. At

27^\circ\mathrm{C}

(

300\ \mathrm{K}

f_1=\dfrac{3\times340}{4\times0.85}=300\ \mathrm{Hz}

. At

51^\circ\mathrm{C}

(

324\ \mathrm{K}

v_2=340\sqrt{324/300}=340\times1.0392=353.3\ \mathrm{m\,s^{-1}}

, so

f_2=300\times1.0392=311.8\ \mathrm{Hz}

. Beats

=311.8-300\approx12\ \mathrm{Hz}

. Using the fundamental gives about

4

Hz, the trap.

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