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 $332 m s ^{-1}$
$(B)$ 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
IIT 2018, Diffcult
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For $1^{\text {st }}$ resonance
$\frac{3 V}{4\left(L_1+e\right)}=f_0$ $. . . . . . (1)$
For $2^{\text {nd }}$ resonance
$\frac{5 V }{4\left( L _2+ e \right)}= f _0$ $. . . . . . (2)$
$\frac{3 V }{4\left( L _1+ e \right)}=\frac{5 V }{4\left( L _2+ e \right)}$
$3\left( L _2+ e \right)=5\left( L _1+ e \right)$
$2 e =3 L _2-5 L _1$
$e =\frac{3 \times 83.9-5 \times 50.7}{2}=\frac{251.7-253.5}{2}=\frac{-1.8}{2}=-0.9$
$\frac{3 V }{4\left( L _1+ e \right)}= f _0$
$V =\frac{500 \times 4 \times 49.8}{3 \times 100}=20 \times 16.6=332 m / s$
$\lambda=\frac{ V }{ f _0}=\frac{332}{500} \times 100=66.4 cm$
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