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A source producing the sound of frequency 170 Hz is approaching a stationary observer with a velocity of 17 ms^–1. The apparent change in the wavelength of sound heard by the observer is (speed of sound in air = 340 ms^–1)

0.1m

0.2m

0.4m

0.5m

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Explanation

$λ = \frac{v}{n} = \frac{340}{170} = 2 m, n' = \frac{340}{340 - 17} \times 170 \Rightarrow n' = 178 .9 H z$

Now $λ^{'} = \frac{v}{n^{'}} = \frac{340}{178 .9} = 1 .9$

⇒ $λ - λ^{'} = 2 - 1 .9 = 0 .1$

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An observer moves towards a stationary source of sound with a speed 1/5^th of the speed of sound. The wavelength and frequency of the sound emitted are λ and f respectively. The apparent frequency and wavelength recorded by the observer are respectively :

$1.2 f, λ$

$f, 1 .2 λ$

$0.8 f, 0 .8 λ$

$1 .2 f, 1 .2 λ$

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Explanation

$n' = \frac{v + v_{0}}{v} . f = \frac{v + \frac{v}{5}}{v} . f = 1 .2 f$

and since the source is stationary, so wavelength remains unchanged for observer.

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The equation of displacement of two waves are given as $y_{1} = 10 \sin (3 π t + \frac{π}{3})$ ; $y_{2} = 5 (\sin 3 π t + \sqrt{3} \cos 3 π t)$ . Then what is the ratio of their amplitudes ?

1 : 2

2 : 1

1 : 1

None of these

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Explanation

$y_{1} = 10 \sin (3 π t + \frac{π}{3})$ ...(i)

and $y_{2} = 5 [\sin 3 π t + \sqrt{3} \cos 3 π t]$

$= 5 \times 2 [\frac{1}{2} \times \sin 3 π t + \frac{\sqrt{3}}{2} \times \cos 3 π t]$

$= 10 [\cos \frac{π}{3} \sin 3 π t + \sin \frac{π}{3} \cos 3 π t]$

$= 10 [\sin (3 π t + \frac{π}{3})]$ ... (ii)

(âˆµ sin(A + B) = sinA cosB + cosA sinB)

Comparing equation (i) and (ii) we get ratio of amplitude 1 : 1.

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Consider ten identical sources of sound all giving the same frequency but having phase angles which are random. If the average intensity of each source is I₀, the average of resultant intensity I due to all these ten sources will be :

I = 100 I₀

I = 10 I₀

I = I₀

$I = \sqrt{10} I_{0}$

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Explanation

In case of interference of two waves resultant intensity

$I = I_{1} + I_{2} + 2 \sqrt{I_{1} I_{2}} \cos ϕ$

If Ï• varies randomly with time, so ${(\cos ϕ)}_{a v} = 0$

⇒ $I = I_{1} + I_{2}$

For n identical waves, $I = I_{0} + I_{0} + . . . . . . . = n I_{0}$

Here $I = 10 I_{0} .$

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41 forks are so arranged that each produces 5 beats per sec when sounded with its near fork. If the frequency of the last fork is double the frequency of the first fork, then the frequencies of the first and last fork are respectively :

200, 400

205, 410

195, 390

100, 200

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Explanation

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