Questions OBJECTIVE - I and Answer (Sound Waves) HC Verma Part 1 (9-16)

 Q#9

An open organ pipe of length L vibrates in its fundamental mode. The pressure variation is maximum
(a) at the two ends
(b) at the middle of the pipe
(c) at distance L/4 from inside the ends
(d) at distance L/8 from inside the ends.

Answer: (b)
The fundamental mode of vibration is the minimum frequency at which a standing wave is formed in the pipe. For an open organ pipe, if the fundamental mode of vibration is set up, pressure nodes are found at the open ends because the air molecules at these ends are free to vibrate. It results in the formation of pressure antinodes at the midway between these two nodes i.e. at the middle of the pipe and here the pressure variation is maximum. Hence option (b).

Q#10
An organ pipe open at both ends contains
(a) longitudinal stationary waves
(b) longitudinal traveling waves
(c) transverse stationary waves
(d) transverse traveling waves.

Answer: (a)
An organ pipe open at both ends contains sound waves which are longitudinal waves, so the option (c) and (d) are not true. When the waves enter through one end it gets reflected from the other end with a phase change of π. The reflected wave is again reflected by the first end. With repeated reflections, if the length of the pipe is a multiple of the half wavelength, stationary waves are formed. Thus option (a).

Q#11
A cylindrical tube open at both ends has a fundamental frequency ν. The tube is dipped vertically in water so that half of its length is inside the water. The new fundamental frequency is
(a) ν/4
(b) ν/2
(c) ν
(d) 2ν.

Answer: (c)
When a tube of length L open at both ends has a fundamental frequency ν, then

v = V/2L

where V is the velocity of the sound wave.
When this tube is half dipped in water, the length of the air column becomes L/2 and it becomes a tube closed at one end. In such closed pipe the frequency of the fundamental mode ν' = V/4L'. Here L' = L/2, so

ν' = V/(4L/2) = V/2L = ν

Hence the option (c).

Q#12
The phenomenon of beats can take place
(a) for longitudinal waves only
(b) for transverse waves only
(c) for both longitudinal and transverse waves
(d) for sound waves only.

Answer: (c)
The phenomenon of beats can take place for both longitudinal and transverse waves only conditions to be followed is the difference in the frequencies should be slight and the amplitudes equal.

Q#13
A tuning fork of frequency 512 Hz is vibrated with a sonometer wire and 6 beats per second are heard. The beat frequency reduces if the tension in the string is slightly increased. The original frequency of vibration of the string is
(a) 506 Hz
(b) 512 Hz
(c) 518 Hz
(d) 524 Hz.

Answer: (a)

The frequency of beats

ν = |ν₁ - ν₂| = 6 Hz

where ν₁ = 512 Hz, ν₂ = frequency of sonometer. Thus ν₂ is either 506 or 518 Hz.
When the tension in the string is slightly increased the frequency of the sonometer slightly increases. Since the beat frequency reduces it means ν₂<ν₁ so ν₂ is 506 Hz. Option (a).

Q#14
The engine of a train sounds a whistle at a frequency ν. The frequency heard by a passenger is
(a) > ν
(b) < ν
(c) = 1/ν
(d) = ν

Answer: (d)
Since the source and the observer both move with the same speed there is no relative motion. Thus there is no Doppler's effect and no change in apparent frequency. Hence the option (d).

Q#15
The change in frequency due to the Doppler effect does not depend on
(a) the speed of the source
(b) the speed of the observer
(c) the frequency of the source
(d) the separation between the source and the observer.

Answer: (d)
The apparent frequency due to the Doppler effect 
ν' = {V/(V – U)}ν₀ 

or ν' = {(V + U)/V}ν₀

depending upon the source moves towards the observer or the observer moves towards the source. Thus the change in frequency due to Doppler effect depends on V, U and ν₀ only. It does not depend on the separation between the source and the observer. So the option (d).

Q#16
A small source of sound moves on a circle as shown in figure (16-Q1) and an observer is sitting at O. Let ν₁ > ν₂ > ν₃ be the frequencies heard when the source is at A, B, and C respectively.
(a) ν₁ > ν₂ > ν₃
(b) ν₁ = ν₂ > ν₃
(c) ν₂ > ν₃ > ν₁
(d) ν₁ > ν₃ > ν₂.

Answer: (c)
When the source is at C the source speed is perpendicular to the line joining the observer and the source. Thus at this instant, the separation between the two is not changing and the frequency heard ν₃ is same as the source. When the source is at A the separation between them is increasing, so due to the Doppler effect ν₁ < ν₃. But when the source is at B, the separation is decreasing and due to the Doppler effect ν₃ < ν₂. So ν₂ > ν₃ > ν₁. Hence the option (c).

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