JEE Advanced PYQ Musical Sound and Doppler’s Effect

Q 1 :

A police car with a siren of frequency 8 kHz is moving with uniform velocity 36 km/hr towards a tall building which reflects the sound waves. The speed of sound in air is 320 m/s. The frequency of the siren heard by the car driver is [2011]

8.50 kHz
8.25 kHz
7.75 kHz
7.50 kHz

1

2

3

4

(1)

Q 2 :

A police car moving at 22 m/s, chases a motorcyclist. The police man sounds his horn at 176 Hz, while both of them move towards a stationary siren of frequency 165 Hz. Calculate the speed of the motorcycle, if it is given that he does not observes any beats. [2003]

33 m/s
22 m/s
zero
11 m/s

1

2

3

4

(2)

$f_{1} =$ frequency of the police car heard by motorcyclist,

$f_{2} =$ frequency of the siren heard by motorcyclist.

$f_{1} = \frac{330 - v}{330 - 22} \times 176; f_{2} = \frac{330 + v}{330} \times 165;$

$∵$ $f_{1} - f_{2} = 0$ $∴$ $v = 22 m/s$

Q 3 :

A siren placed at a railway platform is emitting sound of frequency 5 kHz. A passenger sitting in a moving train A records a frequency of 5.5 kHz while the train approaches the siren. During his return journey in a different train B he records a frequency of 6.0 kHz while approaching the same siren. The ratio of the velocity of train B to that train A is [2002]

$\frac{242}{252}$
$2$
$\frac{5}{6}$
$\frac{11}{6}$

1

2

3

4

(2)

$Using the formula f' = f (\frac{v_{A} + v}{v})$

$\frac{v_{A} + v}{v} = \frac{5.5}{5}$ $and$ $\frac{V_{B} + V}{V} = \frac{6}{5}$

$∴$ $\frac{v_{B}}{v_{A}} = 2$

Q 4 :

A source (S) of sound has frequency 240 Hz. When the observer (O) and the source move towards each other at a speed $v$ with respect to the ground (as shown in Case 1 in the figure), the observer measures the frequency of the sound to be 288 Hz. However, when the observer and the source move away from each other at the same speed $v$ with respect to the ground (as shown in Case 2 in the figure), the observer measures the frequency of sound to be $n$ Hz. The value of $n$ is _____. [2024]

(200)

$Using Doppler's effect, frequency received by observer$

$f_{0} = (\frac{C \pm V_{0}}{C \pm V_{s}}) f_{s}, C = speed of sound$

$Case-1: f_{1} = (\frac{C + V}{C - V}) f_{s}$

$\Rightarrow 288 = (\frac{C + V}{C - V}) 240 .....(i)$

$Case-2: f_{2} = (\frac{C - V}{C + V}) f_{s}$

$or, n = (\frac{C - V}{C + V}) 240 .....(ii)$

Multiplying eq. (i) & (ii)

$(288) (n) = (240) (240)$

$∴$ $n = \frac{240 \times 240}{288} = 200$

Q 5 :

A train S1, moving with a uniform velocity of 108 km/h approaches another train S2 standing on a platform. An observer O moves with a uniform velocity of 36 km/h towards S2 as shown in figure. Both the trains are blowing whistles of same frequency 120 Hz. When O is 600 m away from S2 and distance between S1 and S2 is 800 m, the number of beats heard by O is _____

[Speed of the sound = 330 m/s] [2019]

(8.13)

$Apparent frequency heard by observer due to source S_{1}$

$ν_{1} = ν [\frac{v + v_{0}}{v}] = 120 [\frac{330 + 10}{330}] = 120 \times \frac{34}{33} = 123.636 Hz$

$Apparent frequency heard by observer due to source S_{2}$

$ν_{2} = ν [\frac{v + v_{0}}{v - v_{s}}] = 120 [\frac{330 + 10 \cos 53 °}{330 - 30 \cos 37 °}]$

$∴$ $ν_{2} = 120 [\frac{330 + 10 \times 0.6}{330 - 30 \times 0.8}] = 120 [\frac{336}{306}] = 131.764 Hz$

$∴$ $Beat frequency, ν_{b} = ν_{2} - ν_{1} = 131.764 - 123.636 = 8.125 Hz$ $\approx 8.13 Hz$

Q 6 :

Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed 1.0 ${ms}^{- 1}$ and the man behind walks at a speed 2.0 ${ms}^{- 1}$ . A third man is standing at a height 12 m above the same horizontal line such that all three men are in a vertical plane. The two walking men are blowing identical whistles which emit a sound of frequency 1430 Hz. The speed of sound in air is 330 ${ms}^{- 1}$ . At the instant, when the moving men are 10 m apart, the stationary man is equidistant from them. The frequency of beats in Hz, heard by the stationary man at this instant, is ____________. [2018]

(5)

$Apparent frequency at O due to source at - A$

$ν_{A} = ν [\frac{v}{v - 2 \cos θ}]$

$Apparent frequency at O due to source at - B$

$ν_{B} = ν [\frac{v}{v + \cos θ}]$

$∴$ $Beat frequency, ν_{b} = ν [\frac{v}{v - 2 \cos θ}] - ν [\frac{v}{v + \cos θ}]$

$= ν v [\frac{1}{v - 2 \cos θ} - \frac{1}{v + \cos θ}]$

$= 1430 \times 330 [\frac{1}{330 - 2 \times \frac{5}{13}} - \frac{1}{330 + \frac{5}{13}}]$

$= 1430 \times 330 \times 13 [\frac{1}{330 \times 13 - 10} - \frac{1}{330 \times 13 + 5}]$

$= 1430 \times 330 \times 13 [\frac{1}{4280} - \frac{1}{4295}] \approx 5 Hz$

Q 7 :

A stationary source emits sound of frequency $f_{0}$ = 492 Hz. The sound is reflected by a large car approaching the source with a speed of 2 ${ms}^{- 1}$ . The reflected signal is received by the source and superposed with the original. What will be the beat frequency of the resulting signal in Hz?

(Given that the speed of sound in air is 330 ${ms}^{- 1}$ and the car reflects the sound at the frequency it has received.) [2017]

(6)

$Frequency observed at car$

$ν_{1} = ν_{0} (\frac{v + v_{c}}{v})$

$Frequency of reflected sound as observed at the source$

$ν_{2} = ν_{1} (\frac{v}{v - v_{c}}) = ν_{0} (\frac{v + v_{c}}{v - v_{c}})$

$∴$ $Beat frequency = ν_{2} - ν_{0}$

$= ν_{0} [\frac{v + v_{c} - v + v_{c}}{v - v_{c}}] = ν_{0} [\frac{2 v_{c}}{v - v_{c}}]$

$= 492 [\frac{2 \times 2}{330 - 2}] = \frac{492 \times 4}{328} = 6 Hz$

Q 8 :

Four harmonic waves of equal frequencies and equal intensities $I_{0}$ have phase angles $0, \frac{π}{3}, \frac{2 π}{3}$ and $π$ . When they are superposed, the intensity of the resulting wave is $n I_{0}$ . The value of $n$ is [2015]

(3)

$Resultant of amplitude, A_{R}$

$= \sqrt{I_{0}} [\sin 0 + \sin \frac{π}{3} + \sin \frac{2 π}{3} + \sin π]$

$= \sqrt{I_{0}} [\frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2}] = \sqrt{3} \sqrt{I_{0}}$

$∴$ $I_{R} = A_{R}^{2} = 3 I_{0}$ $∴$ $n = 3$

Q 9 :

A stationary source is emitting sound at a fixed frequency $f_{0}$ , which is reflected by two cars approaching the source. The difference between the frequencies of sound reflected from the cars is 1.2% of $f_{0}$ . What is the difference in the speeds of the cars (in km per hour) to the nearest integer? The cars are moving at constant speeds much smaller than the speed of sound which is 330 ${ms}^{- 1}$ . [2010]

(7)

Firstly, car will be treated as an observer which is approaching the source. Then it will be treated as a source, which is moving in the direction of sound.

Frequency of sound reflected by the car $C_{1}$

$f_{1} = f_{0} (\frac{v + v_{1}}{v - v_{1}})$

And frequency of sound reflected by the car $C_{2}$

$f_{2} = f_{0} (\frac{v + v_{2}}{v - v_{2}})$

$∴$ $f_{1} - f_{2} = (\frac{1.2}{100}) f_{0} = f_{0} [\frac{v + v_{1}}{v - v_{1}} - \frac{v + v_{2}}{v - v_{2}}]$

$or$ $(\frac{1.2}{100}) f_{0} = \frac{2 v (v_{1} - v_{2})}{(v - v_{1}) (v - v_{2})} f_{0}$

$(v - v_{1}) = (v - v_{2}) \approx v as v_{1} and v_{2} are very very less than v .$

$∴$ $(\frac{1.2}{100}) f_{0} = \frac{2 (v_{1} - v_{2})}{v} f_{0}$

$or$ $(v_{1} - v_{2}) = \frac{v - 1.2}{200} = \frac{330 \times 1.2}{200} = 1.98 {m s}^{- 1} \approx 7 {km h}^{- 1}$

Q 10 :

A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved with a speed of 2 ${ms}^{- 1}$ in front of the open end of the pipe and parallel to it, the length of the pipe should be changed for the resonance to occur with the moving tuning fork. If the speed of sound in air is 320 ${ms}^{- 1}$ , the smallest value of the percentage change required in the length of the pipe is __________. [2020]

(0.62 to 0.63)

$Let ℓ_{1} = initial length of pipe$

$ℓ_{2} = new length of pipe$

$V_{T} = Speed of tuning fork$

$In closed organ pipe, f = \frac{V}{4 ℓ_{1}}$

$When tuning fork is moved, f' = f (\frac{V}{V - V_{T}}) = \frac{V}{4 ℓ_{2}}$

$\Rightarrow \frac{V}{4 ℓ_{1}} (\frac{V}{V - V_{T}}) = \frac{V}{4 ℓ_{2}}$ $\Rightarrow \frac{V - V_{T}}{V} = \frac{ℓ_{2}}{ℓ_{1}}$

$\Rightarrow \frac{ℓ_{2}}{ℓ_{1}} - 1 = \frac{V - V_{T}}{V} - 1$ $\Rightarrow \frac{ℓ_{2} - ℓ_{1}}{ℓ_{1}} = \frac{- V_{T}}{V}$

$Percentage change required in the length of the pipe$

$\frac{ℓ_{2} - ℓ_{1}}{ℓ_{1}} \times 100 = \frac{- 2}{320} \times 100 = - 0.625 %$

Hence, smallest value of percentage change required in the length of pipe is 0.625%

Q 11 :

$S_{1}$ and $S_{2}$ are two identical sound sources of frequency $656 Hz$ . The source $S_{1}$ is located at O and $S_{2}$ moves anticlockwise with a uniform speed $4 \sqrt{2} {ms}^{- 1}$ on a circular path around O, as shown in the figure. There are three points P, Q and R on this path such that P and R are diametrically opposite while Q is equidistant from them. A sound detector is placed at point P. The source $S_{1}$ can move direction OP.

[Given: The speed of sound in air is $324 {ms}^{- 1}$ ]

Q. When only $S_{2}$ is emitting sound and it is at Q, the frequency of sound measured by the detector in Hz is __________. [2023]

(648)

$Given: velocity of sound V = 324 m/s$

$Frequency of source f_{0} = 656 Hz$

$Velocity of source S_{2}, V_{s} = 4 \sqrt{2} \cos 45 °$

$From Doppler's effect, apparent frequency measured by the detector,$ $f = (\frac{V}{V + V_{s}}) f_{0}$

$= \frac{324}{324 + 4 \sqrt{2} \times \frac{1}{\sqrt{2}}} \times 656$

$= 648 Hz$

Q 12 :

$S_{1}$ and $S_{2}$ are two identical sound sources of frequency $656 Hz$ . The source $S_{1}$ is located at O and $S_{2}$ moves anticlockwise with a uniform speed $4 \sqrt{2} {ms}^{- 1}$ on a circular path around O, as shown in the figure. There are three points P, Q and R on this path such that P and R are diametrically opposite while Q is equidistant from them. A sound detector is placed at point P. The source $S_{1}$ can move direction OP.

[Given: The speed of sound in air is $324 {ms}^{- 1}$ ]

Q. Consider both sources emitting sound. When $S_{2}$ is at R and $S_{1}$ approaches the detector with a speed 4 ${ms}^{- 1}$ , the beat frequency measured by the detector is __________ Hz. [2023]

(8.2)

$f_{0} = 656 Hz$

$v = 324 m/s$

$Frequency heard due to movement of (S_{1})$

$f_{1} = (\frac{v}{v - u_{s}}) f_{0}$

$f_{1} = \frac{324}{320} \times 656$

$And frequency heard due to movement of (S_{2})$

$f_{2} = 656 Hz$

$∴$ $Beat frequency Δ f = f_{1} - f_{2}$

$= 656 (\frac{324}{320} - 1)$

$Δ f = 8.2$

Topic Question Set

Q 1 :

A police car with a siren of frequency 8 kHz is moving with uniform velocity 36 km/hr towards a tall building which reflects the sound waves. The speed of sound in air is 320 m/s. The frequency of the siren heard by the car driver is [2011]

(1)

Q 2 :

A police car moving at 22 m/s, chases a motorcyclist. The police man sounds his horn at 176 Hz, while both of them move towards a stationary siren of frequency 165 Hz. Calculate the speed of the motorcycle, if it is given that he does not observes any beats. [2003]

Q 8 :

Four harmonic waves of equal frequencies and equal intensities $I_{0}$ have phase angles $0, \frac{π}{3}, \frac{2 π}{3}$ and $π$ . When they are superposed, the intensity of the resulting wave is $n I_{0}$ . The value of $n$ is [2015]

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Topic Question Set

Q 1 : A police car with a siren of frequency 8 kHz is moving with uniform velocity 36 km/hr towards a tall building which reflects the sound waves. The speed of sound in air is 320 m/s. The frequency of the siren heard by the car driver is [2011]

(1)

Q 2 : A police car moving at 22 m/s, chases a motorcyclist. The police man sounds his horn at 176 Hz, while both of them move towards a stationary siren of frequency 165 Hz. Calculate the speed of the motorcycle, if it is given that he does not observes any beats. [2003]

Q 8 : Four harmonic waves of equal frequencies and equal intensities I0 have phase angles 0, π3, 2π3 and π. When they are superposed, the intensity of the resulting wave is nI0. The value of n is [2015]

Want Us to Email you About Special Offers & Updates?

Q 1 :

A police car with a siren of frequency 8 kHz is moving with uniform velocity 36 km/hr towards a tall building which reflects the sound waves. The speed of sound in air is 320 m/s. The frequency of the siren heard by the car driver is [2011]

Q 2 :

A police car moving at 22 m/s, chases a motorcyclist. The police man sounds his horn at 176 Hz, while both of them move towards a stationary siren of frequency 165 Hz. Calculate the speed of the motorcycle, if it is given that he does not observes any beats. [2003]

Q 8 :

Four harmonic waves of equal frequencies and equal intensities $I_{0}$ have phase angles $0, \frac{π}{3}, \frac{2 π}{3}$ and $π$ . When they are superposed, the intensity of the resulting wave is $n I_{0}$ . The value of $n$ is [2015]