Tute W4: DOPPLER EFFECT 1

Transcription

1 Tute W4: DOPPLER EFFECT 1 A Doppler effect occurs wheneer there is relatie motion between a source and the receier. When the source and receier moe towards each other, the frequency detected by the receier is higher than the frequency of the source. Conersely, when the receier and source moe away from each other, the receier detects a frequency which is lower than the source frequency. There are three possible motions: (1) Receier Moing & Source Stationary ( f r = f 1 ± ) ( ) r ± r = f (2) Source Moing & Receier Stationary ( ) ( ) 1 f r = f = f 1 s s (3) Receier & Source Moing ( ) ± r f r = f s where f and are the frequency and elocity of the wae emitted by the source, and r and s are the elocities of the receier and the source respectiely. Note that the upper sign corresponds to motion towards each other (approach) and the lower sign is used for motion away from each other (recession). 1 January 28,

2 1. A supertrain traelling at 50 m s 1 sounds its whistle (with a frequency of 400 Hz). (a) What frequencies for the whistle would a farmer hear when the supertrain approaches him and then passes by? (b) What frequencies would be heard by a passenger in a second supertrain traelling at 55 m s 1 in the opposite direction? (c) If the farmer sounds the same (fixed) whistle, what frequencies would be heard by a passenger on the first supertrain? Assume the speed of sound in air is 340 m s 1. [ Answer: (a) 469 Hz & 349 Hz (b) 545 Hz & 292 Hz (c) 458 Hz & 341 Hz ] 2. A funny car is accelerating down a 400 m long drag strip. Its supercharger emits a high pitched sound with frequency 10 khz. On approaching the finishing line the funny car is moing with a speed of 115 m s 1. What maximum and minimum frequencies of the supercharger does a racing official near the finishing line hear? What waelengths do these frequencies correspond to? [ Answer: f r = khz and 7.47 khz λ = 2.25 cm and 4.55 cm ] 3. An FSC student uses a 30 khz whistle to call his dog. The dog does not react and so the student decides to check whether the whistle works. He asks a friend to blow the whistle from a moing car so that the sound will be Doppler shifted to 20 khz (which he can hear). How fast must the car be moing? [ Answer: 170 m s 1 (612 km hr 1 ) ] 4. A jet has loud turbines which whine with a frequency of 16 khz. If this jet is flying close to the ground and a stationary student detects a turbine sound frequency of 60 khz, how fast is the jet moing? What frequency would be heard by the student after the jet has passed by? [ Answer: m s 1 (898 km hr 1 ) 9231 Hz ] 2

3 5. As a train traels towards a tunnel at 25 m s 1 the drier sounds a whistle with a frequency of 1400 Hz. What does a passenger on the train hear for the frequency of the sound reflected from the front wall of the tunnel? [ Answer: 1622 Hz ] 6. Proe that the sound frequency heard by the passenger (f p ) in the preious question can be written in terms of the elocity of the train ( t ) as ( ) + t f p = f t where f is the sound frequency of the whistle. 7. A train approaches a tunnel and sounds its whistle. The echo of this sound from the cliff aboe the tunnel returns to the train and is measured to hae twice the frequency of the original whistle. Find the speed of the train. [ Answer: m s 1 (408 km hr 1 ) ] 8. A fixed motion detector sends out sound waes with a frequency of 150 khz towards a truck approaching it at a speed of 45 m s 1. (a) What sound frequency is detected by the speeding truck drier? (b) What is the frequency of the sound waes reflected back to the detector? [ Answer: (a) khz khz ] 9. A police car has a siren which emits sound with a frequency of 500 Hz. If the police car is chasing a speeding motorist and both are traelling in the same direction at 45 m s 1, what frequency is heard by (a) the drier of the speeding car? (b) a pedestrian standing right behind the police car? 3

4 Assume the speed of sound in air is 340 m s 1. [ Answer: (a) 500 Hz (b) Hz ] 10. As a bat flies towards a stationary insect, it emits a sound wae with a frequency of 60 khz. The bat then receies an echo of this sound at a frequency of 61.8 khz. Assuming that the speed of sound in air is 340 m s 1, at what speed is the bat flying towards the insect? [ Answer: 5.02 m s 1 ] 11. A battery powered siren with frequency f is dropped from rest oer the edge of a cliff. Show that after falling any distance x the frequency f r heard at the top of the cliff is gien by the general expression ( ) f r = f + 2gx What distance would the siren hae to fall before the sound frequency at the cliff edge drops to half its original alue? Derie an alternatie expression for the frequency heard at the top of the cliff in terms of the time t since the siren was dropped oer the edge of the cliff. [ Answer: 5.9 km ; f r = f/( + gt) ] 12. A siren continuously emits a sound with a frequency of 500 Hz. If the siren is swung around in a horizontal circle of radius 1 m making 3 reolutions per second, what are the maximum and minimum frequencies heard by a stationary listener? [ Answer: f max = 529 Hz f min = 474 Hz ] 13. A flying saucer from another planet is seen to be moing almost directly at a frightened obserer. Its nuclear powered engines are heard to emit an eerie high pitched sound with a frequency of 17.5 khz. After the flying saucer passes by, the 4

5 frequency of its engine sound drops to 50 Hz. Assuming the speed of sound in air is 340 m s 1, with what speed is this alien craft traelling? [ Answer: 338 m s 1 (1217 km hr 1 ) ] Further Interesting Problems: 1. Bats also use the frequency change of the sound echo to determine the flight direction of the bug. While hoering in the air (not moing), the bat emits a sound of 40.0 khz. The frequency of the echo is 40.4 khz. (a) Is the bug moing towards the bat or away from it? Explain. (b) What is the speed of the bug? (Remember, the bat is not moing.) What is the frequency of the echo if the bat is now flying towards the bug with a speed of 1 m/s while emitting the 40 khz sound? The bug is still following its original flight path? [ Answer: moing towards ; 1.69 m/s ; khz ] 2. A car traelling at a constant speed of 80 km/hr passes a police-car which has its siren running. The siren emits a frequency of f. Find the frequency of the siren as detected by the speeding car if the police car is : (a) stationary. (b) chasing the speeding car with a speed of 60 km/hr. (c) chasing the speeding car with a speed of 80 km/hr. (d) chasing the speeding car with a speed of 90 km/hr. [ Answer: 0.93f ; 0.89f ; f ; 1.15f ] 5

6 3. While attempting to tune the note C at 523 Hz, a piano tuner hears 2 beats/s between a reference oscillator and the string. (a) What are the possible frequencies of the string? (b) When she tightens the string slightly, she hears 3 beats/s. What is the frequency of the string now? (c) By what percentage should the piano tuner now change the tension in the string to bring it into tune? [ Answer: f = 521 Hz or 525 Hz ; f = 526 Hz ; reduced by 1.14% ] 4. A block with a speaker bolted to it is connected to a spring haing spring constant k = 20.0 N/m. The total mass m of the block and speaker is 5.00 kg, and the amplitude of this unit s motion is m. (a) If the speaker emits sound waes of frequency 440 Hz, determine the highest and lowest frequencies heard by the person k to the right of the speaker if ω =. (b) If the maximum sound leel heard by m the person is 60.0 db when he is closest to the speaker, 1.00 m away, what is the minimum sound leel heard by the obserer? The speed of sound is 343 m/s. [ Answer: 441 Hz, 439 Hz ; 54 db ] 5. A bungee jumper jumps from rest and screams with a frequency of 589 Hz. What is the frequency heard by the people on the ground below when she has fallen a distance of 11 m? Assume that the bungee cord is 15 long. [ Answer: 615 Hz ] 6. A microphone is attached to a spring that is suspended from the ceiling. Directly below on the floor is a stationary 40 Hz source of sound. The microphone ibrates up and down in simple harmonic motion with a period of 2 s. The difference between the maximum and minimum sound frequencies detected by the microphone is 2.1 Hz. Ignoring any reflections of sound in the room and using 343 m/s for the speed of sound, determine the amplitude of the simple harmonic motion. [ Answer: 9 m ] 6