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# JEE Main 2014 Physics - Wave Motion Part II

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### JEE Main 2014 Physics - Wave Motion Part II

1. 1. 9011041155 / 9011031155 • Live Webinars (online lectures) with recordings. • Online Query Solving • Online MCQ tests with detailed solutions • Online Notes and Solved Exercises • Career Counseling 1
2. 2. 9011041155 / 9011031155 Wave Motion Important definations 1. Wave Wave is a mode of energy transfer through material elastic medium in the form of disturbance propagation through the medium due to periodic, vibrational motion of particle about their mean position. 2. Amplitude (a) The maximum displacement of particle of medium from its mean position is called the amplitude (a) of the wave. (unit is m or cm) 3. Wavelength (λ) The distance between two nearest particles of medium which are in the same phase is called the wavelength (λ) of wave. (unit is m or cm or A0) 2
3. 3. 9011041155 / 9011031155 OR The distance travelled by the wave in one periodic time (T) is called wavelength of wave. 4. Period (T) The time taken by the particle to complete one vibration is called the periodic time (T) or period of the wave (unit is second) 5. Frequency (n) The number of vibrations completed in one second / unit time is called frequency of wave. v n 3
4. 4. 9011041155 / 9011031155 Simple harmonic progressive wave Definition A wave which travel continuously on particular direction with the particle of medium performing simple harmonic (periodic) vibration is called simple harmonic wave. Progressive waves are doubly periodic. i.e. periodic in time and space. When a progressive wave travels through the medium, it can be observed that the particle of the medium repeats the motion after fixed time interval time to time. That means progressive wave is periodic in time. Also it can be observed that at any instant the shape of the wave repeats at equal distances. That means progressive wave is periodic in space. 4
5. 5. 9011041155 / 9011031155 Equation of simple harmonic progressive wave y a s in t For the particle at distance λ from origin log by angle 2π The particle at distance ‘X’ from origin will log behind by angle 2 x {For λ = 2π for x = δ = ? δ = 2 5 x }
6. 6. 9011041155 / 9011031155 Different forms of equation of S.H. Progressive wave : i. y a sin t 2 ii. we know that y a sin x 2 2 T t 2 x T y a sin 2 t x T iii. Also n y 1 T a s in 2 nt x 6
7. 7. 9011041155 / 9011031155 iv. As y = a sin 2 π n t x v ∴ Now, 2 π n = ω & v = nλ y a sin t x n 7
8. 8. 9011041155 / 9011031155 Multiple Choice Questions 1. The equation of a transverse wave is given by where x and y are in m and t is in sec. The frequency of the wave is (a) 1 Hz (b) 2 Hz (c) 3 Hz (d) 4 Hz 2. A wave equation is given by , where x and y are in metres and t is the time in seconds. Which one of the following is a wrong statement? (a) Of amplitude 10-4m traveling in the negative x direction (b) Of wavelength π meter (c) Of frequency hertz (d) Traveling with a velocity of 30 m/s, in the positive x direction 8
9. 9. 9011041155 / 9011031155 3. The equation of a transverses wave traveling along a string is given by y=5 cos π (100t-x) cm. Its wavelength is (a) 10 cm (b) 5 cm (c) 3 cm (d) 2 cm 4. Sound waves having their frequencies below 20 Hz (the audible range) are known as (a) Ultrasonic waves (b) Infrasonic waves (c) Audible waves (d) Supersonic waves 9
10. 10. 9011041155 / 9011031155 Solutions 1. (b) 2 Hz 2. (d) Traveling with a velocity of 30 m/s, in the positive x direction 10
11. 11. 9011041155 / 9011031155 ∴ The wave is traveling in the –ve direction of the x axis. ∴ Statement (a), (b) and (c) are correct and (d) is wrong statement ∴ Traveling with a velocity of 30 m/s, in the positive x direction is the wrong statement 11
12. 12. 9011041155 / 9011031155 3. (d) 2 cm 4. (b) Infrasonic waves Sound waves having their frequencies below 20 Hz are known as infrasonic waves. 12
13. 13. 9011041155 / 9011031155 Superposition of waves Ans: When two or more waves arrive at a point travelling through the medium simultaneously each wave produces its own displacement independently at that point as if other waves are absent. Hence the resultant displacement at that point is the vector sum of all the displacements, produced by the waves. Interference of waves constructive interference:-amplitude is maximum As intensity of the wave is directly proportional to the square of its amplitude, the intensity is also maximum (loudness of sound). It is called waxing. 13
14. 14. 9011041155 / 9011031155 Destructive interference:- resultant displacement is minimum and hence the intensity is also minimum. It is called waning. Beats : When two sound waves of very nearly equal frequency (differing by a few vibrations) and of equal or comparable amplitude travel through the same region simultaneously, the resultant intensity changes with time. It alternates between maximum of sound (waxing) and minimum of sound (waning). This phenomenon of alternate waxing and waning is called beat formation. For beats : Period : The time interval between two successive waxings or wanings is called period of the beat. Frequency : The number of beats completed per second is called the frequency of the beat. 14
15. 15. 9011041155 / 9011031155 Conditions for audible beats 1. The amplitudes of two sound waves producing beats must be nearly equal or comparable. 2. The frequency of two sound waves must be nearly equal or differ only by few vibrations. The audible beats are produced only if frequency difference is not more than 7/second (seven per second) 3. The two waves sound pass through the same region simultaneously. For the superposition of two waves the resultant displacement is given by 15
16. 16. 9011041155 / 9011031155 Frequency of the beat The two sound waves of equal amplitude ‘a’ and slightly different frequencies ‘n1’, and ‘n2’ are represented as y1 = a sin 2πn1t and y2 = a sin2π n2t These two waves interfere to produce beats. Thus the resultant displacement ‘y’ is given by, y = y1 + y2 ∴ y = asin2πn1t + asin2πn2t ∴ y = a{sin(2πn1t) + sin (2πn2t) We know, sin C sin D 2 sin C D co s 2 C D 2 sin (2πn1t) + sin (2πn2t) 16
17. 17. 9011041155 / 9011031155 2 n 1t 2 sin 2 n2 t co s 2 n 1t 2 y 2a sin 2 2 n1 n2 t co s 2 n1 2 y 2a co s 2 n1 n1 2 a co s n2 t 2 n2 t sin 2 2 Let 2 n2 t n1 n2 t 2 n2 t A 2 n1 n 2 y A sin 2 t Where A is resultant amplitude. For waxing For A = ±2 a intensity is maximum i.e. waxing n1 co s 2 n2 t 1 2 2 n1 n2 t 0 , , 2 , 3 ........... 2 17
18. 18. 9011041155 / 9011031155 i.e. t = O, 1 n1 2 , n2 n1 , 3 n 2 n1 n2 Thus the period of beats is 1 n1 2 0 n2 n1 1 n2 ∴Period of beats = n1 1 n2 n1 n2 1 n1 n2 Now, frequency of beats = 1 P e rio d o f b e a ts 1 = 1 n1 n2 ∴ Frequency of beats = (n1 - n2) Thus, frequency of beats is equal to the difference between frequencies of two waves. Also, period of beat is the reciprocal of difference of two frequencies. 18
19. 19. 9011041155 / 9011031155 Application of Beats 1. The phenomenon of beats is used to determine an unknown frequency. The sound note of unknown frequency is sounded simultaneously with a note of known frequency which can be changed. The known frequency is so adjusted that beats are heard. The further adjustment is made till the beats reduce to zero i.e. the frequency of the two notes become equal. 2. Musical instrument can be tuned by noting the beats produced when two different instruments are sounded together. By adjusting the frequency of sound of one of the instruments, the number of beats is reduced to zero. When this is done, both the instruments are emitting sound of the same frequency. The instruments are then in unison with each other. 19
20. 20. 9011041155 / 9011031155 3. The phenomenon of beats can be used to produce low frequency notes used in Jazz orchestra or western music. 4. Beats are used to detect the presence of dangerous gases in mines. Air from a reservoir is blown through a pipe or flute. Air from the mine is blown through another similar pipe or flute. Speed of sound is different in different gases. Hence reservoir with different concentrations than air will sound slightly different frequencies. If no dangerous gases are present in air from the mine, the two flutes will produce notes of the same frequency and beats will not be heard. 20
21. 21. 9011041155 / 9011031155 However if the air from the mine is polluted with presence of dangerous gases, the two flutes will produce notes of slightly different frequencies, giving rise to beats, which serves as an early warning system for safety. 5. In modern days incoming frequency of radio receiver is superposed with a locally produced frequency to produce intermediate frequency which is always constant. This makes tuning of the receiver very simple. superheterodyne oscillators. 21 This is used in
22. 22. 9011041155 / 9011031155 Multiple Choice Questions 1. Wavelengths of two sound notes in air are and respectively. Each note produces 4 beats/sec, with a third note of a fixed frequency. What is the velocity of sound in air? (a) 300 m/s (b) 320 m/s (c) 310 m/s (d) 350 m/s 2. A tuning fork A produces 4 beats/second with another tuning fork of frequency 246 Hz. When the prongs of A are filed a little, the number of beats heard is 6 per second. What is the original frequency of the fork A? (a) 242 Hz (b) 240 Hz (c) 250 Hz (d) 252 Hz 22
23. 23. 9011041155 / 9011031155 Solutions 1. (b) 320 m/s Let η be the frequency of the third note and n1> n> n2 ∴ n1- n=4 and n- n2=4 23
24. 24. 9011041155 / 9011031155 2. (c) 250 Hz The frequency of A may be (246±4) Hz or 250 Hz or 242 Hz, as there are 4 beats/sec. On filing the prongs of A, its frequency is increased. If nA =242, if n’A> 242, then the number of beats will decrease. But if nA =250 Hz, then it is possible to get 6 beats/sec, when its frequency becomes 252 Hz, ∴The frequency of A = 250 Hz 24
25. 25. 9011041155 / 9011031155 Quincke’s Tube Experiment Intensity of sound heard by the listener becomes zero when the path difference = λ / 2. Sound intensity is maximum when the path difference= λ. In general whenever the path difference becomes 0, λ, 2λ etc. or nλ where n = 0, 1, 2 ……., the intensity of sound becomes maximum and whenever the path 25
26. 26. 9011041155 / 9011031155 difference becomes , 3 2 2 , 5 2 or n 1 2 , where n = 0, 1, 2, 3 ……..; the intensity of sound at E becomes minimum (zero). Reflection of transverse and longitudinal waves 26
27. 27. 9011041155 / 9011031155 Reflection of Waves 27
28. 28. 9011041155 / 9011031155 Doppler effect : Definition : The phenomenon of apparent change in frequency of sound waves due to relative motion between the observer and source of the sound is called ‘Doppler effect.’ The general formula for apparent frequency is 28
29. 29. 9011041155 / 9011031155 nA V V n V0 Vs Where, n = actual frequency v = velocity of the sound v0 = velocity of observer. vs = velocity of source Case (I) : O nA V n V Vs Case (II) : O nA nA S V n V n S Vs V V0 V Case (III) : S O 29
30. 30. 9011041155 / 9011031155 nA n V V0 V Case (IV) :S nA n V O V0 V Case (V) : nA n V V0 V VS Applications of Doppler effect 1) RADAR 30
31. 31. 9011041155 / 9011031155 2) Speed of sun 3) Colour Doppler sonography 31
32. 32. 9011041155 / 9011031155 • Ask Your Doubts • For inquiry and registration, call 9011041155 / 9011031155. 32