Beat frequency = f f = f. f = f. = f. = f. = f

Transcription

1 Beat frequency = f f = f v vm f = f v vb v vm f v vb = f = f = f ( v v ( vv m m )( v v ( v ) ( v v b ){( vv ( v vm)( v ( v vb ) ( v v ) b b v ) ) ( vv b b ) m b )( v v )} b ) Ex.17 A source of sound is moving along circular orbit of radius 3 meters with an angular velocity of 10 rad/s. A sound detector located far away from the source is executing linear simple harmonic motion along the line BD as shown in the figure with an amplitude BC = CD = 6 meters. The frequency of oscillation of the detector is 5/ per second. The source is at the point A when the detector is at the point B. If the source emits a continuous sound wave of frequency 340 Hz, find the maximum and the minimum frequencies recorded by the detector. Sol.: Speed of source, v s = r = 3 10 = 30 m/s Maximum velocity of detector v 0 = A v 0 = A f = 6 (5/) = 60 m/s Actual frequency of source n = 340 Hz The frequency recorded by the detector is maximum when source and detector travel along same direction. n max = v v 0 v n = v s = 44 Hz The frequency recorded by detector will be minimum when source and detector are travelling in opposite directions. n min = v v 0 v n = v s = 55 Hz 33030

2 OBJECTIVE QUESTIONS 1. The displacement of a wave is represented by y = sin (500 t 0.05 x) where all the quantities are in their S.I. unit. The maximum particle velocity of the medium is. 0.5 ms ms ms 1 (d) ms 1. If the ratio of the amplitudes of two interfering beams be : 3, the ratio of the minimum and maximum intensities of sound would be 4 : 9 :3 1 : 5 (d) 1 : 5 3. In the interference of waves from two sources of intensities I 0 and 4I 0, the intensity at a point where the phase difference is is 5I 0 3I 0 I 0 (d) I 0 4. In stationary waves pressure nodes and antinodes coincide with displacement nodes and antinodes velocity nodes and antinodes coincide with displacement nodes and antinodes pressure nodes and antinodes coincide with velocity nodes and antinodes (d) none of these 5. A transverse wave is described by the equation y = y 0 sin (ft x/) The maximum particle velocity is equal to four times the wave velocity if y 0 4 y 0 = y 0 (d) = y 0 6. The fundamental frequency of a vibrating organ pipe is 00 Hz. The first overtone is 400 Hz The first overtone may be 400 Hz The first overtone may be 600 Hz (d) 600 Hz is an overtone. 7. In the equation y = a sin k(x pt), where k, the speed of the wave is p k p k/p (d) p/k 8. The speed of sound through a gaseous medium bears a constant ratio with the rms speed of its molecules That constant ratio is (d) 9. The temperature at which the velocity of sound is double its velocity at 0 C is 546 C 819 C 109 C (d) 73 C

3 10. In the fourth harmonic of an open organ pipe, there are N, 3 A 3N, 4A 4N, 5A (d) 5N, 4A 11. The length of an open pipe is 5 cm and the velocity of sound in air is 350 ms 1. The frequency in Hz of the second harmonic of the pipe is (d) A stretched string vibrates with a frequency of 30 Hz in its fundamental mode when the supports are 60 cm apart. The velocity of transverse waves in the string is 9 ms ms 1 36 ms 1 (d) 7 ms A string B has twice the length, twice the diameter, twice the tension and twice the density of another string A. Which of the following alternatives expresses the relation between the frequencies of A and B? n B = 4n A n A = 4n B n A = n B (d) n B = n A 14. The equation y = A sin²(kx t) represents a wave motion with amplitude A frequency / amplitude A/, frequency / amplitude A, frequency /4 (d) does not represent a wave motion. 15. A wave pulse, travelling on a two-piece string, gets partially reflected and partially transmitted at the junction. The reflected wave is inverted in shape as compared to the incident one. If the incident wave has wavelength and the transmitted wave, then > = < (d) nothing can be said about the relation of and 16. Two waves represented by y = a sin (t kx) and y = a cos (t kx) are superposed. The resultant wave will have an amplitude a a a (d) Two wires A and B having identical geometrical construction, are stretched from their natural length by equal amount. The Young s modulus of the wires are Y A and Y B whereas the densities are A and B. It is given that Y A > Y B and A > B. A transverse signal started at one end takes a time t 1 to reach the other end for A and t for B. t 1 < t t 1 = t t 1 > t (d) The information is insufficient to find the relation between t 1 and t 18. Two sine waves travel in the same direction in a medium. The amplitude of each wave is A and the phase difference between the two waves is 10. The resultant amplitude will be A A 4A (d) A

4 19. A wave is represented by the equation y = (0.001 mm) sin [(50 s 1 )t + (.0 m 1 )x]. The wave velocity = 100 m/s The wavelength =.0 m. The frequency = 5/ Hz. (d) The amplitude = mm. 0. A standing wave is produced on a string clamped at one end and free at the other. The length of the string must be an integral multiple of /4 must be an integral multiple of / must be an integral multiple of (d) may be an integral multiple of / 1. Mark out the correct options. The energy of an small part of a string remains constant in a travelling wave. The energy of an small part of a string remains constant in a standing wave. The energies of all the small parts of equal length are equal in a travelling wave. (d) energy trapped between two nodes is constant. In a stationary wave, all the particles of the medium vibrate in phase all the antinodes vibrate in phase the alternate antinodes vibrate in phase (d) all the particles between consecutive nodes vibrate in phase. 3. An open organ pipe of length L vibrates in its fundamental mode. The variation is maximum at the two ends at the middle of the pipe at distance L/4 inside the ends (d) at distance L/8 inside the ends. 4. A cylindrical tube, open at both ends, has a fundamental frequency v. The tube is dipped vertically in water so that half of its length is inside the water. The new fundamental frequency is. v/4 v/ v (d) v 5. A small source of sound moves on a circle as shown in figure and an observer is sitting at O. Let v 1, v, v 3 be the frequencies heard when the source is at A, B and C respectively. A C O v 1 > v > v 3 v 1 = v > v 3 B v > v 3 > v 1 (d) v 1 > v 3 > v The intensity of a plane progressive wave of frequency 1 khz is 10 watt/m. If the density of air is 1.3 kg/m 3 and the speed of sound is equal to 330 m/s, then pressure amplitude of the wave is : N/m N/m N/m (d) N/m

5 7. A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of 9 Kg is suspended from the wire. When this mass is replaced by a mass M, the wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. The value of M is 5 Kg 5 Kg 1.5 Kg (d) 1/5 Kg. 8. Equation of a stationary and travelling waves are as follows y 1 = a sin kx cos t and y = a sin (t kx). 3 The phase difference between two points x1 and x is 1 in the standing wave (y 3k k 1 ) and is 1 in travelling wave (y ) then ratio is 1 5/6 3/4 (d) 6/7 9. In a sonometer wire, the tension is maintained suspending a 50.7 kg mass from the free end of the wire. The suspended mass has a volume of m 3. The fundamental frequency of the wire is 60 Hz. If the suspended mass is completely submerged in water, the fundamental frequency will become : 40 Hz 0 Hz 30 Hz (d) 80 Hz. 30. The beat frequency produced by two tuning forks when sounded together is observed to be 4 Hz. One of the forks makes 384 vibrations per second. When the other fork is loaded with a small piece of wax, the beats disappear 1 st. The frequency of the second tuning fork is 388 Hz 380 Hz more than 388 Hz (d) less than 380 Hz. MORE THAN ONE CORRECT CHOICE 31. In a resonance tube experiment, a close organ pipe of length 10 cm. resonates when tune with a turning fork of frquency 340 Hz. If water is poured in the pipe then (given v air = 340 m/sec) minimum length of water column to have the resonance is 45 cm. the distance between two successive nodes is 50 cm. the maximum length of water column to create the resonance is 95 cm. (d) none of these 3. A wave equation is given as y cos(500t 70x), where y is in mm, x in m and t is in sec. Then : the wave must be a transverse propagating wave the speed of the wave is 50/7 m/s the frequency of oscillations 100Hz (d) two closest points which are in same phase have separation 0/7 cm 33. Three tuning forks are available. One fork marked A produces a 440 Hz tone. The other forks are marked X and Y. The frequency of Y is less than the frequency of X. When forks A and X are sounded together a beat frequency of 4 Hz is heard. For forks A and Y, the beat frequency is 7 Hz. For forks X and Y. The beat frequency is 3 Hz. The frequency of X is 436 Hz The frequency of X is 444 Hz The frequency of Y is 447 Hz. (d) The frequency of Y is 433 Hz

6 34. A transverse sinusoidal wave of amplitude a, wavelength and frequency f is traveling on a stretched string. The maximum speed of any point on the string is v/10 where v is the speed of propagation of the wave. If a = 10 3 m and v = 10 ms 1, then and f are given by = 10 m = 10 3 m f = 10 3 /()Hz (d) f = 10 4 Hz 35. Standing waves are produced on a stretched string of length L with fixed ends. When there is a node at a distance L/3 from one end, then : minimum and next higher number of nodes excluding the ends are, 5 respectively minimum and next higher number of nodes excluding the ends are, 4 respectively frequency produced may be V/3L (d) frequency produced may be 3V/L [V = Velocity of waves in the string] 36. A car moves towards a hill with speed v c. It blows a horn of frequency f which is heared by an observer following the car with speed v 0. The speed of sound in air is v. the wavelength of sound reaching the hill is v/f the wavelength of sound reaching the hill is v v c f the beat frequency observed by the observer is (d) the beat frequency observed by the observer is v v v v 0 c f v (v v )f c 0 v v c 37. A string of length L is stretched along the x-axis and is rigidly clamped at its two ends. It undergoes transverse vibration. If n is an integer, which of the following relations may represents the shape of the string at any time t? nx y Asin cost L nx y A cos cost L (d) nx y Asin sin t L nx y A cos sin t L 38. The vibration of a string fixed at both ends are described by Y = sin(x)sin(100t) where Y is in mm, x is in cm, t in sec, then : Maximum displacement of the particle at x = 1/6 cm would be 1mm. velocity of the particle at x = 1/6 cm at time t = 1/600 sec will be 1573 mm/s If the length of the string be 10 cm, number of loop in it would be 5 (d) None of these 39. In a wave motion y a sin kx t, y can represent electric field magnetic field displacement (d) pressure. 40. Velocity of sound in air is 30 ms 1. A pipe closed at one end has a length of 1 m. Neglecting end correction, the air column in the pipe can resonate for sound of frequency. 80 Hz 40 Hz 30 Hz (d) 400 Hz

7 MISCELLANEOUS ASSIGNMENT Comprehension-1 Doppler s effect is of great importance in determining the velocity of approach or recession of heavenly bodies towards or away from the earth. When the light coming from a distant star is examined by a spectroscope, the spectrum consists of well defined lines. By Doppler s effect we know that when a source of light moves towards the observer, the frequency of light increases (i.e., the wavelength decreases) and vice versa. When the star approaches the earth, the lines are shifted towards the shorter wavelength i.e., towards the blue end of the spectrum. On the other hand, when the star moves away from the earth, the lines are shifted towards the red end. By measuring the shift, the velocity of the star can be calculated. Let a star be moving with a velocity v with respect to the earth. Now by Doppler s principle, the altered frequency n is given by n = cn c v where c is the velocity of light and n is the true frequency of the light waves. If and are the altered and true wavelengths respectively, then the change in wavelength d = c v. Thus knowing the change in wavelength, v can be calculated. 1. A radar operates at wavelength 50 cm. If the beat frequency between transmitted signal and signal reflected from an aircraft is equal to 1 khz at the radar location, the velocity of approaching aircraft will be 900 m/sec 450 m/sec 1000 m/sec (d) 500 m/sec. A radio wave of frequency 840 MHz is sent towards an aeroplane. The frequency of the radio echo has a frequency.8 khz more than the original frequency. The velocity of the aeroplane is 3 km s 1 km s 1 4 km s 1 (d) 0.1 km s 1 3. A star is moving away from the Earth with a velocity of 10 5 ms 1. The shift in the spectral line of wavelength 5700 Å as observed on Earth is 0.53 Å 1.06 Å 1.90 Å (d) 3.08 Å 4. Radio waves are sent towards aeroplane flying high. Change in frequency will be (d) higher when plane is moving away with speed v compared to change in frequency when plane is moving towards earth with same speed higher when plane is moving towards earth with speed v compared to change in frequency when plane is moving away with same speed. change in frequency is independent of whether plane moving towards or away. none of these

8 Comprehension- Any function y = f (ax ± bt) defines wave provided function is bounded. wave velocity = b/a frequency is b/ Amplitude is defined as maximum displacement of particle. 5. The equation of transverse wave in a string is y = 8 sin [1.56 x 5.1 t] cm, x is in cm. What is the amplitude of the wave? 8 cm 1.56 cm 5.1 cm (d) m 6. The frequency of the wave is 1 Hz Hz 3 Hz (d) 4 Hz 7. is cm 3 cm 4 cm (d) 5 cm 8. The speed of the wave is 5 cm s 1 4 cm s 1 0 cm s 1 (d) 33 m s 1 9. A string of linear mass density kg/m is stretched under a tension of 7 N between two rigid supports 60 cm apart. The string is vibrating in second overtone so that amplitude at one of its antinode is 0.5 cm (values in Column II are in CGS unit) Column I Column II A. maximum velocity of a particle at antinode (p) 150 B. maximum velocity of a particle at 5 cm from any node (q) C. maximum acceleration of a particle at antinode (r) (150 ) D. maximum acceleration of particle at 5 cm from any node (s) Match the columns I and II Column I Column II A. y = 4 sin (5x 4t) + 3 cos (4t 5x + /6) (p) Particles at every position are performing SHM B. x x y 10cos t sin (100) t (q) Equation of travelling wave C. y = 10 sin (x 10t) + 10 cos (10t + x) (r) Equation of standing wave D. y = 10 sin (x 10t) + 8 cos (118t 59/30x) (s) Equation of Beats

9 11. A transverse wave is represented by the equation y = y 0 sin (vt x). Value of is n y 0, when the maximum particle velocity equal to two times the wave velocity. Find the value of n The amplitude of a wave disturbance propagating along positive X-axis is given by, y = (1 x ) 1 and y = 1 (x ) at t = 0 at t = 4sec where x and y in meters. The shape of the wave disturbance does not change with time. The velocity of the wave is n 10 1 m/s. Find the value of n. 13. The equation of a wave is represented by y = 10-4 sin m/s. Find the value of n. x 100t 10 m, then the velocity of wave is 500n 14. The time taken by a particle in reaching from trough to crest in a transverse wave is T/n. (T = time period of wave). Find the value of n. t x 15. Equation of progressive wave is given by y = a sin 4, where t is in seconds an x is in meters. Then the distance through which the wave moves in 8 seconds is 4n meter. Find the value of n. 16. A uniform rope of length 10m and mass 15 kg hangs vertically from a rigid support. A block of mass 5 kg is attached to the free end of the rope. A transverse pulse of wavelength 0.08 m is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is 4n cm.. Find the value of n. 5kg 15kg 17. The density of the material of a wire used in sonometer is kg/m 3. If the stress on the wire is N/m, the speed of transverse wave in the wire is 100 n m/s. Find the value of n. 18. The length of a copper wire is 5m and its radius is 1mm. A force of 31.4 N is applied at each end of the wire. The Young's modules of elasticity for copper is N/m. The density of copper is 8900 kg/ m 3. The velocity of transverse waves is the wire is 6.7n m/s. Find the value of n. 19. The displacement equations for two waves undergoing super position are respectively y 1 = 4 sin t and y = 3 sin (t + /). Find the amplitude of the resultant wave. 0. When a body of mass 5Kg is suspended from a sonometer wire then it vibrates with frequency 00 Hz. If the volume of the body is m 3 and it is immersed in then the frequency of vibration of the wire is 40 n Hz.. Find the value of n.

10 PREVIOUS YEAR QUESTIONS IIT-JEE/JEE-ADVANCE QUESTIONS 1. A transverse wave is described by the equation Y = Y 0 sin (f t x/). The maximum particle velocity is equal to four times the wave velocity if = Y 0 /4 = Y 0 / = Y 0 (d) = Y 0. A wave equation which gives the displacement along the Y direction is given by y = 10 4 sin (60 t + x), where x and y are in metres and t is time in seconds. This represents a wave. (d) travelling with a velocity of 30 m/s in the negative x direction. of wavelength metre of frequency 30/ hertz of amplitude 10 4 metre travelling along the negative x direction. 3. A wave is represented by the equation y = A sin (10 x + 15 t + /3) where x is in meters and t is in seconds. The expression represents a wave travelling in the positive x-direction with a velocity 1.5 m/s. a wave travelling in the negative x-direction with a velocity 1.5 m/s. a wave travelling in the negative x-direction having a wave-length 0. m. (d) a wave travelling in the positive x-direction having a wave-length 0. m. 4. A wave represented by the equation y = a cos (kx t) is superposed with another wave to form a stationary wave such that the point x = 0 is a node. The equation for the other wave is a sin (kx + t) a cos (kx + t) a cos (kx t) (d) a sin (kx t) 5. Two identical straight wires are stretched so as to produce 6 beats per second when vibrating simultaneously. On changing the tension slightly in one of them, the beat frequency remains unchanged. Denoting by T 1, T the higher and the lower initial tensions in the strings, then it could be said that while making the above changes in tension T was decreased T was increased T 1 was increased (d) T 1 was decreased 6. Two sound waves of equal intensity I generates beats. The maximum intensity of sound produced in beats will be I 4 I I (d) I/ 7. A whistle giving out 450 Hz approaches a stationary observer at a speed of 33 m/s. The frequency heard by the observer in Hz is (d) A train moves towards a stationary observer with speed 34 m/s. The train sounds a whistle and its frequency registered by the observer is f 1. If the train s speed is reduced to 17 m/s, the frequency registered is f. If the speed of sound is 340 m/s, then the ratio f 1 /f is 18/19 1/ (d) 19/18

11 9. A siren placed at a railway platform is emitting sound of frequency 5 khz. A passenger sitting in a 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 of train A is 4/5 5/6 (d) A cylindrical tube, open at both ends, has a fundamental frequency f in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air column is now f / 3f /4 f (d) f. 11. An air column in a pipe, which is closed at one end, will be in resonance with a vibrating tuning fork of frequency 64 Hz if the length of the column in cm is (d) A tube closed at one end and containing air, produces, when excited, the fundamental note of frequency 51 Hz. If the tube is open at both ends, the fundamental frequency that can be excited is (in Hz) (d) The displacement of particles in a string stretched in the X-direction is represented by y. Among the following expressions for y, those describing wave motion are cos kx sin t k x t cos (k x + t) (d) cos (k x t ) 14. An organ pipe P 1 closed at one end vibrating in its first harmonic and another pipe P one at both ends vibrating in its third harmonic are in resonance with a given tuning fork. The ratio of the length of P 1 to that of P is 8/3 3/8 1/ (d) 1/3 15. An open pipe is suddenly closed at one end with the result that the frequency of third harmonic of the closed pipe is found to be higher by 100 Hz than the fundamental frequency of the open pipe. The fundamental frequency of the open pipe is 00 Hz 300 Hz 40 Hz (d) None 16. In a resonance column experiment, the first resonance is obtained when the level of the water in the tube is at 0 cm from the open end. Resonance will also be obtained when the water level is at a distance of 40 cm from the open end 60 cm from the open end 80 cm from the open end (d) 100 cm from the open end 17. Two vibrating strings of the same material but lengths L and L have radii r and r respectively. Thy are stretched under the same tension. Both the strings vibrate in their fundamental modes, the one of length L with frequency n 1 and the other with frequency n. The ratio n 1 /n is given by 4 8 (d) The ends of a stretched wire of length L are fixed at x = 0 and x = L. In one experiment, the displacement of the wire is y 1 = A sin (x/l) sin t and energy is E 1 and in another experiment its displacement is y = A sin (x/l) sin t and energy E. Then E = E 1 E = E 1 E = 4E 1 (d) E = 16 E 1

12 19. Two pulses in a stretched string whose centres are initially 8 cm. apart the moving towards each other as shown in the figure. The speed of each pulse is cm/s. After seconds, the total energy of the pulses will be (d) zero purely kinetic purely potential partly kinetic and partly potential 0. A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of 9 kg is suspended from the wire. When this mass is replaced by a mass M, the wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. The value of M is 5 kg 5 kg 1.5 kg (d) 1/5 kg 1. If the radius of earth were to shrink by one percent, its mass remaining the same, the acceleration due to gravity on the earth s surface would decrease remains unchanged increase (d) none of these Comprehension-III Two waves y 1 = A cos(0.5 placed along x-axis. x 100 t ) and y = A cos(0.46 x 9. Find the number of times intensity is maximum in time interval of 1 sec (d) Find wave velocity of louder sound t ) are travelling in a pipe 96 m/s 100 m/s 19 m/s (d) 00 m/s 4. Find the number of times y 1 + y = 0 at x = 0 in 1 sec (d) In the experiment to determine the speed of sound using a resonance column, prongs of the tuning fork are kept in a vertical plane prongs of the tuning fork are kept in a horizontal plane in one of the two resonances observed, the length of the resonating air column is close to the wavelength of sound in air (d) in one of the two resonances observed, the length of the resonating air column is close to half the wavelength of sound in air Paragraph Two trains A and B are moving with speeds 0 m/s and 30 m/s respectively in the same direction on the same straight tack, with B ahead of A. The engines are at the front ends. The engine of train A blows a long whistle. Assume that the sound of the whistle is composed of components varying in frequency from f 1 = 800 Hz to f = 110 Hz, as shown in the figure. The spread in the frequency (highest frequency lowest frequency) is thus 30 Hz. The speed of sound in still air is 340 m/s. [

13 6. The speed of sound of the whistle is 340 m/s for passengers in A and 310 m/s for passengers in B 360 m/s for passengers in A and 310 m/s for passengers in B 310 m/s for passengers in A and 360 m/s for passengers in B (d) 340 m/s for passengers in both the trains 7. The distribution of the sound intensity of the whistle as observed by the passengers in train A is best represented by (d) 8. The spread of frequency as observed by the passengers in train B is 310 Hz 330 Hz 350 Hz (d) 90 Hz 9. A vibrating string of certain length l under a tension T resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length 75 cm inside a tube closed at one end. The string also generates 4 beats per second when excited along with a tuning fork of frequency n. Now when the tension of the string is slightly increased the number of beats reduces to per second. Assuming the velocity of sound in air to be 340 m/s, the frequency n of the tuning fork in Hz is (d) A transverse sinusoidal wave moves along a string in a positive x-direction at a speed of 10 cm/s. The wavelength of the wave is 0.5 m and its amplitude is 10 cm. At a particular time t, the snap-shot of the wave is shown in figure. The velocity of point P when its displacement is 5 cm is y 3 ˆ 3 j m/s ˆ j m/s P 3 ˆ 3 i m/s (d) i ˆ m/s The x t graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at t = 4/3s is 3 3 cms cms 3 (d) cm/ s cms x(cm) t(s) x

14 3. When two progressive waves y 1 = 4 sin(x 6t) and y = 3 sin x 6t amplitude of the resultant wave is are superimposed, the 33. 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.% 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 A hollow pipe of length 0.8 m is closed at one end. At its open end a 0.5 m long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is 50 N and the speed of sound is 30 ms 1, the mass of the string is 5 grams 10 grams 0 grams (d) 40 grams 35. 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 30 m/s. The frequency of the siren heard by the car driver is 8.50 khz 8.5 khz 7.75 khz (d) 7.50 khz 36. A point mass is subjected to two simultaneous sinusoidal displacements in x-direction, x 1 (t) = A sin t and x (t) = A sin t. Adding a third sinusoidal displacement x 3 3 (t) = B sin (t + ) brings the mass to a complete rest. The values of B and are 3 A, 4 4 A, A, (d) A, Column I shows four systems, each of the same length L, for producing standing waves. The lowest possible natural frequency of a system is called its fundamental frequency, whose wavelength is denoted as f. Match each system with statements given in Column II describing the nature and wavelength of the standing waves. Column I Column II A. Pipe closed at one end (p) Longitudinal waves 0 L B. Pipe open at both ends (q) Transverse waves 0 L C. Stretched wire clamped at both ends (r) f = L 0 L D. Stretched wire clamped at both ends and at mid-point (s) f = L 0 L L/ (t) f = 4L

15 38. A person blows into open end of a long pipe. As a result, a high-pressure pulse of air travels down the pipe. When this pulse reaches the other end of the pipe. (d) a high-pressure pulse starts travelling up the pipe, if the other end of the pipe is open a low-pressure pulse starts travelling up the pipe, if the other end of the pipe is open a low-pressure pulse starts travelling up the pipe, if the other end of the pipe is closed a high-pressure pulse starts travelling up the pipe, if the other end of the pipe is closed 39. A student is performing the experiment of Resonance Column. The diameter of the column tube is 4 cm. The frequency of the tuning fork is 51 Hz. The air temperature is 38 C in which the speed of sound is 336 m/s. The zero of the meter scale coincides with the top end of the Resonance Column tube. When the first resonance occurs, the reading of the water level in the column is ] 14.0 cm 15. cm 16.4 cm (d) 17.6 cm 40. A horizontal stretched string, fixed at two ends, is vibrating in its fifth harmonic according to the equation, y (x, t) = (0.01m) sin [(6.8 m 1 ) x] cos [(68 s 1 )t]. Assuming = 3.14, the correct statement(s) is (are) The number of nodes is 5 (d) The length of the string is 0.5 m The maximum displacement of the midpoint of the string, from its equilibrium position is 0.01 m The fundamental frequency is 100 Hz 41. Two vehicles, each moving with speed u on the same horizontal straight road, are approaching each other. Wind blows along the road with velocity w. One of these vehicles blows a whistle of frequency f 1. An observer in the other vehicle hears the frequency of the whistle to be f. The speed of sound in still air is V. The correct statement (s) is (are) If the wind blows from the observer to the source, f > f 1. If the wind blows from the source of the observer, f > f 1. If the wind blows from observer to the source, f < f 1. (d) If the wind blows from the source to the observer, f < f A student is performing an experiment using a resonance column and a tuning fork of frequency 44 s 1. He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas). If the minimum height at which resonance occurs is (0.350 ± 0.005) m, the gas in the tube is (Useful information: 167 RT 640 J mole ; 140 RT 590 J mole M in grams are given in the options. Take the value of 1/ 1/ 1/ 1/. The molar masses 10 M for each gas as given there) Neon (M = 0, 10 7 ) Nitrogen (M = 8, ) 8 5 Oxygen (M = 3, 10 9 ) (d) Argon (M = 36, ) 36 3

16 43. One end of a taut string of length 3m along the x axis is fixed at x = 0. The speed of the waves in the string is 100 ms 1. The other end of the string is vibrating in the y direction so that stationary waves are set up in the string. The possible waveform (s) of thee stationary waves is(are) x 50t y( t) A sin cos 6 3 5x 50t y( t) A sin cos 6 3 (d) x 100t y( t) Asin cos 3 3 5x y( t) Asin cos 50t DCE QUESTIONS 1. A source is moving along a circle of radius 10 metres with angular speed of 0 radians per second. It is emitting sound of frequency 30 Hz. What is the minimum frequency heard by a stationary observer standing at large distance from centre? Take velocity of sound as 340 ms (d) 350. Which wavelengths are reflected by the ionosphere? = 10 m = 7.5 m = 5 m (d) =.5 m 3. If = 6000 Å & = 4Å, then velocity of star is 10 5 m/s m/s m/s (d) none of these cm is a wavelength corresponding to the spectrum of Infra-red rays Ultra-violet rays Microwaves (d) X-rays 5. The wavelength of a particle is 99 cm and that of other is 100 cm. Speed of sound is 396 m/s. The number of beats heard is (d) 8 6. Two waves are propagating with same amplitude and nearly same frequency. They result in beats stationary wave resonance (d) none of these 7. Following two wave trains are approaching each other. y 1 = a sin 00 t y = a sin 08 t The number of beats heard per second is (d) 0 8. What does not change when sound enters from one medium to another? Wavelength Speed Frequency (d) none of these

17 9. Fundamental frequency of a sonometer wire is n, if the tension, length and diameter are increased 3 times, what is the new frequency? n n 3 n (d) 3 3 n 10. What is the number of beats heard by the driver of a taxi which is approaching a wall at a speed 30 km/hr and emitting a sound of frequency 300 Hz? Velocity of sound = 330 m/s (d) A person is standing on a railway platform and a train is approaching, what is the maximum wavelength of sound he can hear? Given wavelength of whistle = 1 m; speed of sound in air = 330 m/s; speed of the train = 36 km/hr. 1 m 3/33 m 33/3 m (d) 1/13 m 1. Velocity of sound in open-ended tube is 330 m/s, the frequency of waves is 1.1 khz and the length of tube = 30 cm, which harmonic will it emit? nd 3 rd 4 th (d) 5 th 13. If a particle is travelling with a speed of 0.9 of the speed of sound and emitting radiations of frequency 1 khz and moving towards the observer what is its apparent frequency? (d) 10 kilohertzs 14. In case of a transverse wave, frequency is proportional to T 1 T 1 T (d) T 15. A string is tied on a sonometer. Second end is hanging downward through a pulley with tension T. The velocity of the transverse wave produced is proportional to 1 T T T (d) 1 T = 100 cm, = 90 cm and velocity = 396 m/s. The number of beats are (d) One musical instrument has frequency 90 Hz; velocity of source = 1/10th of the velocity of sound towards observer. What is the frequency of sound as heard by the observer? 100 Hz 10 4 Hz 900 Hz (d) 10 4 Hz 18. Sound waves in air are always longitudinal because the density of air is very small (d) this is an inherent characteristics of sound waves in all media air does not have a modulus of rigidity air is a mixture of several gases

18 19. Equation of a progressive wave is given by y = sin {(t/5 x/9) + /6}, where x + y are in cm. Then which of the following is correct? V = 5 cm/sec. = 18 cm A = 0.04 cm (d) f = 50 Hz 0. A train is approaching with velocity 5 m/sec towards a pedestrian standing on track, frequency of horn of train is 1 KHz. Frequency heard by the pedestrian (v = 350 m/s) Hz 986 Hz (d) 945 Hz 1. Motion of two particles is given by y 1 = 0.5 sin (310 t); y = 0.5 sin (316 t). Find beat frequency (d) 6. Given above is the snapshot of a standing wave. What is the phase difference between a and b. y 90º 180º 360º (d) 0º MAINS QUESTIONS a b x 1. The displacement y of a particle in a medium can be expressed as y = 10 6 sin(100t + 0x + /4)m where t is in second and x in meter. The speed of the wave is 0 m/s 5 m/s 000 m/s (d) 5 m/s. When two tuning forks (fork 1 and fork ) are sounded simultaneously, 4 beats per second are head. Now, some tape is attached on the prong of the fork. When the tuning forks are sounded again, 6 beats per second are heard. If the frequency of fork 1 is 00 Hz, then what was the original frequency of fork? 196 Hz 04 Hz 00 Hz (d) 0 Hz 3. An observer moves towards a stationary source of sound, with a velocity one-fifth of the velocity of sound. What is the percentage increase in the apparent frequency? 5 % 0 % zero (d) 0.5 % 4. A whistle producing sound waves of frequencies 9500 Hz and above is approaching a stationary person with speed v ms 1. The velocity of sound in air is 300 ms 1. If the person can hear frequencies upto a maximum of 10,000 Hz, the maximum value of v upto which he can hear the whistle is 15 ms 1 15 / ms 1 15 ms 1 (d) 30 ms 1 5. A string is stretched between fixed points separated by 75.0 cm. It is observed to have resonant frequencies of 40 Hz and 315 Hz. There are no other resonant frequencies between these two. Then, the lowest resonant frequency for this string is 105 Hz 1.05 Hz 1050 Hz (d) 10.5 Hz 6. A sound absorber attenuates the sound level by 0 db. The intensity decreases by a factor of

19 (d) A wave travelling along the x-axis is described by the equation y(x, t) = cos (x t). If the wavelength and the time period of the wave are 0.08 m and.0 s, respectively, then and in appropriate units are , , 1.50, (d) = 5.00, =.0 8. While measuring the speed of sound by performing a resonance column experiment, a student gets the first resonance condition at a column length of 18 cm during winter. Repeating the same experiment during summer, she measures the column length to be x cm for the second resonance. Then x > > x > > x > 18 (d) 18 > x 9. Three sound waves of equal amplitudes have frequencies (v 1), v, (v + 1). They superpose to give beats. The number of beats produced per second will be 3 1 (d) The transverse displacement y(x, t) of a wave on a string is given by ( ax bt ab xt y( x, t) e ). This represents a wave moving in +x direction with speed a b wave moving in x direction with speed b a standing wave of frequeny b (d) standing wave of frequency 1 b 11. A cylindrical tube, open at both ends, has a fundamental frequency, f in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air-column is now: f f / 3f /4 (d) f 1. A sonometer wire of length 1.5 m is made of steel. The tension in it produces an elastic strain of 1%. What is the fundamental frequency of steel if density and elasticity of steel are kg/m 3 and N/m respectively? 00.5 Hz 770 Hz Hz (d) 178. Hz 13. A pipe of length 85 cm is closed from one end. Find the number of possible natural oscillations of air column in the pipe whose frequencies lie below 150 Hz. The velocity of sound in air is 340 m/s (d) 8

20 BASIC LEVEL ASSIGNMENT 1. The following equation gives the displacement y at time t for a particle at a distance x. where all are in S.I. unit. Find y = 0.01 sin 500 (t x/30) (d) the wavelength, the speed of the wave the velocity amplitude of the particles of the medium. the acceleration amplitude of the particles of the medium.. A stationary wave is given by x y = 5 sin 3 cos 40 t where x and y are in cm and t is in seconds. What are the amplitude and velocity of the component waves whose superposition can give rise to this vibration? What is the distance between the nodes? What is the velocity of a particle of the string at the position x = 1.5 cm when t = 9/8 s? 3. The loudest painless sound produces a pressure amplitude of 8 Nm. Calculate the intensity of this sound wave at STP. Density of air at STP = 1.3 kg m 3 and speed of sound at STP = 33 m s A stationary observer receives sound waves from two tuning forks, one of which approaches and the other recedes with the same velocity. As this takes place, the observer hears beats of frequency Hz. Find the velocity of each tuning fork if their oscillation frequency v 0 = 680 Hz and the velocity of sound in air is v s = 340 ms A string of length 5 cm and mass.5 g is under tension. A pipe closed at one end is 40 cm long. When the string is set vibrating in its first overtone and the air in the pipe in its fundamental frequency, 8 beats per second are heard. It is observed that decreasing the tension in the string decreases the beat frequency. If the speed of sound in air 30 ms 1, find the tension in the string. 6. A copper wire is held at the two ends by rigid supports. At 30 C the wire is just taut, with negligible tension. Find the speed of transverse waves along the wire at 10 C. Density of copper = 9 10³ kgm 3, Young s modulus = Nm and coefficient of linear expansion = C 1.

21 7. A tube closed at one end has a vibrating diaphragm at the other end, which may be assumed to be a displacement node. It is found that when the frequency of the diaphragm is 00 Hz, a stationary wave pattern is set up in which the distance between adjacent nodes is 8 cm. When the frequency is gradually reduced, the stationary wave pattern disappears but another stationary wave pattern reappears at a frequency of 1600 Hz. Calculate. (d) the speed of sound in air, the distance between adjacent nodes at a frequency of 1600 Hz., the distance between the diaphragm and the closed end, the next lower frequencies at which stationary wave patterns will be obtained. 8. A travelling wave is produced on a long horizontal string by vibrating an end up and down sinusoidally. The amplitude of vibrating is 1.0 cm and the displacement becomes zero 00 times per second. The linear mass density of the string is 0.10 kg/m and it is kept under a tension of 90 N. Find the speed and the wavelength of the wave. Assume that the wave moves in the positive x-direction and at t = 0, the end x = 0 is at its positive extreme position. Write the wave equation. Find the velocity and acceleration of the particle at x = 50 cm at time t = 10 ms 9. The equation of a standing wave, produced on a string fixed at both ends, is y = (0.4 cm) sin[(0.314 cm 1 )x] cos [(600 s 1 )t]. What could be the smallest length of the string? 10. Figure shows an aluminium wire of length 60 cm joined to a steel wire of length 80 cm and stretched between two fixed supports. The tension produced is 40 N. The cross-sectional area of the steel wire is 1.0 mm and that of the aluminium wire is 3.0 mm². What could be the minimum frequency of a tuning fork which can produce standing waves in the system with the joint as a node? The density of aluminium is.6 g/cm³ and that of steel is 7.8 g/cm³. 80cm Steel 60cm Aluminium 11. A small source of sound S of frequency 500 Hz is attached to the end of a light string and is whirled in a vertical circle of radius 1.6 m. The string just remains tight where the source is at the highest point. S An observer is located in the same vertical plane at a large distance and at the same height as the centre of the circle. The speed of sound in air = 330 m/s and g = 10 m/s². Find the maximum frequency heard by the observer.

22 An observer is situated at a large distance vertically above the centre of the circle. Find the frequencies heard by the observer corresponding to the sound emitted by the source when it is at the same height as the centre. 1. A source emitting a sound of frequency f is placed at a large distance from an observer. The source starts moving towards the observer with a uniform acceleration a. Find the frequency heard by the observer corresponding to the wave emitted just after the source starts. The speed of sound in the medium is v. 13. A sound wave of frequency 100 Hz is travelling in air. The speed of sound in air is 350 m/s. By how much is the phase changed at a given point in.5 ms? What is the phase difference at a given instant between two points separated by a distance of 10.0 cm along the direction of propagation? 14. The equation of a travelling sound wave is y= 6.0 sin (600 t 1.8 x) where y is measured in 10 5 m, t in second and x in metre. Find the ratio of the displacement amplitude of the particles to the wavelength of the wave. Find the ratio of the velocity amplitude of the particles to the wave speed. 15. Two point sources of sound are kept at a separation of 10 cm. They vibrate in phase to produce waves of wavelength 5.0 cm. What would be the phase difference between the two waves arriving at a point 0 cm from one source on the line joining the sources and on the perpendicular bisector of the line joining the sources? 16. The absolute temperature of air in a region increases linearly from T 1 to T in a space of width d. Find the time taken by a sound wave to go through the region in terms of T 1, T, d and the speed v of sound at 73 K. Evaluate this time for T 1 = 80 K, T = 310 K, d = 33 m and v = 330 m/s. 17. A tuning fork vibrating at frequency 800 Hz produces resonance in a resonance column tube. The upper end is open and the lower end is closed by the water surface which can be varied. Successive resonances are observed at lengths 9.75 cm, 31.5 cm and 5.75 cm. Calculate the speed of sound in air from these data. 18. The fundamental frequency of a closed organ pipe is equal to the first overtone frequency of an open organ pipe. If the length of the open pipe is 60 cm, what is the length of the closed pipe? 19. The sound level at a point is increased by 30 db. By what factor is the pressure amplitude increased?

23 ADVANCE LEVEL ASSIGNMENT 1. In a stationary wave that is formed as a result of reflection of waves from an obstacle, the ratio of the amplitude at an antinode to the amplitude at node is 6. What percentage of energy is transmitted.. A standing wave y = a sin kx cos t is maintained in a homogeneous rod with cross-sectional area S and density. Find the total mechanical energy confined between the sections corresponding to the adjacent nodes. 3. A siren creates a sound level of 60 db at a location 500 m from the speaker. The siren is powered by a battery that delivers a total energy of 1.0 kj. Assuming that the efficiency of siren is 30%, determine the total time for which the siren can sound. 4. A stationary observer receives a sound from a source of frequency 000 Hz moving with a constant velocity. The apparent frequency varies with time as shown in figure. Find speed of source (v s ) maximum value of apparent frequency f m. (speed of sound is v = 300 m/s) f(hz) fm t(s) 5. A source of sonic oscillations with frequency f Hz and a receiver are located at the same point. At t = 0, the source starts receding from the receiver with constant acceleration a. Assuming the velocity of sound to be c, find the oscillation frequency registered by stationary receiver t 0 second after the start up of the motion. 6. A metallic rod of length 1 m is rigidly clamped at its mid point. Longitudinal stationary waves are set up in the rod in such a way that there are two nodes on either side of the mid point. The amplitude of an antinode is 10 6 m. Write the equation of motion at a point cm from the mid point and equation of the constituent waves in the rod. (Young s Modulus of the material of the rod = Nm ; density = 800 kgm 3 ) 7. An aluminium wire of cross-sectional area 10 6 m² is joined to a steel wire of the same cross-sectional area. This compound wire is stretched on a sonometer pulled by a weight of 10 kg. The total length of the compound wire between the bridges is 1.5 m of which the aluminium wire is 0.6 m and the rest is steel wire. Transverse vibrations are set up in the wire by using an external source of variable frequency. Find the lowest frequency of excitation for which the standing waves are formed such that the joint in the wire is a node. What is the total number of nodes at this frequency? The density of aluminium is.6 10³ kg/m³ and that of steel is kg/m³. (g = 10 m/s²)

24 8. A 3m long organ pipe open at both ends is driven to third harmonic standing wave. If the amplitude of pressure oscillations is 1 percent of mean atmospheric pressure (P 0 = 10 5 N/m²). Find the amplitude of particle displacement. Speed of sound v = 33 m/s and density of air = 1.03 kg/m³ 9. A boat is travelling in a river with a speed 10 m/sec along the stream flowing with a speed m/sec. From this boat, a sound transmitter is lowered into the river through a rigid support. The wavelength of the sound emitted from the transmitter inside the water is mm. Assume that attenuation of sound in water and air is negligible. What will be the frequency detected by a receiver kept inside the river downstream? Transmitter and the receiver are now pulled up into air. The air is blowing with a speed 5 m/sec in the direction opposite the river stream. Determine the frequency of the sound detected by the receiver. (Temperature of the air and water = 0 C; Density of river water = 10³ kg/m³; Bulk modulus of the water = Pa; Gas constant, R = 8.31 J/mol-K; Mean molecular mass of air = kg/mol; C p / C V for air = 1.4) 10. A 3.6 m long pipe resonates with a source of frequency 1.5 Hz when water level is at certain heights in the pipe. Find the heights of water level (from the bottom of the pipe) at which resonances occur. Neglect end correction. Now the pipe is filled to a height H (3.6 m). A small hole is drilled very close to its bottom and water is allowed to leak. Obtain an expression for the rate of fall of water level in the pipe as a function of H. If the radii of the pipe and the hole are 10 m and m respectively, calculate the time interval between the occurrence of first two resonances. Speed of sound in air is 340 m/s and g = 10 m/s². 11. A long wire PQR is made by joining two wires PQ and QR of equal radii. PQ has a length 4.8 m and mass 0.06 kg. QR has length.56 m and mass 0. kg. The wire PQR is under a tension of 80 N. A sinusoidal wave pulse of amplitude 3.5 cm is sent along the wire PQ from the end P. No power is dissipated during the propagation of the wave pulse. Calculate: the time taken by the wave pulse to reach the other end R, and the amplitude of the reflected and transmitted wave pulse after the incident wave pulse crosses the joint Q.

25 1. The air column in a pipe closed at one end is made to vibrate in its second overtone by tuning fork of frequency 440 Hz. The speed of sound in air is 330 m/s. End corrections may be neglected. Let P 0 denote the mean pressure at any point in the pipe, and P 0 the maximum amplitude of pressure variation. Find the length L of the air column. What is the amplitude of pressure variation at the middle of the column? What are the maximum and minimum pressures at the open end of the pipe? (d) What are the maximum and minimum pressures at the closed end of the pipe? 13. The first overtone of an open organ pipe beats with the first overtone of a closed organ pipe with a beat frequency of. Hz. The fundamental frequency of the closed organ pipe is 110 Hz. Find the lengths of the pipes. Speed of sound in air v = 330 m/s.

26 ANSWERS Objective Questions 1.. (d) 3. (d) (b, c, d) (d) (d) (c, d) (d). (c, d) (d) (d) (a,b,c) 3. (a,b,d) 33. (a,d) 34. (a,c) 35. (a,d) 36. (b,d) 37. (a,b) 38. (b,c) 39. (a,b,c,d) 40. (a,b,d) Miscellaneous Assignment 1.. (d) (d) 7. (d) A-(r); B-(p); C-(s); D-(q) 10. A-(p),(q); B-(s); C-(p),(r); D-(s) 11. (1) 1. (5) 13. () 14. () 15. (4) 16. (4) 17. () 18. (5) 19. (5) 0. (4) Previous Year Questions IIT-JEE/JEE-ADVANCE QUESTIONS 1.. (a, b, c, d) 3. (b, c) (b,d) (d) 8. (d) (a, c) (a, c) (d) (d) (d) (d) 3. (5) 33. (7) A-(p),(t); B-(p),(s); C-(q),(s); D-(q),(r) 38. (b,d) (b,c) 41. (a,b) 4. (d) 43. (a,c,d)