PHYSICS Class
II IIT-JEE Achiever 2016-17 (ITP) Intensive Revision Program Worksheet – 15
Topics
Waves
Date:
21-12-2016
Multiple choice questions with one correct alternative 1. An open of sufficient length is dipping in water with a speed V vertically. If at any instant l is length of tube above water then the rate at which fundamental frequency of pipe changes is (C is the speed of sound in air) CV (A) (B) 2l 2 CV (C) (D) 2v 2 t 2
CV 4l 2 CV 4v 2 t 2
2. A bird is singing on a tree and a man is hearing at a distance ‘r’ from the bird. Calculate the displacement of the man towards the bird so that the loudness heart by man increases by 20 dB [Assume that the motion of man is along the line joining the bird and the man] 9r r 3r 4r (A) (B) (C) (D) 10 10 5 5 3. Two loudspeakers are emitting sound waves of wavelength λ with an initial phase difference of
π 2
. At
what minimum distance from O on line AB will one hear a maxima? (A) 25 λ 100 λ
(Β)
(C)
15 25 λ
3
(D) 50 λ 4. A man standing in front of a vertical wall at a certain distance beats a drum at regular intervals. The drumming rate is gradually increased and he finds that the echo is not heard distinctly when the drumming rate becomes 2 beats in 3 seconds. He then moves nearer to the wall by 90 m and finds the echo is again not heart when the drumming rate becomes 1 per second. From this data, the velocity of sound must be (A) 330 m/s
(B) 340 m/s
(C) 360 m/s
(D) 180 m/s
5. A wave with a frequency of 30 Hz travels along a string at a speed of 36 meters per second and reflects off a free end. How far is the first node from the end of the string? (A) 0.2 metre
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(B) 0.3 metre
(C) 0.4 metre
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(D) 0.6 metre
6. The mathematical forms for three sinusoidal traveling waves are given by, Wave 1: y(x, t) = (2 cm) sin (3x – 6t) Wave 2: y (x, t) = (3 cm) sin (4x – 12t) Wave 3: y (x, t) = (4 cm) sin (5x – 11t) where x is in meters and t is in seconds. Of these waves: (A) wave 1 has the greatest wave speed and the greatest maximum transverse string speed. (B) wave 2 has the greatest wave speed and wave 1 has the greatest maximum transverse string speed. (C) wave 3 has the greatest wave speed and the greatest maximum transverse string speed. (D) wave 2 has the greatest wave speed and wave 3 has the greatest maximum transverse string speed. 7. A sinusoidal wave travelling in the positive direction of x on a stretched sting has amplitude 2.0 cm, wavelength 1 m and wave velocity 5.0 m/s. At x = 0 and t = 0, it is given that displacement y = 0 and ∂y < 0 . Express the wave function correctly in the form y = f(x, t): ∂t (A) y = (0.02 m) sin 2 π (x – 5t)
(B) y = (0.02 cm) cos 2π (x – 5t)
(C) y = (0.02 m) sin 2 π x − 5t +
1
4
(D) y = (0.02 cm) cos 2π x − 5t +
8. The equation of a plane progressive wave is y = 0.02 sin 8π t −
1
4
x . When it is reflected at a rarer 20
medium (medium with higher velocity) at x = 0, its amplitude becomes 75% of its previous value. The equation of the reflected wave is: x (A) y = 0.02 sin 8π t − 20
(C) y = +0.015 sin 8π t +
(B) y = 0.02 sin 8π t +
x 20
x 20
(D) y = −0.015 sin 8π t +
x 20
9. String 1 is connected with string 2. The mass per unit length in string 1 is µ1 and the mass per unit length in string 2 is 4µ1. The tension in the strings is T. A travelling wave is coming from the left. What fraction of the energy in the incident wave goes into string 2?
(A)
1 8
(B)
4 9
(C)
2 3
(D)
8 9
10. A string of length l is fixed at both ends. It is vibrating in its third overtone with maximum amplitude ‘a’. The amplitude at a distance (A)
a 2
l
16
from one end is:
(B) 0
(C)
3a 2
(D)
a 2
11. A small pulse travelling with speed v in a sting is shown at t = 0, moving towards free end. Which of these is not correctly matched:
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(i) (ii) (iii)
t= t= t=
L
(p)
v 2L
(q)
v 3L
(r)
v
(A) (i)
(B) (ii)
(C) (iii)
(D) None of these
12. Two strings, A and B, of lengths 4L and L respectively and same mass M each, are tied together to form a knot ‘O’ an stretched under the same tension. A transverse wave pulse is sent along the composite string from the side A, as shown to the right. Which of the following diagrams correctly shows the reflected and transmitted wave pulses near the knot ‘O’?
(A)
(B)
(C)
(D)
–7
13. Two slits separated by a distance of 1 mm are illuminated with red light of wavelength 6.5 ant 10 m. The interference fringes are observed on a screen placed 1m from the slits. The distance between the third dark fringe and the fifth bright fringe is equal to: (A) 0.65 mm
(B) 1.625 mm
(C) 3.25 mm
(D) 0.975 mm
14. String I and II have identical lengths and linear mass densities, but string I is under greater tension than string II. The accompanying figure shows four different situations, (a) to (d), in which standing wave patterns exist on the two strings. In which situation it is possible that strings I and II are oscillating at the same resonant frequency? String I
String II
(A) (B) (C) (D)
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Multiple choice questions with one or more than one correct alternative/s 2
2
2
15. A source is moving across a circle given by the equation x + y = R , with constant speed
330 π 6 3
m/s, in
anti-clockwise sense. A detector is at rest at point (2R, 0) w.r.t. the centre of the circle. It the frequency emitted by the source is f and the speed of sound, c = 330 m/s. Then
(A) the position of the source when the detector records the maximum frequency is +
3 2
R, −
R
2
(B) the co-ordinate of the source when the detector records minimum frequency is (0, R) 6 3
(C) the minimum frequency recorded by the detector is
π+6 3
(D) the maximum frequency recorded by the detector is
f
6 3 6 3 − π
f
16. A pipe of length 1 m is closed at one end. The velocity of sound in air is 300 ms − . The air column in the 1
pipe will resonate for sound of frequencies (A) 75 Hz
(B) 225 Hz
(C) 275 Hz
(D) 375 Hz
17. When an ultrasonic wave travels from air to water (A) the wave bends away from the normal (B) the wave bends towards the normal (C) Speed of the ultrasonic wave is greater in water than the speed in air (D) Speed of the ultrasonic wave is greater in air than the speed in water 18. In a longitudinal stationary wave (A) there is no transfer of energy in the direction of propagation of wave but the phase of oscillation travels in the forward direction (B) maximum increase of pressure occurs at displacement nodes (C) maximum decrease of pressure occurs at displacement nodes (D) the pressure at displacement antinodes is same as what it would be in absence of the wave 4
19. A sonar works with a signal of frequency 4 × 10 Hz. The velocity of the wave in air and water of 320 m/s and 1400 m/s and the wavelength are λa and λw respectively. The echo of a signal is heard after 0.8 second, after reflection from the bed. Then (A) λa = λw
(B) λa = 0.23 λw
(C) depth of bed = 560 m
(D) depth of bed = 1020 m
20. In a mixture of gases, the average number of degree of freedom is f. With rms speed of molecules being c, the velocity of sound will be (A) c
f +2 3f
(C) cannot be related without temperature
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(B) c
f 6
(D) for a diatomic behaviour, v = c
4
7 15
21. A narrow steel rod of length ‘L’ is rigidly clamped at its mid-point and transverse standing waves of frequency ‘f’ are set up in it. The speed of transverse waves in the rod is ‘c’. Then: (A) The free ends of the rod must be antinodes (B) The fundamental frequency ‘f’ of the rod is c/(L) (C) The second overtone frequency of the rod is 5c/(2L) (D) ‘f’ can be any integral multiple of the fundamental frequency 22. A pulse is started at a time t = 0 along the +x direction on a long, taut string. The shape of the pulse at t = 0 is given by function f (x) with: x 4 + 1 for −4 < x ≤ 0 f ( x ) = − x + 1 for 0 < x < 1
0
otherwise
here f and x are in centimeters. The linear mass density of the string is 50 g/m and it is under a tension of 5N (A) The shape of the string is drawn at t = 0 then the area of the pulse enclosed by the string and the x-axis is 2.5 cm
2
(B) The shape of the string is drawn at t = 0 then the area of the pulse enclosed by the string and the x-axis is 5 cm
2
(C) The transverse velocity of the particle at x = 13 cm and t = 0.015 s will be – 250 cm/s (D) The transverse velocity of the particle at x = 13 cm and t = 0.015 s will be 250 cm/s 23. String of length L whose one end is x = 0 vibrates according to the relations given by different equations. Choose the correct statements: πx (A) y = A sin sinωt has 1 antinode, 2 nodes L πx (B) y = A cos sin ωt has 2 antinodes, 1 node L 2πx sin ωt has 3 nodes, 2 antinodes (C) y = A sin L 2πx s sin ωt has 3 antinodes, 2 nodes (D) y = A cos L 24. The equation of a transverse wave travelling on a string is y = 2cos (0.5πx − 150πt) Where x and y are in cm and t is in second. Which of the following are true? (A) The velocity of the wave is 3 ms
−1
(C) The wavelength of the wave is 4 cm
(B) The period of the wave is 0.1 is (D) The amplitude of the wave is 2 cm
25. Two particle P and Q have a phase difference of π when a sine wave passes through the region (A) P oscillates at half the frequency of Q (B) P and Q move in opposite directions (C) P and Q must be separated by half of the wavelength (D) the displacement of P and Q have equal magnitudes
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26. Two pulses travelling on same string are described by equation y1 = y2 =
5 (3x − 4 t ) 2 + 2
and
−5 . Mark out the correct statements. (3x + 4 t − 6)2 + 2
(A) At t = (B) At t =
4 3 3 4
s the two waves cancel each other every where s the two waves cancel each other every where
(C) At x = 1 the two waves cancel each other at all times. (D) At all point two waves cancel each other 27. y(x,t) =
0.8
( 4x + 5t )2 + 5
represents a moving pulse, where x and y are in metre and t in second. Then
(A) pulse is moving in + x direction
(B) in 2 it will travel a distance of 2.5 m
(C) its maximum displacement is 0.16 m
(D) it is a symmetric pulse
28. A mechanical wave propagates in a medium along the x-axis. The particles of the medium (A) may move on the x-axis
(B) may move on the y-axis
(C) must move on the y-axis
(D) must move on the x-axis
29. The plane wave represented by an equation of the form y = (x −vt) implies the propagation along the positive x-axis without change of shape with constant velocity v
dy (A) = −v dt dx dy
d2 y = −v 2 (B) dt dx dy
d2 y d2 y d2 y = − v 2 (D) 2 = v 2 (C) 2 dt dt dx dx d2 y
2
30. Which one of the following represent a travelling wave? (A) y = A ( x − vt 31. Y ( x , t ) =
0.8
( 4 x + 5 t )2 + 5
(B) y = A cos (ax + bt) (C) y = A log (x −vt)
2
2
(D) y = f (x − vt )
m represents a moving pulse, where x, y are in metre and t in second
(A) It is a non-periodic travelling pulse
(B) t is travelling along −ve x-axis
(C) Maximum displacement is 0.16 m
(D) Velocity of motion of pulse is −12.5 ms−
1
32. In a wave motion y = a sin (kx − ωt), y can represent (A) electric field
(B) magnetic field
(C) displacement
(D) pressure change
33. Two waves of equal frequency f and speed v travel in opposite directions along the same path. The waves have amplitudes A and 3A. Then (A) the amplitude of the resulting wave varies with position between maxima of amplitude 4A and minima of zero amplitude (B) the distance between a maxima and adjacent minima of amplitude is
ν 2f
(C) at a point on the path the average displacement is zero ⇒ Average displacement of medium particle at any point is zero. (D) the position of a maxima or minima of amplitude does not change with time
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Read the passage given below and answer questions by choosing the correct alternative In an organ pipe (may be closed or open) of length 1m standing wave is setup, whose equation 2π for longitudinal displacement is given by ξ = (0.1 mm) cos ( y ) cos ( 400) t where y is 0.8 measured from the top of the tube in meters and t in second. 34. The upper end and the lower ends of the tube are respectively (A) open – closed
(B) closed – open
(C) open – open
(D) closed – closed
(C) Third harmonic
(D) Fundamental mode
35. The air column is vibrating in (A) First overtone
(B) Second overtone
5
2
36. Equation of the standing wave in terms of excess pressure is (Bulk modulus of air B = 5 × 10 N/m ) 2π 2π 2 2 (A) Pex = (125pN / m ) sin (B) Pex = (125πN / m ) cos ( y) cos ( 400 t ) ( y ) sin ( 400t ) 0.8 0.8 2π 2π 2 2 c os ( 200t ) (C) Pex = ( 225π N / m ) sin (D) Pex = ( 225πN / m ) cos ( y ) co ( y ) sin ( 200t ) 0.8 0.8
Read the passage given below and answer questions by choosing the correct alternative A and B are two identical point sources of acoustic waves, each producing sound of 1 kHz. They are located in two trains, one in each of the engines of the trains which runs on two close parallel tracks. An observer is standing in between the tracks and listening to the sound received from each of the sources. 37. Firstly, the two trains are approaching the stationary observer from opposite sides with the same speed which is 1% of the velocity of sound in air. The number of beats that he will hear between the frequencies of A and B as heard by him will be (per second) (A) 10
(B) 2
(C) 0
(D) greater than 10
38. If one of the trains remain stationary while the other train recedes from the observer with a speed which is 1% of the velocity of sound in air, the number of beats heard by the observer per second will be (in Hz) (A) 0
(B) greater than 10
(C) 3.5
(D) 9.9
39. If both the trains remain stationary, but the observer runs towards one train parallel to the track with such a speed that he hears exactly 10 beats per seconds, his speed (given the speed of sound is 350 m/s) is (A) 6.3 km/h
(B) 4.5 km/h
(C) 12.6 km/h
(D) 3.6 km/h
Read the passage given below and answer questions by choosing the correct alternative A piano creates sound by gently striking a taut wire with a soft hammer when a key on the piano is pressed. All piano wires in a given piano are approximately of the same length. However, each wire is tied down at two points, the bridge and the agraffe. The length of the wire between the bridge and the agraffe is called the speaking length. The point of the wire struck by the hammer is displaced perpendicularly to the wire’s length. A standing wave is generated by the hammer strike, v =
T
µ
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[where v is the velocity, T is the tension in the wire, and µ is the mass per unit length of the wire]. Tuning a piano involves adjustment of the tension in the wires until just the right pitch is achieved. Correct pitch is achieved by listening to the beat frequency between the piano and a precalibrated tuning fork. 40. A piano note is compared to a tuning fork vibrating at 440 Hz. Three beats per second are discerned by the piano tuner. When the tension in the string is increased slightly, the beat frequency increases. What was the initial frequency of the piano wire? (A) 434 Hz
(B) 437 Hz
(C) 443 Hz
(D) 446 Hz
41. Sound waves move through air at approximately 30 m/s. A piano wire with a 90 cm speaking length resonates at a frequency of 360 Hz. What is the wavelength of the resulting sound wave? (A) 0.94 m
(B) 1.06 m
(C) 4 m
(D) 1.8 m
Read the passage given below and answer questions by choosing the correct alternative A metallic rod of length 1 m has one end free and other end rigidly clamped. Longitudinal stationary waves are set-up in the rod in such a way that there are total six antinodes present along the rod. The amplitude of –6
2
3
3
an antinodes is 4 × 10 N/m and 4 × 10 kg/m respectively. Consider the free end to be at origin and at t = 0 particles at free and are at positive extreme. 42. The equation describing displacements of particles about their mean positions is 11π 11π 6 3 −6 3 cos x cos ( 22 22π × 10 t) x sin ( 22 2 2π × 10 t ) (A) s = 4 × 10 co (B) s = 4 × 10 cos 2 2 (C) s = 4 × 10 6 cos ( 5πx ) cos ( 20 π ×103 t )
(D) s = 4 × 10 6 cos ( 5πx ) sin ( 20 π ×103 t )
−
−
43. The equation describing stress developed in t he rod is 11 11 4 3 4 s in πx + π cos ( 22 2 2π × 103 t) (A) 140.8π × 10 cos πx + x cos ( 22π × 10 t ) (B) 140.8π × 10 si 2 2 4
3
(C) 128π× 10 cos (5πx + π) cos (20π× 10 t)
4
3
(D) 128π× 10 sin (5πx + π) cos (20 π× 10 t)
44. The magnitude of strain at midpoint of the rod at t = 1 sec is −6
(A) 11 3π × 10
(B) 11 2π × 10−6
−6
(C) 10 3π × 10
(D) 10 2π × 10−6
umerical problems 45. Sachin and his mother have a tin whistle each. The pipe length of Sachin’s tin whistle is 52 cm long while the pipe length of mother’s tin whistle is 50 cm long. They both play at the same time, sounding the whistles at their fundamental resonant frequencies. The note that they played are not in tune with each other. The velocity of sound in a air is 325 m/s. Assume the whistle is a pipe with one end open find the beat frequency (in Hz) that is heard when both whistles are plying simultaneously. (Round off to nearest integer) 46. If we run towards a resonance column apparatus vibrating in the fundamental mode with a speed of 5 m/s, we hear a frequency of 165 Hz. When we run away from the apparatus with a speed of 5 m/s, we hear a frequency of 160 Hz. What is the length of the resonance column (in cm) that we see? Neglect end correction.
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47. Two tuning forks A and B each of the natural frequency 85 Hz move with velocity 10 m/s relative to stationary observer ‘O’. Fork A moves away from the observer while the fork B moves towards him as shown in the figure. A wind with a speed 10 m/s is blowing in the direction of motion of fork A. Find the beat frequency measured by the observer in Hz. [Take speed of sound in air as 340 m/s]
48. A source emits sounds waves of frequency 1000 Hz. The source moves to the right with a speed of 32 ms− relative to ground. On the right a reflecting surface moves towards left with a speed of 64 ms − 1
1
relative to the ground. The speed of sound in air is 332 ms − . If the wavelength of sound in head of the 1
source is n, find 10 n? 49. A source emits sounds waves of frequency 1000 Hz. The source moves to the right with a speed of 32 ms− relative to ground. On the right a reflecting surface moves towards left with a speed of 64 ms − 1
1
−1
relative to the ground. The speed of sound in air is 332 ms . If the number of waves arriving per second which meets the reflecting surface is n, find n/20. 50. A source emits sounds waves of frequency 1000 Hz. The source moves to the right with a speed of 32 ms− relative to ground. On the right a reflecting surface moves towards left with a speed of 64 ms − 1
1
relative to the ground. The speed of sound in air is 332 ms − . If the speed of reflected waves is n, find 1
n/4? 51. A source emits sounds waves of frequency 1000 Hz. The source moves to the right with a speed of −1
32 ms relative to ground. On the right a reflecting surface moves towards left with a speed of 64 ms
−1
relative to the ground. The speed of sound in air is 332 ms − . If the wavelength of reflected waves is n, 1
find 100n? 52. Two loudspeaker blow in phase at 170 Hz. An observer sits at 8 m from one speaker and 11 m from the other. The intensity level from either speaker acting alone is 60 dB. The speed of sound is 340 ms − . 1
Find the observed intensity level when both speakers are ON together. 53. Two loudspeaker blow in phase at 170 Hz. An observer sits at 8 m from one speaker and 11 m from the other. The intensity level from either speaker acting alone is 60 dB. The speed of sound is 340 ms − . 1
Find the observed intensity level when both speakers are ON together but one has some technical change so that the speakers are 180 ° out of phase. 54. Two loudspeaker blow in phase at 170 Hz. An observer sits at 8 m from one speaker and 11 m from the −1
other. The intensity level from either speaker acting alone is 60 dB. The speed of sound is 340 ms . Find the observed intensity level when both speakers are ON and in phase but the frequency is 180 HZ. 55. Two coherent monochromatic sources P and Q emit light of wavelength 563.3 nm. The distance between P and Q is 5 µm. If the detector is moved along a line QR perpendicular to PQ and starting from Q, what is the total number of maxima observed?(Don’t count the maxima at infinity).
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56. If the figure given below shows the displacement-time curve of common medium particle for two sound waves A and B propagating in the same medium, the ratio of their intensities I A /IB = X. Find the value of X.
57. A string will break apart if it is placed under too much tensile stress. One type of steel has density ρsteel = 104 kg / m3 and breaking stress σ = 9 × 10 8 N/m2. We make a guitar string from (4π) gram of this type of steel. It should be able to withstand (900 π) N without breaking. The highest possible fundamental frequency (in Hz) of standing waves on the string if the entire length of the string vibrates is n then find n/25.
Column Matching 58. In closed organ pipe Column I
Column II
(i)
third overtone frequency is x times fundamental frequency then x is
(P)
3
(ii)
number of nodes in second overtime
(Q)
4
(iii)
number of antinodes in second overtime
(R)
5
(S)
7
(iv)
th
n harmonic does not exists, where n is
59. Three successive resonance frequencies in an organ pipe are 1310, 1834 and 2358 Hz. Velocity of sound in air is 340 m/s. Column I
Column II
(i)
Length of the pipe (in cm.)
(P)
262
(ii)
Fundamental frequency (in Hz)
(Q)
786
(iii)
th
(R)
32.4
st
(S)
1310
(iv)
Frequency of 5 harmonic (in Hz) Frequency of 1 overtone (in Hz)
60. are propagating along a taut string that coincides with the x-axis. The wave function for the the two waves are y1 = A cos [K (x −vt)] and y2 = A cos [K (x + vt) + δ) Column − I
Column − II
(i)
Beats
(P)
Redistribution of energy
(ii)
Standing waves
(Q)
Nearly equal frequencies
(iii)
Interference
(R)
Varying amplitude
(iv)
Echo
(S)
Reflection from a rigid support
(T)
Multiple Reflection
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