JEE Physics Waves MCQs Set 15

Answer: (a) \( \frac67 \)

Explanation:
1. Travelling wave: \( \phi_2=k(x_2-x_1)=\frac{3\pi}{2}-\frac\pi3=\frac{7\pi}{6} \).
2. Standing wave: \( \sin kx_1=\sin\frac\pi3\gt 0 \) and \( \sin kx_2=\sin\frac{3\pi}2\lt 0 \). The two points lie in different loops (the node at \( kx=\pi \) is between them), so they are in opposite phase: \( \phi_1=\pi \).
3. \( \frac{\phi_1}{\phi_2}=\frac{\pi}{7\pi/6}=\frac67 \).

Teacher's Note:
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107. .

Answer: (b) increases with an increase in humidity

Explanation:
1. \( v=\sqrt{\frac{\gamma P}{\rho}} \). At constant temperature, a change in pressure changes the density in the same proportion, so v is independent of pressure.
2. Moist air is less dense than dry air (water vapour is lighter than nitrogen and oxygen), so the speed increases with humidity. It also increases with temperature.

Teacher's Note:
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Answer: (c) there is no transfer of heat

Explanation:
1. Compressions and rarefactions follow each other so quickly that heat has no time to flow between them: the process is adiabatic.
2. So \( PV^\gamma \) is constant, not PV (Boyle's law); the density changes between compressions and rarefactions; and the bulk modulus \( \gamma P \) stays constant.

Teacher's Note:
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109. .

Answer: (d) \( y=-0.4\sin2\pi\left(t+\frac x2\right) \)

Explanation:
1. The reflected amplitude is \( \frac23\times0.6=0.4 \).
2. The reflected wave travels in the opposite direction, so \( x\) has a plus sign. Reflection from a denser medium reverses the phase (a change of \( \pi \)), which gives a minus sign: \( y=-0.4\sin2\pi\left(t+\frac x2\right) \).

Teacher's Note:
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87. (
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95. .

Answer: (a) a constant higher value while the train approaches, followed by an abrupt fall to a constant lower value when it has passed

Explanation:
1. For an approaching source \( n'=\frac{nv}{v-V} \) (constant), and for a receding source \( n'=\frac{nv}{v+V} \) (constant).
2. The change occurs abruptly at the instant the train passes, so the graph is a step down.

Teacher's Note:
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157. .

Answer: (b) they vibrate with the same frequency, in the same phase, but with different amplitudes

Explanation:
1. In a standing wave every point between two nodes oscillates as \( \cos(120\pi t) \): the same frequency and phase.
2. The amplitude \( 0.06\left|\sin\frac{2\pi x}{3}\right| \) depends on x, so amplitudes (and energies) differ.

Teacher's Note:
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97. .

Answer: (c) I and II only

Explanation:
1. The wind carries the medium, so the speed of sound relative to the ground is \( 340+10=350 \) m/s.
2. The source and the observer are both at rest relative to the ground, so no Doppler shift occurs; the wavelength simply grows to \( \frac{350}{400} \) m, and the frequency heard remains 400 Hz.

Teacher's Note:
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165. .

Answer: (d) \( f_3=2n_2=3f_1=150\ \text{Hz} \)

Explanation:
1. \( v=\sqrt{\frac{100}{0.01}}=100 \) m/s. Both ends fixed: \( f_n=\frac{nv}{2L}=50n \), so \( f_1=50,\ f_2=100,\ f_3=150 \) Hz.
2. One end fixed: \( n_k=\frac{(2k-1)v}{4L}=25(2k-1) \), so \( n_1=25,\ n_2=75,\ n_3=125 \) Hz.
3. \( f_3=150=2n_2=3f_1 \). The other relations do not hold.

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Answer: (a) \( 1600\ \text{cm}^3 \)

Explanation:
1. Fundamental: \( L_1=\frac\lambda4=\frac{320}{4\times1000}=8 \) cm. The next resonating length is \( L_2=\frac{3\lambda}{4}=24 \) cm.
2. The air column must lengthen by 16 cm: volume drained \( =100\times16=1600 \) cm³.

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120. .

Answer: (b) 2.5 cm

Explanation:
1. Third harmonic of an open pipe: \( f=\frac{3v}{2L}=\frac{600}{7.8\pi} \) Hz, so \( \omega=2\pi f=\frac{1200}{7.8}=153.8 \) rad/s.
2. \( \Delta p_0=\rho v\omega s_0 \), so \( s_0=\frac{1000}{1.3\times200\times153.8}=0.025 \) m \( =2.5 \) cm.

Teacher's Note:
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119. .

Answer: (c) \( \frac L5 \)

Explanation:
1. Tensions: \( T_{AB}=mg\frac{L-x}{L} \) and \( T_{CD}=mg\frac xL \). Frequency \( \propto\sqrt T \).
2. First harmonic of AB = second harmonic of CD: \( \sqrt{T_{AB}}=2\sqrt{T_{CD}} \), so \( T_{AB}=4T_{CD} \).
3. \( L-x=4x \), so \( x=\frac L5 \).

Teacher's Note:
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107. .

Answer: (d) \( \frac{\sqrt{42}}{5} \)

Explanation:
1. \( v=\sqrt{\frac{\gamma RT}{M}} \), with \( M_{H_2}=2 \) and \( M_{He}=4 \).
2. \( \frac{v_H}{v_{He}}=\sqrt{\frac{7/5}{5/3}\cdot\frac{4}{2}}=\sqrt{\frac{21}{25}\cdot2}=\frac{\sqrt{42}}{5} \).

Teacher's Note:
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77. .

Answer: (a) 200 Hz

Explanation:
1. \( f=\frac1{2L}\sqrt{\frac T\mu} \), with \( \mu\propto r^2 \).
2. Length doubled: factor \( \frac12 \). Radius halved: \( \mu \) falls to \( \frac14 \), so \( \sqrt{1/\mu} \) is doubled. Tension \( \frac19 \): factor \( \frac13 \).
3. \( f=600\times\frac12\times2\times\frac13=200 \) Hz.

Teacher's Note:
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11.
12. c
13. h
14. a
15. n
16. g
17. e
18.
19. a
20. s
21.
22. a
23.
24. m
25. u
26. l
27. t
28. i
29. p
30. l
31. y
32. i
33. n
34. g
35.
36. f
37. a
38. c
39. t
40. o
41. r
42.
43. o
44. n
45.
46. t
47. h
48. e
49.
50. f
51. r
52. e
53. q
54. u
55. e
56. n
57. c
58. y
59. .

Answer: (b) 5 Hz

Explanation:
1. Five segments means five loops: \( L=5\frac\lambda2 \), so \( \lambda=4 \) m.
2. \( f=\frac v\lambda=\frac{20}{4}=5 \) Hz.

Teacher's Note:
1. E
2. a
3. c
4. h
5.
6. l
7. o
8. o
9. p
10.
11. o
12. f
13.
14. a
15.
16. s
17. t
18. a
19. n
20. d
21. i
22. n
23. g
24.
25. w
26. a
27. v
28. e
29.
30. i
31. s
32.
33. h
34. a
35. l
36. f
37.
38. a
39.
40. w
41. a
42. v
43. e
44. l
45. e
46. n
47. g
48. t
49. h
50.
51. l
52. o
53. n
54. g
55. .

Answer: (c) 1.5 cm

Explanation:
1. \( k=\frac{2\pi}{3} \) rad/cm gives \( \lambda=3 \) cm.
2. Adjacent nodes are half a wavelength apart: 1.5 cm.

Teacher's Note:
1. N
2. o
3. d
4. e
5. s
6.
7. o
8. f
9.
10. a
11.
12. s
13. t
14. a
15. n
16. d
17. i
18. n
19. g
20.
21. w
22. a
23. v
24. e
25.
26. a
27. r
28. e
29.
30. s
31. e
32. p
33. a
34. r
35. a
36. t
37. e
38. d
39.
40. b
41. y
42.
43. h
44. a
45. l
46. f
47.
48. a
49.
50. w
51. a
52. v
53. e
54. l
55. e
56. n
57. g
58. t
59. h
60. .

Answer: (d) 5

Explanation:
1. The waves have the same frequency and wave number, and a phase difference of \( \frac\pi2 \).
2. \( A=\sqrt{4^2+3^2+2(4)(3)\cos\frac\pi2}=5 \).

Teacher's Note:
1. W
2. i
3. t
4. h
5.
6. a
7.
8. q
9. u
10. a
11. r
12. t
13. e
14. r
15. -
16. c
17. y
18. c
19. l
20. e
21.
22. p
23. h
24. a
25. s
26. e
27.
28. d
29. i
30. f
31. f
32. e
33. r
34. e
35. n
36. c
37. e
38.
39. t
40. h
41. e
42.
43. a
44. m
45. p
46. l
47. i
48. t
49. u
50. d
51. e
52. s
53.
54. a
55. d
56. d
57.
58. l
59. i
60. k
61. e
62.
63. p
64. e
65. r
66. p
67. e
68. n
69. d
70. i
71. c
72. u
73. l
74. a
75. r
76.
77. v
78. e
79. c
80. t
81. o
82. r
83. s
84. .

Answer: (a) \( \lambda=\frac{\pi y_0}{2} \)

Explanation:
1. Maximum particle velocity \( =2\pi fy_0 \); wave velocity \( =f\lambda \).
2. \( 2\pi fy_0=4f\lambda \), so \( \lambda=\frac{\pi y_0}{2} \).

Teacher's Note:
1. T
2. h
3. e
4.
5. f
6. r
7. e
8. q
9. u
10. e
11. n
12. c
13. y
14.
15. c
16. a
17. n
18. c
19. e
20. l
21. s
22. ,
23.
24. s
25. o
26.
27. t
28. h
29. e
30.
31. c
32. o
33. n
34. d
35. i
36. t
37. i
38. o
39. n
40.
41. r
42. e
43. l
44. a
45. t
46. e
47. s
48.
49. o
50. n
51. l
52. y
53.
54. t
55. h
56. e
57.
58. w
59. a
60. v
61. e
62. l
63. e
64. n
65. g
66. t
67. h
68.
69. a
70. n
71. d
72.
73. t
74. h
75. e
76.
77. a
78. m
79. p
80. l
81. i
82. t
83. u
84. d
85. e
86. .

Answer: (b) 540 Hz

Explanation:
1. The wall receives \( f_1=480\times\frac{340}{340-20} \) Hz.
2. The wall re-emits \( f_1 \), and the approaching observer hears \( f_2=f_1\frac{340+20}{340}=480\times\frac{360}{320}=540 \) Hz.

Teacher's Note:
1. A
2.
3. m
4. o
5. v
6. i
7. n
8. g
9.
10. c
11. a
12. r
13.
14. r
15. e
16. f
17. l
18. e
19. c
20. t
21. i
22. n
23. g
24.
25. s
26. o
27. u
28. n
29. d
30.
31. o
32. f
33. f
34.
35. a
36.
37. w
38. a
39. l
40. l
41.
42. i
43. s
44.
45. a
46.
47. d
48. o
49. u
50. b
51. l
52. e
53.
54. D
55. o
56. p
57. p
58. l
59. e
60. r
61.
62. s
63. h
64. i
65. f
66. t
67. :
68.
69. f
70. i
71. r
72. s
73. t
74.
75. t
76. h
77. e
78.
79. w
80. a
81. l
82. l
83.
84. a
85. s
86.
87. o
88. b
89. s
90. e
91. r
92. v
93. e
94. r
95. ,
96.
97. t
98. h
99. e
100. n
101.
102. t
103. h
104. e
105.
106. c
107. a
108. r
109.
110. a
111. s
112.
113. o
114. b
115. s
116. e
117. r
118. v
119. e
120. r
121. .

Answer: (c) Both are true, but Statement II is not a correct explanation of Statement I

Explanation:
1. Intensity \( \propto\omega^2A^2 \). Wave 1: \( \omega^2(2a)^2=4a^2\omega^2 \). Wave 2: \( (2\omega)^2a^2=4a^2\omega^2 \). The intensities are equal: Statement I is true.
2. Statement II is true for waves of the same frequency, but here the frequencies differ, so it does not explain why the intensities are equal.

Teacher's Note:
1. I
2. n
3. t
4. e
5. n
6. s
7. i
8. t
9. y
10.
11. d
12. e
13. p
14. e
15. n
16. d
17. s
18.
19. o
20. n
21.
22. b
23. o
24. t
25. h
26.
27. t
28. h
29. e
30.
31. s
32. q
33. u
34. a
35. r
36. e
37.
38. o
39. f
40.
41. t
42. h
43. e
44.
45. a
46. m
47. p
48. l
49. i
50. t
51. u
52. d
53. e
54.
55. a
56. n
57. d
58.
59. t
60. h
61. e
62.
63. s
64. q
65. u
66. a
67. r
68. e
69.
70. o
71. f
72.
73. t
74. h
75. e
76.
77. f
78. r
79. e
80. q
81. u
82. e
83. n
84. c
85. y
86. .

Answer: (d) \( \text{wave moving in the } -x \text{ direction with speed } \sqrt{\frac ba} \)

Explanation:
1. \( ax^2+bt^2+2\sqrt{ab}\,xt=\left(\sqrt a\,x+\sqrt b\,t\right)^2 \), so \( y=f\left(x+\sqrt{\frac ba}\,t\right) \).
2. A function of \( x+vt \) is a wave moving in the \( -x \) direction with speed \( v=\sqrt{\frac ba} \).

Teacher's Note:
1. A
2. n
3. y
4.
5. f
6. u
7. n
8. c
9. t
10. i
11. o
12. n
13.
14. o
15. f
16.
17. \
18. (
19.
20. x
21. \
22. p
23. m
24.
25. v
26. t
27.
28. \
29. )
30.
31. i
32. s
33.
34. a
35.
36. t
37. r
38. a
39. v
40. e
41. l
42. l
43. i
44. n
45. g
46.
47. w
48. a
49. v
50. e
51. ,
52.
53. s
54. o
55.
56. l
57. o
58. o
59. k
60.
61. f
62. o
63. r
64.
65. a
66.
67. p
68. e
69. r
70. f
71. e
72. c
73. t
74.
75. s
76. q
77. u
78. a
79. r
80. e
81.
82. i
83. n
84.
85. t
86. h
87. e
88.
89. e
90. x
91. p
92. o
93. n
94. e
95. n
96. t
97. .

Answer: (a) 103 Hz

Explanation:
1. Only the component of the source velocity along the line to the observer matters: \( 19.4\cos60^\circ=9.7 \) m/s towards O.
2. \( f'=100\times\frac{330}{330-9.7}=103 \) Hz.

Teacher's Note:
1. U
2. s
3. e
4.
5. t
6. h
7. e
8.
9. c
10. o
11. m
12. p
13. o
14. n
15. e
16. n
17. t
18.
19. o
20. f
21.
22. t
23. h
24. e
25.
26. s
27. o
28. u
29. r
30. c
31. e
32.
33. v
34. e
35. l
36. o
37. c
38. i
39. t
40. y
41.
42. a
43. l
44. o
45. n
46. g
47.
48. t
49. h
50. e
51.
52. l
53. i
54. n
55. e
56.
57. j
58. o
59. i
60. n
61. i
62. n
63. g
64.
65. t
66. h
67. e
68.
69. s
70. o
71. u
72. r
73. c
74. e
75.
76. a
77. n
78. d
79.
80. o
81. b
82. s
83. e
84. r
85. v
86. e
87. r
88. .

Answer: (b) 120 cm

Explanation:
1. Closed pipe: \( f=\frac{v}{4(0.2)}=\frac{v}{0.8} \). The second overtone of an open pipe is its third harmonic: \( \frac{3v}{2L} \).
2. \( \frac{v}{0.8}=\frac{3v}{2L} \), so \( L=1.2 \) m \( =120 \) cm.

Teacher's Note:
1. T
2. h
3. e
4.
5. s
6. e
7. c
8. o
9. n
10. d
11.
12. o
13. v
14. e
15. r
16. t
17. o
18. n
19. e
20.
21. i
22. s
23.
24. t
25. h
26. e
27.
28. t
29. h
30. i
31. r
32. d
33.
34. h
35. a
36. r
37. m
38. o
39. n
40. i
41. c
42.
43. f
44. o
45. r
46.
47. a
48. n
49.
50. o
51. p
52. e
53. n
54.
55. p
56. i
57. p
58. e
59. ,
60.
61. b
62. u
63. t
64.
65. t
66. h
67. e
68.
69. f
70. i
71. f
72. t
73. h
74.
75. h
76. a
77. r
78. m
79. o
80. n
81. i
82. c
83.
84. f
85. o
86. r
87.
88. a
89.
90. c
91. l
92. o
93. s
94. e
95. d
96.
97. p
98. i
99. p
100. e
101. .

Answer: (c) \( \frac1n=\frac1{n_1}+\frac1{n_2}+\frac1{n_3} \)

Explanation:
1. For a string of fixed tension and mass per length, \( n\propto\frac1L \).
2. \( L=L_1+L_2+L_3 \) gives \( \frac1n=\frac1{n_1}+\frac1{n_2}+\frac1{n_3} \).

Teacher's Note:
1. F
2. r
3. e
4. q
5. u
6. e
7. n
8. c
9. y
10.
11. i
12. s
13.
14. i
15. n
16. v
17. e
18. r
19. s
20. e
21. l
22. y
23.
24. p
25. r
26. o
27. p
28. o
29. r
30. t
31. i
32. o
33. n
34. a
35. l
36.
37. t
38. o
39.
40. l
41. e
42. n
43. g
44. t
45. h
46. ,
47.
48. s
49. o
50.
51. l
52. e
53. n
54. g
55. t
56. h
57. s
58.
59. a
60. d
61. d
62.
63. a
64. s
65.
66. r
67. e
68. c
69. i
70. p
71. r
72. o
73. c
74. a
75. l
76. s
77.
78. o
79. f
80.
81. f
82. r
83. e
84. q
85. u
86. e
87. n
88. c
89. i
90. e
91. s
92. .

Answer: (d) 6

Explanation:
1. \( f_k=(2k-1)\frac{v}{4L}=(2k-1)\frac{340}{3.4}=100(2k-1) \) Hz.
2. The condition \( 100(2k-1)\lt 1250 \) gives \( 2k-1\le11 \): the frequencies are 100, 300, 500, 700, 900, 1100 Hz. There are 6.

Teacher's Note:
1. A
2.
3. c
4. l
5. o
6. s
7. e
8. d
9.
10. p
11. i
12. p
13. e
14.
15. s
16. u
17. p
18. p
19. o
20. r
21. t
22. s
23.
24. o
25. n
26. l
27. y
28.
29. o
30. d
31. d
32.
33. m
34. u
35. l
36. t
37. i
38. p
39. l
40. e
41. s
42.
43. o
44. f
45.
46. i
47. t
48. s
49.
50. f
51. u
52. n
53. d
54. a
55. m
56. e
57. n
58. t
59. a
60. l
61.
62. f
63. r
64. e
65. q
66. u
67. e
68. n
69. c
70. y
71. .

Answer: (a) 1412 Hz

Explanation:
1. The motorcyclist moves at 10 m/s towards the car, which moves in the same direction (away from him) at 5 m/s.
2. \( f'=1392\times\frac{343+10}{343+5}=1392\times\frac{353}{348}=1412 \) Hz.

Teacher's Note:
1. T
2. a
3. k
4. e
5.
6. t
7. h
8. e
9.
10. s
11. i
12. g
13. n
14.
15. o
16. f
17.
18. e
19. a
20. c
21. h
22.
23. v
24. e
25. l
26. o
27. c
28. i
29. t
30. y
31.
32. r
33. e
34. l
35. a
36. t
37. i
38. v
39. e
40.
41. t
42. o
43.
44. t
45. h
46. e
47.
48. d
49. i
50. r
51. e
52. c
53. t
54. i
55. o
56. n
57.
58. f
59. r
60. o
61. m
62.
63. t
64. h
65. e
66.
67. s
68. o
69. u
70. r
71. c
72. e
73.
74. t
75. o
76.
77. t
78. h
79. e
80.
81. o
82. b
83. s
84. e
85. r
86. v
87. e
88. r
89. .

Answer: (b) \( y=\sin(x-2t) \)

Explanation:
1. \( k=\frac{2\pi}{\lambda}=1 \) rad/m and \( \omega=2\pi f=2 \) rad/s.
2. A wave in the positive x-direction: \( y=\sin(kx-\omega t)=\sin(x-2t) \).

Teacher's Note:
1. C
2. o
3. m
4. p
5. u
6. t
7. e
8.
9. k
10.
11. a
12. n
13. d
14.
15. o
16. m
17. e
18. g
19. a
20. ,
21.
22. t
23. h
24. e
25. n
26.
27. u
28. s
29. e
30.
31. t
32. h
33. e
34.
35. s
36. i
37. g
38. n
39.
40. c
41. o
42. n
43. v
44. e
45. n
46. t
47. i
48. o
49. n
50.
51. f
52. o
53. r
54.
55. t
56. h
57. e
58.
59. d
60. i
61. r
62. e
63. c
64. t
65. i
66. o
67. n
68.
69. o
70. f
71.
72. t
73. r
74. a
75. v
76. e
77. l
78. .

Answer: (c) 254 Hz

Explanation:
1. The unknown frequency is 254 Hz or 246 Hz.
2. Second harmonic: 508 Hz or 492 Hz. Only 508 gives 5 beats with 513 Hz, so the frequency is 254 Hz.

Teacher's Note:
1. C
2. h
3. e
4. c
5. k
6.
7. e
8. a
9. c
10. h
11.
12. c
13. a
14. n
15. d
16. i
17. d
18. a
19. t
20. e
21.
22. a
23. g
24. a
25. i
26. n
27. s
28. t
29.
30. t
31. h
32. e
33.
34. s
35. e
36. c
37. o
38. n
39. d
40.
41. c
42. o
43. n
44. d
45. i
46. t
47. i
48. o
49. n
50.
51. t
52. o
53.
54. r
55. e
56. m
57. o
58. v
59. e
60.
61. t
62. h
63. e
64.
65. a
66. m
67. b
68. i
69. g
70. u
71. i
72. t
73. y
74.
75. i
76. n
77.
78. t
79. h
80. e
81.
82. b
83. e
84. a
85. t
86.
87. f
88. r
89. e
90. q
91. u
92. e
93. n
94. c
95. y
96. .

Answer: (d) \( \frac{1500}{23}\ \text{cm and}\ \frac{2000}{23}\ \text{cm} \)

Explanation:
1. Frequency is inversely proportional to length, so the lengths are in the ratio \( 1:\frac13:\frac15=15:5:3 \).
2. Total 23 parts: the lengths are \( \frac{1500}{23},\ \frac{500}{23},\ \frac{300}{23} \) cm.
3. The bridges are at \( \frac{1500}{23} \) cm and \( \frac{1500+500}{23}=\frac{2000}{23} \) cm from the end.

Teacher's Note:
1. R
2. e
3. m
4. e
5. m
6. b
7. e
8. r
9.
10. t
11. h
12. a
13. t
14.
15. b
16. r
17. i
18. d
19. g
20. e
21.
22. p
23. o
24. s
25. i
26. t
27. i
28. o
29. n
30. s
31.
32. a
33. r
34. e
35.
36. c
37. u
38. m
39. u
40. l
41. a
42. t
43. i
44. v
45. e
46.
47. d
48. i
49. s
50. t
51. a
52. n
53. c
54. e
55. s
56. ,
57.
58. w
59. h
60. i
61. l
62. e
63.
64. t
65. h
66. e
67.
68. q
69. u
70. e
71. s
72. t
73. i
74. o
75. n
76.
77. f
78. i
79. r
80. s
81. t
82.
83. g
84. i
85. v
86. e
87. s
88.
89. s
90. e
91. g
92. m
93. e
94. n
95. t
96.
97. l
98. e
99. n
100. g
101. t
102. h
103. s
104. .

Answer: (a) 4 beats per second, with intensity ratio 81 : 1 between waxing and waning

Explanation:
1. Frequencies: \( \frac{600\pi}{2\pi}=300 \) Hz and \( 304 \) Hz, so 4 beats per second.
2. Maximum and minimum amplitudes are \( 5+4=9 \) and \( 5-4=1 \), so the intensity ratio is \( 81:1 \).

Teacher's Note:
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