ASSIGNMENT ON WAVES Assignment (Subjective Problems) LEVEL – I
1.
If a wave form has the equation y1 = A1 sin ( t - kx) & y2 = A2 cos ( t - kx), find the equation of the resulting wave on superposition.
2.
A wave train has the equation y = 4 sin (30 t + 0.1x) where x is in cm and t in seconds. What is the frequency of the source? How much time does a wave pulse take to reach a point 30 cm from it?
3.
When the stretching force of a wire in increased by 25 kg-wt, the frequency of the note emitted is changed in the ratio 2/3. Calculate the original stretching force.
4.
A policeman on duty detects a drop of 10% in the pitch of the horn of a moving car as it crosses him. If the velocity of sound is 330 meters per second, calculate the speed of the car.
5.
A steel wire fixed at both ends has a fundamental frequency of 200 Hz. A person can hear sound of maximum frequency 15 KHz. What is the highest harmonic that can be played on this string which is audible to the person?
6.
A wire of length L is fixed at both ends such that F is tension in it. Its mass per unit length is given from one end to other end as = ox where o is constant. Find time taken by a transverse pulse to move from lighter end to its mid point.
7.
The transverse displacement of a string, fixed at both of its ends, is given as y(x, t) = 0.06 sin 2 x cos 120 t where x, y are in meters
By nallamothu
Page 1
ASSIGNMENT ON WAVES and t is in seconds. The length of the string is 1.5 m & its mass m = 3 the (a)wavelength
10-2 kg. Find
(b) frequency. (c) Amplitude of the component of waves.
(d) Maximum velocity of the particle. (e) Amplitude at a distance x = 0.375 m.
8.
Two closed organ pipes A and B have lengths 50 cm and 70 cm respectively. The 5th harmonic (=2nd overtone) of A resonates with nth harmonic of B. Find n.
9.
The displacement of the medium in a sound wave is given by the equation y = A cos(ax+bt) (SI system) Where A, and b are positive constant. (a) What is the wave length and frequency of the incident wave? (b) Find the wave speed and maximum particle speed.
10.
The intensity of sound from a point source is 1.0 x 10-8 W/m2 at a distance of 5.0 m from the source. What will be the intensity at a distance of 25 m from the source?
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ASSIGNMENT ON WAVES LEVEL – II
1.
A loop of rope is whirled at a high angular velocity , so that it becomes a taut circle of radius R. (i) Find the tension in the rope if the linear mass density of the rope is . (ii) A kink develops in the whirling rope. Under what condition does the kink remain stationary relative to an observer on the ground?
2.
AB is cylinder of length 1.0 m, fitted with a flexible diaphragm C at the middle and two thin flexible diaphragms A and B at the ends. The portions AC and BC contain hydrogen and oxygen respectively. The diaphragms A and B are set into vibration of the same frequency. What is the minimum frequency of these vibrations for which the diaphragm C is a node? Under the conditions of the experiment, the velocity of sound in hydrogen is 1100 m/s and in oxygen is 300 m/s.
3.
A long tube contains air at a pressure of P and temperature T. The tube is open at one end and closed at the other end by a movable piston. A tuning fork near the open end is vibrating with a frequency f. Resonance is produced when the piston is at distance L1 and L2 from the open end. Mean molecular mass of the air is M. (a) Find the speed of sound in air. (b) Find the adiabatic constant of the air.
4.
A piston is fitted in a cylindrical tube of small cross section with the other end of the tube open. The tube resonates with a tuning fork of frequency 512 Hz. The piston is gradually pulled out of the tube and it is found that a second resonance occurs when the piston is pulled out through a distance of 32.0 cm. Calculate the speed of sound in the air of the tube.
5.
Three tuning forks with unknown frequencies f1, f2 and f3 are vibrated. 5 beats per second are heard when f1 and f2 are vibrated, 6 beats per second for f1 and f3, while 7 beats per second for f2 and f3. If f2 is loaded with wax, number of beats for f2 and f3 decreases while for f1 and f2 increases. Find tuning forks having maximum frequency and minimum frequency in terms of f2. By nallamothu
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ASSIGNMENT ON WAVES
6.
Two rods AB and CD of equal cross- sectional area A and Young’s constant y1 and y2 are joined and suspended from a fixed support. A block of mass M is attached to the lowest point C. The density of the rods is negligibly small. Find displacement of the point C.
A 1 B 2 C M
7.
A string vibrates according to the equation y = 5 sin ( x/3) cos (40 t) where x and y are in cm and t is in second. (a) What are the amplitude and velocity of the component waves whose superposition can give rise to this vibration? (b) What is the distance between two successive nodes? (c) What is the velocity of a particle of the string at position x = 1.5 cm and t = 9/8 second?
8.
How long will it take sound waves to travel distance between the points A and B if the air temperature between them varies linearly from T1 to T2? The velocity of sound propagation in air is equal v =
T , where
is a constant.
9.
The first overtone of an open organ pipe beats with the first overtone of a closed organ pipe with a beat frequency of 2.2 Hz. The fundamental frequency of the closed organ pipe is 110 Hz. Find the lengths of the pipes.
10.
A whistle emitting a sound of frequency 440 Hz is tied to a string of 1.5 m length and rotated with an angular velocity of 20 rad/s in the horizontal plane. Calculate the range of frequencies heard by an observer stationed at a large distance from the whistle.
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Page 4
ASSIGNMENT ON WAVES Assignment (Objective Problems) LEVEL – I
1.
The amplitude of resulting wave dues to superposition of y 1 = A sin ( t y2 = A sin ( t kx + ) is (A) 2A cos (B) 2A tan ( /2) (C) A cos sin (D) none
2.
A sine wave has an amplitude A and wavelength . The ratio of particle velocity and the wave velocity is equal to (2 A = ) (A) 1 (C) 1
3.
kx) &
(B) = 1 (D) data insufficient.
The equation of a wave pulse moving with a speed 1 m/sec at time t = 0 is given as y = f(x) = (A) y =
(C) y =
1 1 x2 1
. Its equation at time t = 1 second can be given as
1 (1 x )
2
1 1 (x
2
1)
(B) y =
1 1 (1 x )2
1
(D) y = 1
1 1 x2
4.
The velocity of a transverse wave in a string does not depend on (A) tension (B) density of material of string (C) radius of string (D) length of string
5.
The frequency of a tuning fork with an amplitude A = 1 cm is 250 Hz. The maximum velocity of any particle in air is equal to (A) 2.5 m/s (B) 5 m/s (C) 3.30 / m/sec
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(D) none of these
Page 5
ASSIGNMENT ON WAVES 6.
7.
In a resonance column experiment, the first resonance is obtained when the level of the water in tube is 20 cm from the open end. Resonance will also be obtained when the water level is at a distance of (A) 40 cm from the open end.
(B) 60 cm from the open end.
(C) 80 cm from the open end.
(D) data insufficient.
A wire of length having tension T and radius r vibrates with natural frequency f. Another wire of same metal with length 2 having tension 2T and radius 2r will vibrate with natural frequency
8.
9.
10.
(A) f
(B) 2f
(C) 2 2f
(D)
f 2 2
Under the same conditions of pressure and temperature, the velocities of sound in oxygen and hydrogen gases are v0 and vH, then (A) vH= vo
(B) vH = 4vo
(C) vo = 4 VH
(D) vH = 16 vo
A tuning for of frequency 600 Hz produces a progressive travelling wave having wave velocity 300 m/s. Two particles of a medium, separated by 1.5 m, vibrate being affected by the wave (A) In phase
(B) In opposite phase.
(C) 45 out of phase.
(D) None of these
At t=0 source starts falling under gravity and a detector is projected upwards with a velocity 10 m/s. For the vertical upward motion of detector
S
5m
(A) Apparent frequency received by detector = source frequency. (B) Initially apparent frequency > source frequency and finally less than source frequency.
D
(C) Apparent frequency depends only on the detector velocity.
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ASSIGNMENT ON WAVES (D) Data insufficient.
11.
A string is clamped on both ends. Which of the following wave equations is valid for a stationary wave set up on this string? (Origin is at one end of string.) (A) Y = A sin kx. sin t (B) y = A cos kx sin t (C) Y = A cos kx. cos t (D) None of the above.
12.
A string is hanging from a rigid support. A transverse wave pulse is set up at the bottom. The velocity v of the pulse related to the distance covered by it is given as (A) v x (B) v x (C) v 1/x (D) none of these
x
13.
The third overtone of a closed organ pipe is equal to the second harmonic of an open organ pipe. Then the ratio of their lengths is equal to (A) 7/4 (B) 3/5 (C) 3/2 (D) none of these
14.
Standing waves can be produced in (A) solid only (C) Gases only
(B) liquid only (D) all of the above
15.
If the temperature of the medium drops by 1 %, the velocity of sound in that medium (A) Increases by 5 % (B) remains unchanged (C) decreases by 0.5 % (D) decreases by 2 %
16.
The velocity of sound through a diatomic gaseous medium of molecular weight M at 0°C is approximately.
17.
(A)
R M
(B)
3R M
(C)
382 R M
(D)
273 R M
The amplitude of a wave disturbance propagating in the positive x direction is given by y =
1 1 at time t = 0 and by y = at time t = 2 seconds where x and 2 (1 x ) 1 ( x 2)2
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ASSIGNMENT ON WAVES y are in meters. The shape of the wave disturbance does not change during the propagation. The velocity of the wave is
18.
(A) 0.5 m/s
(B) 1 m/sec
(C) 2 m/s
(D) 1.5 m/sec
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) A wave travelling in positive x-direction with a velocity 1.5 m/s. (B) A wave travelling in negative x-direction with a velocity 1.5 m/s. (C) A wave travelling in the negative x-direction having a wavelength 2 m. (D) A wave travelling in positive x-direction having a wavelength 2 m.
19.
20.
A transverse wave is given by A sin( t x) where of wave velocity to maximum particle velocity is
and
are constants. The ratio
(A) A
(B) 1/ A
(C) 1
(D) none of the above.
Two blocks, each of mass m, are connected by a mass less thread Y and A represent Young’s modulus and cross sectional area of wire respectively. The strain developed in the thread is (A)
mg 1 sin 2yA
(B)
mg yA
(C)
mg sin yA
(D)
2mg yA
By nallamothu
m
m
Page 8
ASSIGNMENT ON WAVES LEVEL – II
1.
2.
A wave going in a solid (A) Must be longitudinal (C) Must be transverse
(B) May be longitudinal (D) May be transverse
A wave is represented by the equation y = (0.001 mm) sin [(50 s 1)t + (2.0 m 1)x] (A) The wave velocity = 100 m/s (C) The frequency = 25/ Hz
(B) The wavelength = 2.0 m (D) The amplitude = 0.001 mm
3.
An electrically maintained tuning fork vibrates with constant frequency and constant amplitudes. If the temperature of the surrounding air increases but pressure remains constant, the sound produced will have (A) Larger wavelength (B) Larger frequency (C) Larger velocity (D) Larger time period
4.
The fundamental frequency of a vibrating organ pipe is 200 Hz. (A) The first overtone is 400 Hz (B) The first overtone may be 400 Hz (C) The first over tone may be 600 Hz (D) 600 Hz is an overtone
5.
A listener is at rest with respect to the source of sound. A wind starts blowing along the line joining the source and the observer. Which of the following quantities do not change? (A) Frequency (B) Velocity of sound (C) Wavelength (D) Time period
6.
The figures represent two snaps of a travelling wave on a string of mass per unit length, 1 = 0.25 kg/m. The two snaps are taken at time t = 0 and at t s. Then 24 (A) Speed of wave is 4 m/s (B) The tension in the string is 4 N (C) The equation of the wave is y = 10 sin ( x – 4 t (D) The maximum velocity of the particle
By nallamothu
25
6
)
m/s
Page 9
ASSIGNMENT ON WAVES y(mm)
y(mm)
10 5
10 5 x(m)
–5 –10
–5 –10
1
1 t=24 s
t=0
Figure I 7.
x(m)
Figure II
As a wave propagates, (A) The wave intensity remains constant for a plane wave (B) The wave intensity decrease as the inverse of the distance from the source for a spherical wave (C) The wave intensity decreases as the inverse square of the distance from the source for spherical wave (D) Total intensity of the spherical wave over the spherical surface remains constant at all times, while source is at the centre of spherical surface.
8.
A source is moving across a circle given by the equation x 2+y2 = R2, with constant speed v
330 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. If 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 3 R R, 2 2
(B)The co-ordinate of the source when the detector records minimum frequency is (0, R) (C) The maximum frequency recorded by the detector is (D) The minimum frequency recorded by the detector is
9.
6 3 6 3 6 3 6 3
f f
A wave is represented by the equation: y
(1mm)sin 50 s By nallamothu
1
t (2.0m 1)x + (1mm)cos 50 s
1
t (2.0m 1 )x Page 10
ASSIGNMENT ON WAVES (A) The wave–velocity is zero, since it is a standing wave. (B) A node is formed at x
3 m. 8
(C) The amplitude of the oscillation at the antinodes is 2 mm. (D) Energy transfer occurs along the positive x–axis.
10.
A very light rod AB is initially hung from a point P by means of two identical copper wires of the same length as the rod as shown in the figure. Particles of masses 1 kg and 4 kg are then attached to the ends A and B of the rod. The ratio of the fundamental frequencies of vibration of the wires AP and BP, i.e.,
P
A
B
fA = fB
1 2
(A) 4
(B)
(C) 16
(D) 2.
Comprehension
I.
1.
A narrow tube is bent in the form of a circle of radius R, as shown in the figure. Two small holes S and D are made in the tube at the positions right angle to each other. A source placed at S generates a wave of intensity I0 which is equally divided into two parts: one part travels along the longer path, while the other travels along the shorter path. Both the part waves meet at the point D where a detector is placed.
R S
If a maxima is formed at a detector then, the magnitude of wavelength produced is given by R (A) R (B) 2 R (C) (D) all of these 4 By nallamothu
D
of the wave
Page 11
ASSIGNMENT ON WAVES
2.
If a minima is formed at the detector then, the magnitude of wavelength produced is given by 3 (A) 2 R (B) R 2 2 (C) (D) None of these R 5
3.
The maximum intensity produced at D is given by (A) 4I0 (B) 2I0 (C) I0 (D) 3I0
II.
Speed of a transverse wave depends on mass and tension. Two strings of equal lengths are joined at B. Mass of string BC is four times mass of string AB. If a wave pulse is generated in string AB, which travels towards boundary at B with speed v. Equation of incident pulse is given as yi
of the wave
T en sio n=co n stan t v A x = 0
1
2
B x = L
C x = 2 L
Ai sin( t kx)
Based on above information, answer the following questions. 4.
Amplitude of wave reflected back after incident on boundary at point B Ai 2Ai (A) (B) 3 3 Ai 2A i (C) (D) . 3 3
5.
Speed of transmitted wave on string BC is v 2 (D) None of these.
(A) v
(B)
(C) 2v 6.
Equations of reflected and transmitted waves respectively are : yr
(A) yt yr
(C) yt
Ai sin( t kx) 3 2A i sin( t 2kx) 3 Ai sin( t kx) 3 2A i sin( t 2kx) 3
By nallamothu
yr
(B) yt yr
(D) yt
Ai sin( t kx) 3 2A i sin( t 2kx) 3 Ai sin( t kx) 3 . 2A i k sin( t x) 3 2
Page 12
ASSIGNMENT ON WAVES
Matching 1.
The figure shows a string at a certain moment as a transverse wave passes through it. Three particles A, B and C of the string are also shown. Match the physical quantities in the left column with the description in the column on the right. Column A
A
B C
Column B
(A)
Velocity of A
(p)
Downwards, if the wave is travelling towards right.
(B)
Acceleration of A
(q)
Downwards, if the wave is travelling towards left.
(C)
Velocity of B
(r)
Downwards, no matter which way the wave is travelling.
(D)
Velocity of C
(s)
Zero.
2.
A source of sound in moving along a circular orbit of radius 3 m with angular velocity of 10 rad s–1. A sound detector located for away is executing linear SHM with amplitude 6 m on line BCD as shown. The 5 frequency of detector for oscillation is per
N M
A
B C D
P
second. The source is at A when detector at B at t = 0. Source emits a continuous sound wave of frequency 340 Hz. (velocity of sound = 330 ms–1). Match the column A with B. Column A
Column B
(A)
The frequency of sound recorded by detector at t = 3T/4.
(p)
255 H.
(B)
The frequency of sound recorded by detector at t = T/4.
(q)
1 : 1.
(C)
The ratio of the time period of source and detector (circular motion and
(r)
442 Hz.
By nallamothu
Page 13
ASSIGNMENT ON WAVES SHM). (D)
Maximum velocity of detector/maximum velocity of source.
(s)
2 : 1.
(T is time period of oscillation).
3.
The diagrams in Column I show transverse sinusoidal standing/travelling waveforms on stretched strings. In each case, the string is oscillating in a particular mode, and, its shape and other characteristics are shown at time t = 0. The maximum amplitude (in all the cases) is A, the velocity of the waveform on the string is e, the mass per unit length of the string is and the frequency of vibration is f (angular frequency = ). The kinetic energy of the string (of length L) is represented by the functions in Column II. Match the correct entries in Column II. Column II Column I (A)
F ix e d e n d
F ix e d e n d
(p)
9
2
c2 2 A L
2
c2 2 2 A sin t L
4
A L
(B)
F ix e d e n d
F ree e n d A
(q)
9 4
L
(C)
F ree end
(r)
F ree end A A
9 2 c2 2 2 A sin t 16 L
L
(D)
A
(s)
2
c2 2 2 A sin t 4 L
L T ravellin gw ave
By nallamothu
Page 14
ASSIGNMENT ON WAVES Hints (Subjective):
LEVEL- I
1.
The amplitude of the resulting wave is given as Ar =
A12
A 22
2.
Compare it with the standard wave equation y = A sin ( t + kx)
3.
f1 =
8.
f1 =
1 2L
f2 =
P T ; P = number of loops. 2L m
T
T
, f2 =
2.5
T m
LEVEL- II
1.
y = 2a sin
2
x cos t
compare it with the given equation.
2.
f=
(2n 1)v 4
By nallamothu
n = 1, 2, 3 . . .. . for Ist, 2nd, 3rd . . .. . .overtone.
Page 15
ASSIGNMENT ON WAVES 3.
The centripetal force 2T sin = (dm)
v2 R
4.
y = A cos (ax + bt)
6.
(a) L2 – L1 =
8.
P = I1A1 = I2 A2 = I1 (4 r12 ) = I2 (4 r22 )
v 2f
5.
n1 =
v1 v , n2 = 2 4 4
RT M
(b) v =
Answers (Subjective Assignment): LEVEL –I
1.
A12
A 22 sin { t
kx + tan-1 (A2/A1)}.
2.
(a) 15 Hz.
3. 2 kg wt.
4.
17.4 m/s
5. 75
6.
7. (a)1 m (b) 60 Hz
(c) 0.03 m
1 3 2
o
F
(b)
L
1 10
sec.
32
(d) 7.2 m/s (e) 4.2 cm.
8. 7
9. (a) =
10.
2 a
4 10
(b) f = b/2
10
By nallamothu
(b) b/a, Ab
W/m2. Page 16
ASSIGNMENT ON WAVES LEVEL –II
1. (a) 2f(L2
3.
Mg 1 A y1
L1) (b)
4f 2M(| L 2 L1 |)2 RT
2 y2
5. f1 = f2 + 5, f3 = f2
7
7. (a) 5/2 cm, 120 cm/s (b) 3 cm (c) Zero
2.
f1 = f2 + 5, f3 = f2
4.
327.68 m/s.
6.
Mg 1 A y1
7
2 y2
2
8.
T1
T2
9. 1 = 0.993 m or 1.006 m
10. Range = 403 Hz to 484 Hz
Answers (Objective Assignment): LEVEL –I
1.
(D)
2.
(A)
3.
(B)
4.
(D)
5.
(B)
6.
(B)
7.
(D)
8.
(B)
9.
(A)
10.
(B)
By nallamothu
Page 17
ASSIGNMENT ON WAVES 11.
(A)
12.
(A)
13.
(A)
14.
(D)
15.
(C)
16.
(C)
17.
(B)
18.
(B)
19.
(B)
20.
(A)
LEVEL –II
1.
(B), (D)
2.
(C), (D)
3.
(A), (C)
4.
(B), (C), (D)
5.
(A), (D)
6.
(A), (B), (C), (D)
7.
(A), (C), (D)
8.
(A), (B), (C), (D)
9.
(B), (C)
10.
(D)
1. (D)
2.
(A)
3. (B)
4.
(A)
5.(B)
6.
(C)
COMPREHENSION
MATCH THE FOLLOWING
1.
(A) – (p); (B) – (r); (C) – (q); (D) – (s)
2.
(A) – (r); (B) – (p); (C) – (q); (D) – (s)
3.
(A) – (q); (B) – (r); (C) – (s); (D) – (p)
By nallamothu
Page 18
