waves assignment

7/28/2019 Waves Assignment

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ASSIGNMENT ON WAVES

By nallamothu Page 1

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


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ASSIGNMENT ON WAVES


and t is in seconds. The length of the string is 1.5 m & its mass m = 3 10-2

kg. Find

the

(a)wavelength (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/m

2

at a distance of 5.0m 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 ofthe 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.


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ASSIGNMENT ON WAVES


6. Two rods AB and CD of equal cross- sectional area A and

Youngs 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

B

C

M

1

2

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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ASSIGNMENT ON WAVES


Assignment (Objective Problems)

LEVEL I

1. The amplitude of resulting wave dues to superposition of y1 = A sin ( t kx) &

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 (B) = 1

(C) 1 (D) data insufficient.

3. The equation of a wave pulse moving with a speed 1 m/sec at time t = 0 is given as y

= f(x) =2x1

1. Its equation at time t = 1 second can be given as

(A) y =2

)x1(1

1(B) y =

2)x1(1

1

(C) y =)1x(1

12

(D) y =

2x1

11

1

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 (D) none of these


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ASSIGNMENT ON WAVES


6. 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.

7. 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

(A) f (B) 2f

(C) 2 f2 (D)22

f

8. 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

9. 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

10. 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 ofdetector

(A) Apparent frequency received by detector = source frequency.

(B) Initially apparent frequency > source frequency and finally less

than source frequency.

(C) Apparent frequency depends only on the detector velocity.

5m

S

D


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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 thesex

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 (B) liquid only

(C) Gases 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

0C is approximately.

(A)M

R(B)

M

R3

(C)M

R382(D)

M

R273

17. The amplitude of a wave disturbance propagating in the positive x direction is given

by y =)x1(

12

at time t = 0 and by y =2)2x(1

1at time t = 2 seconds where x and


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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

(A) 0.5 m/s (B) 1 m/sec

(C) 2 m/s (D) 1.5 m/sec

18. 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. A transverse wave is given by A sin( t x) where and are constants. The ratio

of wave velocity to maximum particle velocity is

(A) A (B) 1/ A

(C) 1 (D) none of the above.

20. Two blocks, each of mass m, are connected by a mass

less thread Y and A represent Youngs modulus and cross

sectional area of wire respectively. The strain developed

in the thread is

(A)yA2

sin1mg(B)

yA

mg

(C)yA

sinmg(D)

yA

mg2

mm


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ASSIGNMENT ON WAVES


LEVEL II

1. A wave going in a solid

(A) Must be longitudinal (B) May be longitudinal(C) Must be transverse (D) May be transverse

2. A wave is represented by the equation

y = (0.001 mm) sin [(50 s1)t + (2.0 m

1)x]

(A) The wave velocity = 100 m/s (B) The wavelength = 2.0 m

(C) The frequency = 25/ Hz (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,

= 0.25 kg/m. The two snaps are taken at time t = 0 and at1

t24

s. Then

(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

6

)

(D) The maximum velocity of the particle m / s25


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ASSIGNMENT ON WAVES


10

5

5

10

x(m)

y(mm)

t=0

10

5

5

10

x(m)

y(mm)

t= s1

24

1

Figure I Figure II

7. 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 x2+y

2= R

2, with constant speed

330v6 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 RR,

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 is6 3

f6 3

(D) The minimum frequency recorded by the detector is6 3

f6 3

9. A wave is represented by the equation:

1 1y (1mm) sin 50 s t (2.0 m )x + 1 1(1mm)cos 50s t (2.0m )x


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ASSIGNMENT ON WAVES


(A) The wavevelocity is zero, since it is a standing wave.

(B) A node is formed at3

x m8

.

(C) The amplitude of the oscillation at the antinodes is 2 mm.

(D) Energy transfer occurs along the positive xaxis.

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., A

B

f

f=

A B

P

(A) 4 (B)1

2

(C) 16 (D) 2.

Comprehension

I. 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.

S

D

R

1. If a maxima is formed at a detector then, the magnitude of wavelength of the wave

produced is given by

(A) R (B)R

2

(C)R

4(D) all of these


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ASSIGNMENT ON WAVES


2. If a minima is formed at the detector then, the magnitude of wavelength of the wave

produced is given by

(A) 2 R (B)3

R2

(C)2

R5

(D) None of these

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 awave pulse is generated in string AB, which travels

towards boundary at B with speed v. Equation of

incident pulse is given as

i iy A sin( t kx)

Based on above information, answer the following

questions.

A

v

B C

x=0 x=L x=2

Tension=constant

1 2

4. Amplitude of wave reflected back after incident on boundary at point B

(A)iA

3 (B)i2A

3

(C) iA

3(D) i

2A

3.

5. Speed of transmitted wave on string BC is

(A) v (B)v

2

(C) 2v (D) None of these.

6. Equations of reflected and transmitted waves respectively are :

(A)

ir

it

Ay sin( t kx)

3

2Ay sin( t 2kx)

3

(B)

ir

it

Ay sin( t kx)

3

2Ay sin( t 2kx)

3

(C)

ir

it

Ay sin( t kx)

3

2Ay sin( t 2kx)

3

(D)

ir

it

Ay sin( t kx)

3

2A ky sin( t x)

3 2

.


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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.

AB

C

Column A 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 s1

. A

sound detector located for away is executing linear

SHM with amplitude 6 m on line BCD as shown. The

frequency of detector for oscillation is5

per

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 ms1

).

Match the column A with B.

A B CM

N

P

Column A Column B

(A) The frequency of sound recorded bydetector 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.


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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 IColumn II

(A)

A

L

Fixedend

Fixeend

(p) 2 229 c A

4 L

(B)

L

Fixedend

Freeend

(q) 2 22 29 c A sin t

4 L

(C)

L

Free

end

Free

end

(r) 2 22 29 c A sin t

16 L

(D)A

LTravellingwave

(s) 2 22 2c A sin t

4 L


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ASSIGNMENT ON WAVES


Hints (Subjective):

LEVEL- I

1. The amplitude of the resulting wave is given as

Ar =22

21 AA

2. Compare it with the standard wave equation

y = A sin ( t + kx)

3. f1 =T

, f2 =5.2T

8. f1 =m

T

L2

1

f2 = mTL2P ; P = number of loops.

LEVEL- II

1. y = 2a sin2

x cos t

compare it with the given equation.

2. f =4

v)1n2(n = 1, 2, 3 . . .. . for Ist, 2

nd, 3

rd. . .. . .overtone.


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ASSIGNMENT ON WAVES


3. The centripetal force

2T sin = (dm)R

v2

4. y = A cos (ax + bt) 5. n1 =4

v1 , n2 =4

v2

6. (a) L2 L1 =f2

v(b) v =

M

RT

8. P = I1A1 = I2 A2 = I1 (421r ) = I2 (4

22r )

Answers (Subjective Assignment):

LEVELI

1. 2221 AA sin { t kx + tan

-1(A2/A1)}. 2. (a) 15 Hz. (b)

10

1sec.

3. 2 kg wt. 4. 17.4 m/s

5. 75 6.23o L

F23

1

7. (a)1 m (b) 60 Hz (c) 0.03 m (d) 7.2 m/s (e) 4.2 cm.

8. 7

9. (a) =a

2(b) f = b/2 (b) b/a, Ab

10. 4 1010

W/m2.


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ASSIGNMENT ON WAVES


LEVELII

1. (a) 2f(L2

L1) (b)

RT

|)LLM(|4f 2122

2. f1

= f2

+ 5, f3

= f2 7

3.2

2

1

1

yyA

Mg 4. 327.68 m/s.

5. f1 = f2 + 5, f3 = f2 7 6.2

2

1

1

yyA

Mg

7. (a) 5/2 cm, 120 cm/s (b) 3 cm (c) Zero 8.21 TT

2

9. 1 = 0.993 m or 1.006 m

10. Range = 403 Hz to 484 Hz

Answers (Objective Assignment):

LEVELI

1. (D) 2. (A)

3. (B) 4. (D)

5. (B) 6. (B)

7. (D) 8. (B)

9. (A) 10. (B)


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ASSIGNMENT ON WAVES


11. (A) 12. (A)

13. (A) 14. (D)

15. (C) 16. (C)

17. (B) 18. (B)

19. (B) 20. (A)

LEVELII

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)

COMPREHENSION

1. (D) 2. (A)

3. (B) 4. (A)

5.(B) 6. (C)

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)

waves assignment

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