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ENTHUSIAST & LEADER COURSE
ENTHUSIAST & LEADER COURSE

PHYSICS GR # WAVE ON A STRING

SECTION-I
Single Correct Answer Type 9 Q. [3 M (–1)]
1. A traveling wave is of the form y (x,t) = A cos (kx – wt) + B sin (kx – wt), which can also be written as
y (x,t) = D sin (kx – wt – f) where
(A) D = A + B (B) D = |A + B| (C) D2 = A2 + B2 (D) D = A – B
2. A thin string with linear density µ is joined to a thick string with linear density 2µ. A incident pulse is
sent down the thin string toward the thick string and eventually creates reflected and transmitted pulses.
Which of the following is true ?
(A) The reflected and transmitted pulses are both inverted.

®
(B) Neither the reflected nor transmitted pulses are inverted.
(C) The reflected pulse is inverted, but the transmitted pulse is not inverted.
(D) The transmitted pulse is inverted, but the reflected pulse is not inverted.
3. Here given snap shot of a progressive wave at t = 0 with time period = T. Then the equation of the wave
if wave is going in +ve x-direction and if wave is going in –ve x-direction will be respectively.

æ 2p ö y
ç Here, T = ÷
è wø A
(A) y = A sin (kx + wt), y = A sin (kx – wt)
(B) y = A cos (kx + wt), y = A cos (kx – wt) 0 l/2 l x
(C) y = A sin (wt – kx), y = A sin (wt + kx)
(D) y = A sin (kx – wt), y = A sin (kx + wt)
4. A progressive wave is travelling in a string as shown. Then which of the following statement about KE
and potential energy of the elements A and B is true?
B

(A) For point A : kinetic energy is maximum and potential energy is min.
(B) For point B : kinetic energy is minimum and potential energy is min.
(C) For point A : kinetic energy is minimum and potential energy is max.
(D) For point B : kinetic energy is minimum and potential energy is max.
5. A string consists of two parts attached at x = 0. The right part of the string (x > 0) has mass mr per unit
length and the left part of the string (x < 0) has mass ml per unit length. The string tension is T. If a wave
of unit amplitude travels along the left part of the string, as shown in the figure, what is the amplitude of
the wave that is transmitted to the right part of the string ?

2 2 ml / mr ml / m r - 1
(A) 1 (B) 1 + m / m (C) 1 + m / m (D)
ml / mr + 1
l r l r

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6. Five waveforms moving with equal speeds on the x-axis

p 3p
y1 = 8 sin (wt + kx) ; y2 = 6 sin (wt + + kx) ; y3 = 4 sin (wt + p + kx) ; y4 = 2 sin (wt + + kx);
2 2
p
y5 = 4 2 sin (wt – kx + ) are superimposed on each other. The resulting wave is :
4
p p
(A) 8 2 cos kx sin (wt + ) (B) 8 2 sin (wt – kx + )
4 4
p
(C) 8 2 sin kx cos (wt + ) (D) 8 sin (wt + kx)
4
7. The wave function of a triangular wave pulse is defined by the relation below at time t = 0 sec .

®
ì a y
ïmx for 0 £ x £ Direction of pulse propagation
2
ï
í a
y = ï- m( x - a) for £ x £ a
2
ï a x
î0 every where else, where m<<1

The wave pulse is moving in the +X direction in a string having tension T and mass per unit length m.
The total energy present with the wave pulse is :-

m 2Ta mTa
(A) (B) m2Ta (C) mTa (D)
2 2

8. A plane progressive harmonic wave is given by the equation : j = jmsin(2t – 3x + 4y + p/3), where x
and y are in meters, and t is in seconds. Let n̂ is the unit vector in the direction of wave propagation, and
v is the speed of wave w.r.t. the wave medium, then :-

3 4 4ˆ 3ˆ 2
(A) n̂ = - ˆi + ˆj ; v = 0.4 m / s (B) n̂ = i - j;v = m /s
5 5 5 5 3

3 4 4 3
(C) n̂ = ˆi - ˆj ; v = 0.4 m / s (D) n̂ = - ˆi + ˆj ; v = 0.5 m / s
5 5 5 5
9. The vibrations of a string of length 600 cm fixed at both ends are represented by the equation

æpxö
y = 4 sin çè ÷ø cos (96 pt), where x and y are in cm and t in second. What is the maximum displacement
15

at a point at x=5 cm?

(A) 3 cm (B) 2 3 cm (C) 3 3 cm (D) 4 3 cm

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Multiple Correct Answer Type 9 Q. [4 M (–1)]

10. The figures represent two snaps of a travelling wave on a string of mass per unit length, µ = 0.25 kg/m.
1
The two snaps are taken at time t = 0 and at t = s . Then the possible solution for wave are :
24

y(mm) y(mm)

10 10

5 5

x(m) 1 x(m)
–5 –5
–10 t=0 –10 1
t= — s

®
24
Figure-1 Figure-2

(A) speed of wave is 4 m/s.

(B) the tension in the string is 4 N
æ pö
(C) the equation of the wave is y = 10 sin ç px - 4 pt + ÷
è 6 ø
p
(D) the maximum velocity of the particle = m/s
25
11. y (x, t) = 0.8/[(4x + 5t)2 + 5] represents a moving pulse, where x & y are in meter and t in second .
Then:
(A) pulse is moving in +x direction (B) in 2s it will travel a distance of 2.5 m
(C) its maximum displacement is 0.16 m (D) it is a symmetric pulse.
12. A string of mass 0.2 kg and length 2m is tied at two ends to fixed supports under a tension of 10 N. A point
P on the string is found to travel from one extreme to other in 0.1s. Taking one end as x = 0 and the other
end x = 2m and t = 0 as the time when P is at rest. (Position of P is x)
The CORRECT statements will be
(A) For 0 < t < 0.1 s, energy flows across P in positive x-direction for 0 < x < 1 m
(B) For 0 < t < 0.05 s, energy flows across P in negative x-drection for 0 < x < 1 m
(C) At t = 0.05 s, rate of energy flow through P is zero for x = 0.5 m
(D) At t = 0.1s, rate of energy flow through P for all values of x is zero
13. String of length L whose one end is at x = 0, vibrates according to the relations given by different
equations. Choose the CORRECT statement(s).
px
(A) y = A sin sin wt has 1 antinodes, 2 nodes
L
px
(B) y = A cos sin wt has 2 antinodes, 1 nodes
L
2 px
(C) y = A sin sin wt has 3 nodes, 2 antinodes
L
2 px
(D) y = A cos sin wt has 3 antinodes, 2 nodes
L

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14. A wave disturbance in a medium is described by

y (x, t) = 0.02 cos (50 pt + p/2) cos (10 px),
where x and y are in metres and t in seconds.
(A) A node occurs at x = 0.15 m (B) An antinode occurs at x = 0.3 m
(C) The speed of the wave is 5.0 m/s (D) The wavelength is 0.2 m
æ pö
15. A waveform : y1 = A sin ç 2x - 4t + ÷ is superposed with a second waveform, to produce a standing
è 3ø
wave with a node at x = 0. The equation of the second waveform can be :-
æ 5p ö æ pö
(A) y2 = A sin ç 2x + 4t + ÷ (B) y2 = A sin ç 2x - 4t + ÷
è 3 ø è 3ø

æ pö æ pö
(C) y2 = A sin ç 2x + 4t - ÷ (D) y2 = A sin ç 2x + 4t + ÷

®
è 3ø è 3ø
16. A string fixed at both ends and under tension T vibrates in its 1 overtone with an amplitude A at the
st

antinodes. The total energy of the string is E and the maximum possible speed of a particle of the string
is v. If the same string were to vibrate in its fundamental mode under a tension 4T and with an amplitude
A at the antinode then :-
(A) The total energy of the string will be E
(B) The total energy of the string will be 2E
(C) The maximum possible speed of a particle on the string is v
(D) The maximum possible speed of a particle on the string is 2v
17. A long wire ABC is made by joining two wires AB and BC of equal cross-sectional area. AB has
length 4.80 m and mass 0.12 kg. BC has length 2.56 m and mass 0.4 kg. The wire ABC is under a
tension of 160 N. A sinusoidal wave y = 5.6 (cm) sin (wt – kx) is sent along the wire ABC from the end
A. No power dissipates during the propagation of the wave.
(A) The amplitude of the reflected wave is 2.4 cm A B C
(B) The amplitude of transmitted wave is 3.2. cm
(C) The maximum displacement of the nodes of the stationary wave in the wire AB is 3.2 cm
(D) The fraction of power transmitted from the junction B is approximately 0.816
18. In a travelling one dimensional mechanical sinusoidal wave
(A) potential energy and kinetic energy of an element become maximum simultaneously.
(B) all particles oscillate with the same frequency and the same amplitude
(C) all particles may come to rest simultaneously
(D) we can find two particles, in a length equal to half of a wavelength, which have the same non zero
acceleration simultaneously.
Linked Comprehension Type (2 Para × 3Q.) (1 Para × 2Q.) [3 M (-1)]
(Single Correct Answer Type)
Paragraph for Question Nos. 19 to 21
w
A harmonic oscillator at x = 0, oscillates with a frequency and amplitude a. It is generating waves
2p
at end of a thin string in which velocity of wave is v1 and which is connected to another heavier string in
which velocity of wave is v2 as shown, length of first string is l.
y
v1 v2

l x
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19. If harmonic oscillator oscillates by an equation y = a sinwt. The equation of incident wave in first
string is

æ xö æ xö
(A) y = a sin w ç t - ÷ (B) y = a sin w ç t + v ÷
è v1 ø è 1ø

é æ xö ù é æ xö ù
(C) y = asin êw çè t - v ÷ø + p ú (D) y = asin êw çè t + v ÷ø + p ú
ë 1 û ë 1 û

20. Equation of transmitted wave in second string if its amplitude is at is

æ xö æ lö
(A) y = at sin w ç t - v ÷ (B) y = at sin w ç t - v ÷
è 2ø è 1ø

®
æ l x - lö æ xö
(C) y = at sin w ç t - v - v ÷ (D) y = at sin w ç t - v ÷
è 1 2 ø è 2ø

21. Equation of reflected wave, if it is reflecting at the joint and amplitude of reflected wave is aR

æ xö é æ l l - xö ù
(A) y = aR sin w ç t - ÷ (B) y = aR sin êwçè t - v - v ÷ø + p ú
è v2 ø ë 1 1 û

é æ xö ù é æ 2l + xö ù
(C) y = aR sin êwçèt + v ÷ø + pú (D) y = aR sinêwçèt + v ÷ø + pú
ë 1 û ë 1 û

Paragraph for Question Nos. 22 to 24

A 2m string has tension 1N is fixed at both end and its is vibrating in its third harmonic with antinode
amplitude 3 cm and frequency 100 Hz, then
22. Possible stationary wave equation for the vibration of the string will be (assume origin at left end of the
string and x is measured in meters towards right and t is measured in seconds)

æ 3p ö æ 3p ö
(A) y = (0.06 m) sin ç x ÷ cos (200 pt) (B) y = (0.03 m) sin ç x ÷ cos (200 pt)
è 2 ø è 4 ø

æ 3p ö æ 3p ö
(C) y = (0.06 m) sin ç x ÷ cos (200 pt) (D) y = (0.03 m) sin ç x ÷ cos (200 pt)
è 4 ø è 2 ø
23. Total wave energy on the string will be nearly equal to
(A) 40 mJ (B) 10 mJ (C) 30 mJ (D) 20 mJ
24. At what time from the start (by the answer of first question in this paragraph) string will have maximum
kinetic energy first time (second)

1 1 1 1
(A) (B) (C) (D)
200 100 400 800

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Paragraph for Question No. 25 and 26

A suspension bridge consists of a pair of cables hung between two towers with the roadway suspended
from these cables by means of closely spaced vertical wires of negligible mass as shown in figure. Each
flexible cable of length l (> L) and mass m hangs between two poles and mass of roadway section is M
distributed uniformly along length L. At the ends, the cable makes an angle of a with the walls. (Assume
each cable bears equal load and M>>m)

m
a

M
L

®
Hanging Bridge

25. The shape of any cable is best describe by equation assuming center of a cable is origin, vertical direction
is taken as positive Y-axis and horizontal right taken as positive X-axis?
cot a 2 tan a 2 tan a 2 cot a 2
(A) y = x (B) y = x (C) y = x (D) y = x
2L 2L L L
26. If a small amplitude transverse pulse is passed through the cable, what is the speed of transverse waves
at the middle ?
1 Mgl tan a Mgl tan a 1 Mgl cot a Mgl cot a
(A) (B) (C) (D)
2 m m 2 m m
SECTION-II
Numerical Answer Type Question 1Q.[3M (0)]
(upto second decimal place)
27. A string will break apart if it is placed under too much tensile stress. One type of steel has density.
rsteel = 104 kg/m3 and breaking stress s = 9 × 108 N/m2. We make a guitar string from (4p) gram of this
type of steel. It should be able to withstand (900 p)N without breaking. What is highest possible
fundamental frequency (in Hz) of standing waves on the string if the entire length of the string vibrates.
SECTION-III
Numerical Grid Type (Ranging from 0 to 9) 1 Q. [4 M (0)]
28. A string of length l is fixed at both ends. It is vibrating in its 3 overtone with maximum amplitude
rd

l
a = 2 3 mm. Find the square of amplitude (in mm2) at a distance 3 from one end.

Subjective Type 1 Q. [4 M (0)]

29. A steel of wire of length 25 cm is fixed at its ends to rigid walls. Young’s modulus of steel = 200 GPa,
coefficient of linear thermal expansion = 10–5 / °C. Initially, the wire is just taut at 20°C temperature.
The density of steel = 8.0 g/cc. A tuning fork of frequency 200 Hz is touched to the wire, to execute
oscillations. Simultaneously, the temperature is slowly lowered. At what temperature will resonance
occur corresponding to the third overtone?

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GR # WAVE ON A STRING
SECTION-I
® ANSWER KEY

Single Correct Answer Type 9 Q. [3 M (–1)]

1. Ans. (C) 2. Ans. (C) 3. Ans. (D) 4. Ans. (B) 5. Ans. (C)
6. Ans. (A) 7. Ans. (B) 8. Ans. (C) 9. Ans. (B)
Multiple Correct Answer Type 9 Q. [4 M (–1)]
10. Ans. (A,B,C,D) 11. Ans. (B, C, D) 12. Ans. (C,D) 13. Ans. (A,B,C,D) 14. Ans. (A, B, C, D)
15. Ans. (A, C) 16. Ans. (A, C) 17. Ans. (A,B,C,D) 18. Ans. (A,B,D)
Linked Comprehension Type (2 Para × 3Q.) (1 Para × 2Q.) [3 M (-1)]
(Single Correct Answer Type)
19. Ans. (A) 20. Ans. (C) 21. Ans. (B) 22. Ans. (D) 23. Ans. (B)
24. Ans. (C) 25. Ans. (D) 26. Ans. (A)
SECTION-II
Numerical Answer Type Question 1Q.[3M (0)]
(upto second decimal place)
27. Ans. 375
SECTION-III
Numerical Grid Type (Ranging from 0 to 9) 1 Q. [4 M (0)]
28. Ans. 9
Subjective Type 1 Q. [4 M (0)]
29. Ans. 17.5°

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