DC Pandey Solutions: Superposition of Waves | Physics Class 11 - NEET PDF Download

 Page 1


Introductory Exercise 15.1
Q.1. A string vibrates according to the equation
where x and y are in centimetres and t is in seconds.
(a) What is the speed of the component wave?
(b) What is the distance between the adjacent nodes?
(c) What is the velocity of the particle of the string at the position x = 1.5
cm when t = 9/8 s?
Sol.
(b) Distance between adjacent nodes
Now substitute the given values of x and t.
Q.2. If two waves differ only in amplitude and are propagated in opposite
directions through a medium, will they produce standing waves? Is
energy transported?
Sol. In this case, node points will also oscillate, but standing waves are
Page 2


Introductory Exercise 15.1
Q.1. A string vibrates according to the equation
where x and y are in centimetres and t is in seconds.
(a) What is the speed of the component wave?
(b) What is the distance between the adjacent nodes?
(c) What is the velocity of the particle of the string at the position x = 1.5
cm when t = 9/8 s?
Sol.
(b) Distance between adjacent nodes
Now substitute the given values of x and t.
Q.2. If two waves differ only in amplitude and are propagated in opposite
directions through a medium, will they produce standing waves? Is
energy transported?
Sol. In this case, node points will also oscillate, but standing waves are
formed.
Ques 3: Figure shows different standing wave patterns on a string of
linear mass density 4.0 × 10-2 kg/m under a tension of 100 N. The
amplitude of antinodes is indicated in each figure. The length of the
string is 2.0 m.
(i) Obtain the frequencies of the modes shown in figures (a) and (b).
(ii) W rite down the transverse displacement y as a function of x and f for
each mode. (T ake the initial configuration of the wire in each mode to be
as shown by the dark lines in the figure).
Sol:
Page 3


Introductory Exercise 15.1
Q.1. A string vibrates according to the equation
where x and y are in centimetres and t is in seconds.
(a) What is the speed of the component wave?
(b) What is the distance between the adjacent nodes?
(c) What is the velocity of the particle of the string at the position x = 1.5
cm when t = 9/8 s?
Sol.
(b) Distance between adjacent nodes
Now substitute the given values of x and t.
Q.2. If two waves differ only in amplitude and are propagated in opposite
directions through a medium, will they produce standing waves? Is
energy transported?
Sol. In this case, node points will also oscillate, but standing waves are
formed.
Ques 3: Figure shows different standing wave patterns on a string of
linear mass density 4.0 × 10-2 kg/m under a tension of 100 N. The
amplitude of antinodes is indicated in each figure. The length of the
string is 2.0 m.
(i) Obtain the frequencies of the modes shown in figures (a) and (b).
(ii) W rite down the transverse displacement y as a function of x and f for
each mode. (T ake the initial configuration of the wire in each mode to be
as shown by the dark lines in the figure).
Sol:
Q.4. A 160 g rope 4 m long is fixed at one end and tied to a light string of
the same length at the other end. Its tension is 400 N.
(a) What are the wavelengths of the fundamental and the first two
overtones?
(b) What are the frequencies of these standing waves?
[Hint: In this case, the fixed end is a node and the end tied with the light
string is antinode.]
Sol.
Page 4


Introductory Exercise 15.1
Q.1. A string vibrates according to the equation
where x and y are in centimetres and t is in seconds.
(a) What is the speed of the component wave?
(b) What is the distance between the adjacent nodes?
(c) What is the velocity of the particle of the string at the position x = 1.5
cm when t = 9/8 s?
Sol.
(b) Distance between adjacent nodes
Now substitute the given values of x and t.
Q.2. If two waves differ only in amplitude and are propagated in opposite
directions through a medium, will they produce standing waves? Is
energy transported?
Sol. In this case, node points will also oscillate, but standing waves are
formed.
Ques 3: Figure shows different standing wave patterns on a string of
linear mass density 4.0 × 10-2 kg/m under a tension of 100 N. The
amplitude of antinodes is indicated in each figure. The length of the
string is 2.0 m.
(i) Obtain the frequencies of the modes shown in figures (a) and (b).
(ii) W rite down the transverse displacement y as a function of x and f for
each mode. (T ake the initial configuration of the wire in each mode to be
as shown by the dark lines in the figure).
Sol:
Q.4. A 160 g rope 4 m long is fixed at one end and tied to a light string of
the same length at the other end. Its tension is 400 N.
(a) What are the wavelengths of the fundamental and the first two
overtones?
(b) What are the frequencies of these standing waves?
[Hint: In this case, the fixed end is a node and the end tied with the light
string is antinode.]
Sol.
(a) 1 = ?1/4
Now
Similarly f2 and f3.
Q.5. A string fastened at both ends has successive resonances with
wavelengths of 0.54m for the nth harmonic and 0.48 m for the (n +1)th
harmonic.
(a) Which harmonics are these?
(b) What is the length of the string?
(c) What is the wavelength of the fundamental frequency?
Sol.
or
Page 5


Introductory Exercise 15.1
Q.1. A string vibrates according to the equation
where x and y are in centimetres and t is in seconds.
(a) What is the speed of the component wave?
(b) What is the distance between the adjacent nodes?
(c) What is the velocity of the particle of the string at the position x = 1.5
cm when t = 9/8 s?
Sol.
(b) Distance between adjacent nodes
Now substitute the given values of x and t.
Q.2. If two waves differ only in amplitude and are propagated in opposite
directions through a medium, will they produce standing waves? Is
energy transported?
Sol. In this case, node points will also oscillate, but standing waves are
formed.
Ques 3: Figure shows different standing wave patterns on a string of
linear mass density 4.0 × 10-2 kg/m under a tension of 100 N. The
amplitude of antinodes is indicated in each figure. The length of the
string is 2.0 m.
(i) Obtain the frequencies of the modes shown in figures (a) and (b).
(ii) W rite down the transverse displacement y as a function of x and f for
each mode. (T ake the initial configuration of the wire in each mode to be
as shown by the dark lines in the figure).
Sol:
Q.4. A 160 g rope 4 m long is fixed at one end and tied to a light string of
the same length at the other end. Its tension is 400 N.
(a) What are the wavelengths of the fundamental and the first two
overtones?
(b) What are the frequencies of these standing waves?
[Hint: In this case, the fixed end is a node and the end tied with the light
string is antinode.]
Sol.
(a) 1 = ?1/4
Now
Similarly f2 and f3.
Q.5. A string fastened at both ends has successive resonances with
wavelengths of 0.54m for the nth harmonic and 0.48 m for the (n +1)th
harmonic.
(a) Which harmonics are these?
(b) What is the length of the string?
(c) What is the wavelength of the fundamental frequency?
Sol.
or
...(i)
Similarly ,
or
... (ii)
From Eqs. (i) and (ii) we have,
(b) From Eq. (i)
l = (0.27)n = 0.27 × 8 = 2.16 m
Q.6. If the frequencies of the second and fifth harmonics of a string
differ by 54 Hz. What is the fundamental frequency of the string?
Sol.
5f - 2f=54
3f = 54 f = 18 Hz