Q#1
A sine wave is traveling in a medium. The minimum distance between the two particles, always having the same speed, is(a) λ/4
(b) λ/3
(c) λ/2
(d) λ.
Answer: (c)
In a sine wave, the particles on the string vibrate in simple harmonic motion. The particles having equal displacements from the mean position will have equal speed. So at any instant, there may be up to four particles having the same displacement in a wavelength. But in the next instant, the displacements of the particles do not remain the same. Out of the four particles, a pair of particles will have some displacement while another pair will have some other displacement. And the distance between the particles of the pair will be λ/2. In fact, the particles on a string having separation λ/2 will have always the same displacements and the same speed.
In this diagram particles A, B and C, D have always the same displacements and the same speed.
Q#2
A sine wave is traveling in a medium. A particular particle has zero displacement at a certain instant. The particle closest to it having zero displacement is at a distance
(a) λ/4
(b) λ/3
(c) λ/2
(d) λ.
Answer: (c)
The particles at distance λ/2 have always the same displacement. So the closest zero displacement particle will be at λ/2 distance away.
Q#3
Which of the following equations represents a wave traveling along Y-axis?
(a) x = A sin(ky - ⍵t)
(b) y = A sin(kx - ⍵t)
(c) y = A sin ky cos ⍵t
(d) y = A cos ky sin ⍵t.
Answer:(a)
In a transverse wave, the displacement of a particle is perpendicular to the direction of the wave motion. So, the displacement of the particle in a wave traveling along Y-axis will be along X-axis.
Q#4
The equation y = A sin²(kx - ⍵t) represents a wave motion with
(a) amplitude A, frequency ⍵/2π
(b) amplitude A/2, frequency ⍵/π
(c) amplitude 2A, frequency ⍵/4π
(c) does not represent a wave motion.
Answer: (b)
y = A sin²(kx - ⍵t)
y = ½A{1 - cos(2kx - 2⍵t)}
y' = ½Acos(2kx - 2⍵t)
It represents a wave motion with amplitude A/2 and frequency 2⍵/2π i.e. ⍵/π
Q#5
Which of the following is a mechanical wave?
(a) Radio waves.
(b) X-rays.
(c) Light waves.
(d) Sound waves.
Answer: (d)
Mechanical waves require a medium. In these waves, only the sound waves require a medium.
Q#6
A cork floating in a calm pond executes simple harmonic motion of frequency ν when a wave generated by a boat passes by it. The frequency of the wave is
(a) ν
(b) ν/2
(c) 2ν
(d) √2ν
Answer: (a)
The time taken by the wave in moving one wavelength is exactly the same as in moving the cork one oscillation. Hence the frequency of the wave is equal to the frequency of the cork.
Q#7
Two strings A and B, made of same material, are stretched by same tension. The radius of string A is double of the radius of B. A transverse wave travels on A with speed vₐ and on B with speed vᵦ. The ratio of vₐ/vᵦ is
(a) 1/2
(b) 2
(c) 1/4
(d) 4
Answer:(a)
The velocity of the wave on a string is
v = √(F/µ), where µ = mass per unit length.
So the velocity is inversely proportional to the square root of the mass per unit length of the string.
Here the µₐ/µᵦ = 4πr²/πr² =4
Hence vₐ/vᵦ = √(µᵦ/µₐ) = √(1/4) = 1/2
Q#8
Both the strings, shown in figure (15-Q1), are made of same material and have same cross-section. The pulleys are light. The wave speed of a transverse wave in the string AB is v₁ and in CD v₂. Then v₁/v₂ is
(a) 1
(b) 2
(c) √2
(d) 1/√2
Answer: (d)
Assuming that the system is in equilibrium, the tension in the string over pulleys is the same in each part say = T.
Thus the tension in the string CD = 2T.
So v₁/v₂ = √(F₁µ₂/F₂µ₁),
But µ₁ = µ₂
= √(F₁/F₂) = √(T/2T) = 1/√2
Q#9
The velocity of sound in air is 332 m/s. Its velocity in the vacuum will be
(a) > 332 m/s
(b) = 332 m/s
(c) < 332 m/s
(d) meaningless.
Answer: (d)
Sound is a mechanical wave which requires a medium. Since there is no medium in the vacuum, there is no sound wave. Hence its velocity is meaningless.
Q#10
A wave pulse, traveling on a two-piece string, gets partially reflected and partially transmitted at the junction. The reflected wave is inverted in shape as compared to the incident one. If the incident wave has wavelength 𝜆 and the transmitted wave 𝜆',
(a) 𝜆' > 𝜆
(b) 𝜆' = 𝜆
(c) 𝜆' <𝜆
(d) nothing can be said about the relation of 𝜆 and 𝜆'.
Answer: (c)
Since the reflected wave is inverted, the pulse is traveling from the lighter string to heavier string. Hence the speed of the incident pulse is more than the speed of the transmitted pulse. Let the velocity of the incident pulse = v and the transmitted pulse = v'. The frequency of the pulses will be the same, hence v = νλ and v' = νλ'. Now v > v'
νλ > νλ'
λ > λ'
λ' < λ
Hence option (c).
Q#11
Two waves represented by y = a sin(⍵t - kx) and y = a cos(⍵t - kx) are superimposed. The resultant wave will have an amplitude
(a) a
(b) √2a
(c) 2a
(d) 0.
Answer: (b)
When two waves are superimposed, the displacements of the particle due to each wave is added. Hence the resultant wave is given as
y' = a sin(⍵t - kx) + a cos(⍵t - kx)
y' = a {sin(⍵t - kx) + sin(π/2+⍵t - kx)}
y'= a(2)sin{(⍵t - kx + π/2+⍵t - kx)/2}*cos{(⍵t - kx - π/2 - ⍵t + kx)/2}
y' = 2a sin(⍵t – kx+π/4)*cosπ/4
y' = (2a/√2) sin(⍵t – kx + π/4)
y' =√2a sin(⍵t - kx + π/4)
Clearly the amplitude of the resultant wave is √2a.
A sine wave is traveling in a medium. A particular particle has zero displacement at a certain instant. The particle closest to it having zero displacement is at a distance
(a) λ/4
(b) λ/3
(c) λ/2
(d) λ.
Answer: (c)
The particles at distance λ/2 have always the same displacement. So the closest zero displacement particle will be at λ/2 distance away.
Q#3
Which of the following equations represents a wave traveling along Y-axis?
(a) x = A sin(ky - ⍵t)
(b) y = A sin(kx - ⍵t)
(c) y = A sin ky cos ⍵t
(d) y = A cos ky sin ⍵t.
Answer:(a)
In a transverse wave, the displacement of a particle is perpendicular to the direction of the wave motion. So, the displacement of the particle in a wave traveling along Y-axis will be along X-axis.
Q#4
The equation y = A sin²(kx - ⍵t) represents a wave motion with
(a) amplitude A, frequency ⍵/2π
(b) amplitude A/2, frequency ⍵/π
(c) amplitude 2A, frequency ⍵/4π
(c) does not represent a wave motion.
Answer: (b)
y = A sin²(kx - ⍵t)
y = ½A{1 - cos(2kx - 2⍵t)}
y' = ½Acos(2kx - 2⍵t)
It represents a wave motion with amplitude A/2 and frequency 2⍵/2π i.e. ⍵/π
Q#5
Which of the following is a mechanical wave?
(a) Radio waves.
(b) X-rays.
(c) Light waves.
(d) Sound waves.
Answer: (d)
Mechanical waves require a medium. In these waves, only the sound waves require a medium.
Q#6
A cork floating in a calm pond executes simple harmonic motion of frequency ν when a wave generated by a boat passes by it. The frequency of the wave is
(a) ν
(b) ν/2
(c) 2ν
(d) √2ν
Answer: (a)
The time taken by the wave in moving one wavelength is exactly the same as in moving the cork one oscillation. Hence the frequency of the wave is equal to the frequency of the cork.
Q#7
Two strings A and B, made of same material, are stretched by same tension. The radius of string A is double of the radius of B. A transverse wave travels on A with speed vₐ and on B with speed vᵦ. The ratio of vₐ/vᵦ is
(a) 1/2
(b) 2
(c) 1/4
(d) 4
Answer:(a)
The velocity of the wave on a string is
v = √(F/µ), where µ = mass per unit length.
So the velocity is inversely proportional to the square root of the mass per unit length of the string.
Here the µₐ/µᵦ = 4πr²/πr² =4
Hence vₐ/vᵦ = √(µᵦ/µₐ) = √(1/4) = 1/2
Q#8
Both the strings, shown in figure (15-Q1), are made of same material and have same cross-section. The pulleys are light. The wave speed of a transverse wave in the string AB is v₁ and in CD v₂. Then v₁/v₂ is
(a) 1
(b) 2
(c) √2
(d) 1/√2
Answer: (d)
Assuming that the system is in equilibrium, the tension in the string over pulleys is the same in each part say = T.
Thus the tension in the string CD = 2T.
So v₁/v₂ = √(F₁µ₂/F₂µ₁),
But µ₁ = µ₂
= √(F₁/F₂) = √(T/2T) = 1/√2
Q#9
The velocity of sound in air is 332 m/s. Its velocity in the vacuum will be
(a) > 332 m/s
(b) = 332 m/s
(c) < 332 m/s
(d) meaningless.
Answer: (d)
Sound is a mechanical wave which requires a medium. Since there is no medium in the vacuum, there is no sound wave. Hence its velocity is meaningless.
Q#10
A wave pulse, traveling on a two-piece string, gets partially reflected and partially transmitted at the junction. The reflected wave is inverted in shape as compared to the incident one. If the incident wave has wavelength 𝜆 and the transmitted wave 𝜆',
(a) 𝜆' > 𝜆
(b) 𝜆' = 𝜆
(c) 𝜆' <𝜆
(d) nothing can be said about the relation of 𝜆 and 𝜆'.
Answer: (c)
Since the reflected wave is inverted, the pulse is traveling from the lighter string to heavier string. Hence the speed of the incident pulse is more than the speed of the transmitted pulse. Let the velocity of the incident pulse = v and the transmitted pulse = v'. The frequency of the pulses will be the same, hence v = νλ and v' = νλ'. Now v > v'
νλ > νλ'
λ > λ'
λ' < λ
Hence option (c).
Q#11
Two waves represented by y = a sin(⍵t - kx) and y = a cos(⍵t - kx) are superimposed. The resultant wave will have an amplitude
(a) a
(b) √2a
(c) 2a
(d) 0.
Answer: (b)
When two waves are superimposed, the displacements of the particle due to each wave is added. Hence the resultant wave is given as
y' = a sin(⍵t - kx) + a cos(⍵t - kx)
y' = a {sin(⍵t - kx) + sin(π/2+⍵t - kx)}
y'= a(2)sin{(⍵t - kx + π/2+⍵t - kx)/2}*cos{(⍵t - kx - π/2 - ⍵t + kx)/2}
y' = 2a sin(⍵t – kx+π/4)*cosπ/4
y' = (2a/√2) sin(⍵t – kx + π/4)
y' =√2a sin(⍵t - kx + π/4)
Clearly the amplitude of the resultant wave is √2a.
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