Wave Motion JEE Main - Physics, Solution by DC Pandey Notes | Study DC Pandey Solutions for JEE Physics - JEE

 Page 1


Exercises 
For JEE Main 
  Subjective Questions 
  Wave Equation and Plane Progressive Harmonic Wave 
Q 1.  A certain transverse wave is described by 
   y(x, t )= (6.50 mm) cos 2 ?
xt
28.0cm 0.0360s
??
?
??
??
 
Determine the wave's (a) amplitude, (b) wavelength, (c) frequency, (d) speed of propagation and 
(e) direction of propagation. 
Q 2.  For the wave y = 5 sin 30 ? [t - (x/240)], where x and y are in cm and t is in seconds, find the (a) 
displacement when t = 0 and x = 2 cm, (b) wavelength, (c) velocity of the wave and (d) frequency 
of the wave. 
Q 3.  The equation for a wave travelling in x-direction on a string is 
   y = (3.0 cm)sin[(3.14 cm
-1
)x - (314 s
-1
)t] 
  (a) Find the maximum velocity of a particle of the string. 
  (b) Find the acceleration of a particle at x = 6.0 cm at time t = 0.11s. 
Q 4.  A wave of frequency 500 Hz has a wave velocity of 350 m/s. 
(a) Find the distance between two points which are 60° out of phase. 
(b) Find the phase difference between two displacements at a certain point at times 10
-3
 s apart. 
Q 5.  At t = 0, a transverse wave pulse in wire is described by the function; 
   
2
6
y
x3
?
?
 
where x and y are in metres. Write the function y(x, t) that describes this wave if it is travelling in 
the positive x-direction with a speed of 4.50 m/s. 
Q 6.  A wave is described by the equation; 
   
xt
y (1.0mm)sin
2.0cm 0.01s
??
? ? ?
??
??
 
  (a) Find the time period and the wavelength. 
(b) Write the equation for the velocity of the particles. Find the speed of the particle at x= 1.0 cm 
at time t = 0.01s. 
  (c) What are the speeds of the particles at x = 3.0 cm, 5.0 cm and 7.0 cm at t = 0.01 s ? 
  (d) What are the speeds of the particles at x= 1.0cm at t = 0.011, 0.012 and 0.013 s ? 
Q 7.  A sinusoidal wave travelling in the positive x-direction has an amplitude of 15.0 cm, a wavelength 
of 40.0 cm and a frequency of 3.00 Hz. The vertical displacement of the medium at t = 0 and x = 0 
is also 15.0 cm, as shown in figure. 
  (a) Find the angular wave number k, period T, angular frequency ? and speed v of the wave. 
(b) Write a general expression for the wave function. 
Page 2


Exercises 
For JEE Main 
  Subjective Questions 
  Wave Equation and Plane Progressive Harmonic Wave 
Q 1.  A certain transverse wave is described by 
   y(x, t )= (6.50 mm) cos 2 ?
xt
28.0cm 0.0360s
??
?
??
??
 
Determine the wave's (a) amplitude, (b) wavelength, (c) frequency, (d) speed of propagation and 
(e) direction of propagation. 
Q 2.  For the wave y = 5 sin 30 ? [t - (x/240)], where x and y are in cm and t is in seconds, find the (a) 
displacement when t = 0 and x = 2 cm, (b) wavelength, (c) velocity of the wave and (d) frequency 
of the wave. 
Q 3.  The equation for a wave travelling in x-direction on a string is 
   y = (3.0 cm)sin[(3.14 cm
-1
)x - (314 s
-1
)t] 
  (a) Find the maximum velocity of a particle of the string. 
  (b) Find the acceleration of a particle at x = 6.0 cm at time t = 0.11s. 
Q 4.  A wave of frequency 500 Hz has a wave velocity of 350 m/s. 
(a) Find the distance between two points which are 60° out of phase. 
(b) Find the phase difference between two displacements at a certain point at times 10
-3
 s apart. 
Q 5.  At t = 0, a transverse wave pulse in wire is described by the function; 
   
2
6
y
x3
?
?
 
where x and y are in metres. Write the function y(x, t) that describes this wave if it is travelling in 
the positive x-direction with a speed of 4.50 m/s. 
Q 6.  A wave is described by the equation; 
   
xt
y (1.0mm)sin
2.0cm 0.01s
??
? ? ?
??
??
 
  (a) Find the time period and the wavelength. 
(b) Write the equation for the velocity of the particles. Find the speed of the particle at x= 1.0 cm 
at time t = 0.01s. 
  (c) What are the speeds of the particles at x = 3.0 cm, 5.0 cm and 7.0 cm at t = 0.01 s ? 
  (d) What are the speeds of the particles at x= 1.0cm at t = 0.011, 0.012 and 0.013 s ? 
Q 7.  A sinusoidal wave travelling in the positive x-direction has an amplitude of 15.0 cm, a wavelength 
of 40.0 cm and a frequency of 3.00 Hz. The vertical displacement of the medium at t = 0 and x = 0 
is also 15.0 cm, as shown in figure. 
  (a) Find the angular wave number k, period T, angular frequency ? and speed v of the wave. 
(b) Write a general expression for the wave function. 
 
Q 8.  For the wave shown in figure, find its amplitude, frequency and wavelength if its speed is 300 m/s. 
Write the equation for this wave as it travels out along the +x-axis if its position at t = 0 is as 
shown. 
 
Q 9.  Transverse waves on a string have wave speed 8.00 m/s, amplitude 0.0700 m and wavelength 0.32 
m. The waves travel in the negative x-direction and at t = 0 the x = 0 end of the string has its 
maximum upward displacement. 
  (a) Find the frequency, period and wave number of these waves. 
  (b) Write a wave function describing the wave. 
  (c) The transverse displacement of a particle at x = 0.360 mat time t = 0.150s. 
(d) How much time must elapse from the instant in part (c) until the particle at x = 0.360 m next 
has maximum upward displacement ? 
  Speed of a Transverse Wave on a String 
Q 10.  A copper wire 2.4 mm in diameter is 3 m long and is used to suspend a 2 kg mass from a beam. If 
a transverse disturbance is sent along the wire by striking it lightly with a pencil, how fast will the 
disturbance travel ? The density of copper is 8920 kg/m
3
. 
Q 11.  A uniform rope of length 12 m and mass 6 kg hangs vertically from a rigid support. A block of 
mass 2 kg is attached to the free end of the rope. A transverse pulse of wavelength 0.06 m is 
produced at the lower end of the rope. What is the wavelength of the pulse when it reaches the top 
of the rope? 
Q 12.  A flexible steel cable of total length L and mass per unit length ? hangs vertically from a support 
at one end.  
  (a) Show that the speed of a transverse wave down the cable is v g(L x) ?? , where x is 
measured from the support,  
  (b) How long will it take for a wave to travel down the cable? 
Q 13.  A loop of rope is whirled at a high angular velocity ?, so that it becomes a taut circle of radius R. 
A kink develops in the whirling rope,  
  (a) Show that the speed of the kink in the rope is v = ?R.  
  (b) Under what conditions does the kink remain stationary relative to an observer on the ground? 
Page 3


Exercises 
For JEE Main 
  Subjective Questions 
  Wave Equation and Plane Progressive Harmonic Wave 
Q 1.  A certain transverse wave is described by 
   y(x, t )= (6.50 mm) cos 2 ?
xt
28.0cm 0.0360s
??
?
??
??
 
Determine the wave's (a) amplitude, (b) wavelength, (c) frequency, (d) speed of propagation and 
(e) direction of propagation. 
Q 2.  For the wave y = 5 sin 30 ? [t - (x/240)], where x and y are in cm and t is in seconds, find the (a) 
displacement when t = 0 and x = 2 cm, (b) wavelength, (c) velocity of the wave and (d) frequency 
of the wave. 
Q 3.  The equation for a wave travelling in x-direction on a string is 
   y = (3.0 cm)sin[(3.14 cm
-1
)x - (314 s
-1
)t] 
  (a) Find the maximum velocity of a particle of the string. 
  (b) Find the acceleration of a particle at x = 6.0 cm at time t = 0.11s. 
Q 4.  A wave of frequency 500 Hz has a wave velocity of 350 m/s. 
(a) Find the distance between two points which are 60° out of phase. 
(b) Find the phase difference between two displacements at a certain point at times 10
-3
 s apart. 
Q 5.  At t = 0, a transverse wave pulse in wire is described by the function; 
   
2
6
y
x3
?
?
 
where x and y are in metres. Write the function y(x, t) that describes this wave if it is travelling in 
the positive x-direction with a speed of 4.50 m/s. 
Q 6.  A wave is described by the equation; 
   
xt
y (1.0mm)sin
2.0cm 0.01s
??
? ? ?
??
??
 
  (a) Find the time period and the wavelength. 
(b) Write the equation for the velocity of the particles. Find the speed of the particle at x= 1.0 cm 
at time t = 0.01s. 
  (c) What are the speeds of the particles at x = 3.0 cm, 5.0 cm and 7.0 cm at t = 0.01 s ? 
  (d) What are the speeds of the particles at x= 1.0cm at t = 0.011, 0.012 and 0.013 s ? 
Q 7.  A sinusoidal wave travelling in the positive x-direction has an amplitude of 15.0 cm, a wavelength 
of 40.0 cm and a frequency of 3.00 Hz. The vertical displacement of the medium at t = 0 and x = 0 
is also 15.0 cm, as shown in figure. 
  (a) Find the angular wave number k, period T, angular frequency ? and speed v of the wave. 
(b) Write a general expression for the wave function. 
 
Q 8.  For the wave shown in figure, find its amplitude, frequency and wavelength if its speed is 300 m/s. 
Write the equation for this wave as it travels out along the +x-axis if its position at t = 0 is as 
shown. 
 
Q 9.  Transverse waves on a string have wave speed 8.00 m/s, amplitude 0.0700 m and wavelength 0.32 
m. The waves travel in the negative x-direction and at t = 0 the x = 0 end of the string has its 
maximum upward displacement. 
  (a) Find the frequency, period and wave number of these waves. 
  (b) Write a wave function describing the wave. 
  (c) The transverse displacement of a particle at x = 0.360 mat time t = 0.150s. 
(d) How much time must elapse from the instant in part (c) until the particle at x = 0.360 m next 
has maximum upward displacement ? 
  Speed of a Transverse Wave on a String 
Q 10.  A copper wire 2.4 mm in diameter is 3 m long and is used to suspend a 2 kg mass from a beam. If 
a transverse disturbance is sent along the wire by striking it lightly with a pencil, how fast will the 
disturbance travel ? The density of copper is 8920 kg/m
3
. 
Q 11.  A uniform rope of length 12 m and mass 6 kg hangs vertically from a rigid support. A block of 
mass 2 kg is attached to the free end of the rope. A transverse pulse of wavelength 0.06 m is 
produced at the lower end of the rope. What is the wavelength of the pulse when it reaches the top 
of the rope? 
Q 12.  A flexible steel cable of total length L and mass per unit length ? hangs vertically from a support 
at one end.  
  (a) Show that the speed of a transverse wave down the cable is v g(L x) ?? , where x is 
measured from the support,  
  (b) How long will it take for a wave to travel down the cable? 
Q 13.  A loop of rope is whirled at a high angular velocity ?, so that it becomes a taut circle of radius R. 
A kink develops in the whirling rope,  
  (a) Show that the speed of the kink in the rope is v = ?R.  
  (b) Under what conditions does the kink remain stationary relative to an observer on the ground? 
 
Q 14.  A wire of variable mass per unit length ? = ?0x, is hanging from the ceiling as shown in figure. 
The length of wire is l0. A small transverse disturbance is produced at its lower end. Find the time 
after which the disturbance will reach to the other end. 
 
Q 15.  A non-uniform wire of length L and mass M has a variable linear mass density given by ? = kx, 
where x is distance from one end of wire and k is a constant. Find the time taken by a pulse 
starting at one end to reach the other end when the tension in wire is T. 
Q 16.  One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that 
vibrates at 120 Hz. The other end passes over a pulley and supports a 1.50 kg mass. The linear 
mass density of the rope is 0.0550 kg/m. 
  (a) What is the speed of a transverse wave on the rope ? 
  (b) What is the wavelength ? 
  (c) How would your answers to parts (a) and (b) change if the mass were increased to 3.00 kg? 
  Energy in Wave Motion 
Q 17. A certain 120 Hz wave on a string has an amplitude of 0.160 mm. How much energy exists in an 
80 g length of the string? 
Q 18.  A taut string for which ? = 5.00 × 10
-2
 kg/m is under a tension of 80.0 N. How much power must 
be supplied to the string to generate sinusoidal waves at a frequency of 60.0 Hz and an amplitude 
of 6.00 cm? 
Q 19.  A 200 Hz wave with amplitude 1 mm travels on a long string of linear mass density 6 g/m kept 
under a tension of 60 N. 
  (a) Find the average power transmitted across a given point on the string. 
  (b) Find the total energy associated with the wave in a 2.0 m long portion of the string. 
Q 20.  A transverse wave of amplitude 0.50 mm and frequency 100 Hz is produced on a wire stretched to 
a tension of 100 N. If the wave speed is 100 m/s. What average power is the source transmitting to 
the wire? 
Solutions 
Page 4


Exercises 
For JEE Main 
  Subjective Questions 
  Wave Equation and Plane Progressive Harmonic Wave 
Q 1.  A certain transverse wave is described by 
   y(x, t )= (6.50 mm) cos 2 ?
xt
28.0cm 0.0360s
??
?
??
??
 
Determine the wave's (a) amplitude, (b) wavelength, (c) frequency, (d) speed of propagation and 
(e) direction of propagation. 
Q 2.  For the wave y = 5 sin 30 ? [t - (x/240)], where x and y are in cm and t is in seconds, find the (a) 
displacement when t = 0 and x = 2 cm, (b) wavelength, (c) velocity of the wave and (d) frequency 
of the wave. 
Q 3.  The equation for a wave travelling in x-direction on a string is 
   y = (3.0 cm)sin[(3.14 cm
-1
)x - (314 s
-1
)t] 
  (a) Find the maximum velocity of a particle of the string. 
  (b) Find the acceleration of a particle at x = 6.0 cm at time t = 0.11s. 
Q 4.  A wave of frequency 500 Hz has a wave velocity of 350 m/s. 
(a) Find the distance between two points which are 60° out of phase. 
(b) Find the phase difference between two displacements at a certain point at times 10
-3
 s apart. 
Q 5.  At t = 0, a transverse wave pulse in wire is described by the function; 
   
2
6
y
x3
?
?
 
where x and y are in metres. Write the function y(x, t) that describes this wave if it is travelling in 
the positive x-direction with a speed of 4.50 m/s. 
Q 6.  A wave is described by the equation; 
   
xt
y (1.0mm)sin
2.0cm 0.01s
??
? ? ?
??
??
 
  (a) Find the time period and the wavelength. 
(b) Write the equation for the velocity of the particles. Find the speed of the particle at x= 1.0 cm 
at time t = 0.01s. 
  (c) What are the speeds of the particles at x = 3.0 cm, 5.0 cm and 7.0 cm at t = 0.01 s ? 
  (d) What are the speeds of the particles at x= 1.0cm at t = 0.011, 0.012 and 0.013 s ? 
Q 7.  A sinusoidal wave travelling in the positive x-direction has an amplitude of 15.0 cm, a wavelength 
of 40.0 cm and a frequency of 3.00 Hz. The vertical displacement of the medium at t = 0 and x = 0 
is also 15.0 cm, as shown in figure. 
  (a) Find the angular wave number k, period T, angular frequency ? and speed v of the wave. 
(b) Write a general expression for the wave function. 
 
Q 8.  For the wave shown in figure, find its amplitude, frequency and wavelength if its speed is 300 m/s. 
Write the equation for this wave as it travels out along the +x-axis if its position at t = 0 is as 
shown. 
 
Q 9.  Transverse waves on a string have wave speed 8.00 m/s, amplitude 0.0700 m and wavelength 0.32 
m. The waves travel in the negative x-direction and at t = 0 the x = 0 end of the string has its 
maximum upward displacement. 
  (a) Find the frequency, period and wave number of these waves. 
  (b) Write a wave function describing the wave. 
  (c) The transverse displacement of a particle at x = 0.360 mat time t = 0.150s. 
(d) How much time must elapse from the instant in part (c) until the particle at x = 0.360 m next 
has maximum upward displacement ? 
  Speed of a Transverse Wave on a String 
Q 10.  A copper wire 2.4 mm in diameter is 3 m long and is used to suspend a 2 kg mass from a beam. If 
a transverse disturbance is sent along the wire by striking it lightly with a pencil, how fast will the 
disturbance travel ? The density of copper is 8920 kg/m
3
. 
Q 11.  A uniform rope of length 12 m and mass 6 kg hangs vertically from a rigid support. A block of 
mass 2 kg is attached to the free end of the rope. A transverse pulse of wavelength 0.06 m is 
produced at the lower end of the rope. What is the wavelength of the pulse when it reaches the top 
of the rope? 
Q 12.  A flexible steel cable of total length L and mass per unit length ? hangs vertically from a support 
at one end.  
  (a) Show that the speed of a transverse wave down the cable is v g(L x) ?? , where x is 
measured from the support,  
  (b) How long will it take for a wave to travel down the cable? 
Q 13.  A loop of rope is whirled at a high angular velocity ?, so that it becomes a taut circle of radius R. 
A kink develops in the whirling rope,  
  (a) Show that the speed of the kink in the rope is v = ?R.  
  (b) Under what conditions does the kink remain stationary relative to an observer on the ground? 
 
Q 14.  A wire of variable mass per unit length ? = ?0x, is hanging from the ceiling as shown in figure. 
The length of wire is l0. A small transverse disturbance is produced at its lower end. Find the time 
after which the disturbance will reach to the other end. 
 
Q 15.  A non-uniform wire of length L and mass M has a variable linear mass density given by ? = kx, 
where x is distance from one end of wire and k is a constant. Find the time taken by a pulse 
starting at one end to reach the other end when the tension in wire is T. 
Q 16.  One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that 
vibrates at 120 Hz. The other end passes over a pulley and supports a 1.50 kg mass. The linear 
mass density of the rope is 0.0550 kg/m. 
  (a) What is the speed of a transverse wave on the rope ? 
  (b) What is the wavelength ? 
  (c) How would your answers to parts (a) and (b) change if the mass were increased to 3.00 kg? 
  Energy in Wave Motion 
Q 17. A certain 120 Hz wave on a string has an amplitude of 0.160 mm. How much energy exists in an 
80 g length of the string? 
Q 18.  A taut string for which ? = 5.00 × 10
-2
 kg/m is under a tension of 80.0 N. How much power must 
be supplied to the string to generate sinusoidal waves at a frequency of 60.0 Hz and an amplitude 
of 6.00 cm? 
Q 19.  A 200 Hz wave with amplitude 1 mm travels on a long string of linear mass density 6 g/m kept 
under a tension of 60 N. 
  (a) Find the average power transmitted across a given point on the string. 
  (b) Find the total energy associated with the wave in a 2.0 m long portion of the string. 
Q 20.  A transverse wave of amplitude 0.50 mm and frequency 100 Hz is produced on a wire stretched to 
a tension of 100 N. If the wave speed is 100 m/s. What average power is the source transmitting to 
the wire? 
Solutions 
1.  (b) 
  (c) 
  (d) 
  Since at and kx have opposite signs.  
  Hence, wave is travelling in positive direction. 
2.   (a) Put t = 0, x = 2cm  
  (b) 
  (c) Wave velocity = 
   = 240 cm/s    
  (d) 
3.    (a) v max = ?A          
  = (314) (3.0) 
   = 942 cm/s = 9.42 m/s 
  (b) At given t and x 
   y = (3.0) sin [(3.14 × 6) - (314 × 0.11)]. 
   = 0                                                                  
  Now            a = - ?
2
y = 0                     
4.    (a) 
   
   = 0.116 m  
  (b) ? ? = ( ?) ?t 
    
5.    ?t and kx should have opposite signs. 
    
  ?    Coefficient of t = v                  (coefficient of x) 
   = (4.5) (1) = 4.5 
    
6.    (a)  
   = 0.02 s = 20 ms 
    
Page 5


Exercises 
For JEE Main 
  Subjective Questions 
  Wave Equation and Plane Progressive Harmonic Wave 
Q 1.  A certain transverse wave is described by 
   y(x, t )= (6.50 mm) cos 2 ?
xt
28.0cm 0.0360s
??
?
??
??
 
Determine the wave's (a) amplitude, (b) wavelength, (c) frequency, (d) speed of propagation and 
(e) direction of propagation. 
Q 2.  For the wave y = 5 sin 30 ? [t - (x/240)], where x and y are in cm and t is in seconds, find the (a) 
displacement when t = 0 and x = 2 cm, (b) wavelength, (c) velocity of the wave and (d) frequency 
of the wave. 
Q 3.  The equation for a wave travelling in x-direction on a string is 
   y = (3.0 cm)sin[(3.14 cm
-1
)x - (314 s
-1
)t] 
  (a) Find the maximum velocity of a particle of the string. 
  (b) Find the acceleration of a particle at x = 6.0 cm at time t = 0.11s. 
Q 4.  A wave of frequency 500 Hz has a wave velocity of 350 m/s. 
(a) Find the distance between two points which are 60° out of phase. 
(b) Find the phase difference between two displacements at a certain point at times 10
-3
 s apart. 
Q 5.  At t = 0, a transverse wave pulse in wire is described by the function; 
   
2
6
y
x3
?
?
 
where x and y are in metres. Write the function y(x, t) that describes this wave if it is travelling in 
the positive x-direction with a speed of 4.50 m/s. 
Q 6.  A wave is described by the equation; 
   
xt
y (1.0mm)sin
2.0cm 0.01s
??
? ? ?
??
??
 
  (a) Find the time period and the wavelength. 
(b) Write the equation for the velocity of the particles. Find the speed of the particle at x= 1.0 cm 
at time t = 0.01s. 
  (c) What are the speeds of the particles at x = 3.0 cm, 5.0 cm and 7.0 cm at t = 0.01 s ? 
  (d) What are the speeds of the particles at x= 1.0cm at t = 0.011, 0.012 and 0.013 s ? 
Q 7.  A sinusoidal wave travelling in the positive x-direction has an amplitude of 15.0 cm, a wavelength 
of 40.0 cm and a frequency of 3.00 Hz. The vertical displacement of the medium at t = 0 and x = 0 
is also 15.0 cm, as shown in figure. 
  (a) Find the angular wave number k, period T, angular frequency ? and speed v of the wave. 
(b) Write a general expression for the wave function. 
 
Q 8.  For the wave shown in figure, find its amplitude, frequency and wavelength if its speed is 300 m/s. 
Write the equation for this wave as it travels out along the +x-axis if its position at t = 0 is as 
shown. 
 
Q 9.  Transverse waves on a string have wave speed 8.00 m/s, amplitude 0.0700 m and wavelength 0.32 
m. The waves travel in the negative x-direction and at t = 0 the x = 0 end of the string has its 
maximum upward displacement. 
  (a) Find the frequency, period and wave number of these waves. 
  (b) Write a wave function describing the wave. 
  (c) The transverse displacement of a particle at x = 0.360 mat time t = 0.150s. 
(d) How much time must elapse from the instant in part (c) until the particle at x = 0.360 m next 
has maximum upward displacement ? 
  Speed of a Transverse Wave on a String 
Q 10.  A copper wire 2.4 mm in diameter is 3 m long and is used to suspend a 2 kg mass from a beam. If 
a transverse disturbance is sent along the wire by striking it lightly with a pencil, how fast will the 
disturbance travel ? The density of copper is 8920 kg/m
3
. 
Q 11.  A uniform rope of length 12 m and mass 6 kg hangs vertically from a rigid support. A block of 
mass 2 kg is attached to the free end of the rope. A transverse pulse of wavelength 0.06 m is 
produced at the lower end of the rope. What is the wavelength of the pulse when it reaches the top 
of the rope? 
Q 12.  A flexible steel cable of total length L and mass per unit length ? hangs vertically from a support 
at one end.  
  (a) Show that the speed of a transverse wave down the cable is v g(L x) ?? , where x is 
measured from the support,  
  (b) How long will it take for a wave to travel down the cable? 
Q 13.  A loop of rope is whirled at a high angular velocity ?, so that it becomes a taut circle of radius R. 
A kink develops in the whirling rope,  
  (a) Show that the speed of the kink in the rope is v = ?R.  
  (b) Under what conditions does the kink remain stationary relative to an observer on the ground? 
 
Q 14.  A wire of variable mass per unit length ? = ?0x, is hanging from the ceiling as shown in figure. 
The length of wire is l0. A small transverse disturbance is produced at its lower end. Find the time 
after which the disturbance will reach to the other end. 
 
Q 15.  A non-uniform wire of length L and mass M has a variable linear mass density given by ? = kx, 
where x is distance from one end of wire and k is a constant. Find the time taken by a pulse 
starting at one end to reach the other end when the tension in wire is T. 
Q 16.  One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that 
vibrates at 120 Hz. The other end passes over a pulley and supports a 1.50 kg mass. The linear 
mass density of the rope is 0.0550 kg/m. 
  (a) What is the speed of a transverse wave on the rope ? 
  (b) What is the wavelength ? 
  (c) How would your answers to parts (a) and (b) change if the mass were increased to 3.00 kg? 
  Energy in Wave Motion 
Q 17. A certain 120 Hz wave on a string has an amplitude of 0.160 mm. How much energy exists in an 
80 g length of the string? 
Q 18.  A taut string for which ? = 5.00 × 10
-2
 kg/m is under a tension of 80.0 N. How much power must 
be supplied to the string to generate sinusoidal waves at a frequency of 60.0 Hz and an amplitude 
of 6.00 cm? 
Q 19.  A 200 Hz wave with amplitude 1 mm travels on a long string of linear mass density 6 g/m kept 
under a tension of 60 N. 
  (a) Find the average power transmitted across a given point on the string. 
  (b) Find the total energy associated with the wave in a 2.0 m long portion of the string. 
Q 20.  A transverse wave of amplitude 0.50 mm and frequency 100 Hz is produced on a wire stretched to 
a tension of 100 N. If the wave speed is 100 m/s. What average power is the source transmitting to 
the wire? 
Solutions 
1.  (b) 
  (c) 
  (d) 
  Since at and kx have opposite signs.  
  Hence, wave is travelling in positive direction. 
2.   (a) Put t = 0, x = 2cm  
  (b) 
  (c) Wave velocity = 
   = 240 cm/s    
  (d) 
3.    (a) v max = ?A          
  = (314) (3.0) 
   = 942 cm/s = 9.42 m/s 
  (b) At given t and x 
   y = (3.0) sin [(3.14 × 6) - (314 × 0.11)]. 
   = 0                                                                  
  Now            a = - ?
2
y = 0                     
4.    (a) 
   
   = 0.116 m  
  (b) ? ? = ( ?) ?t 
    
5.    ?t and kx should have opposite signs. 
    
  ?    Coefficient of t = v                  (coefficient of x) 
   = (4.5) (1) = 4.5 
    
6.    (a)  
   = 0.02 s = 20 ms 
    
   = 4.0 cm  
  (b) 
    
  Put x = 1 cm and t = 0.01 s in above equation We get, 
   v p = 0 
  (c)  and (d) Putting the given value in the above equation we get the answer. 
7.    (a)       
    
    
  (b) Amplitude is 15 cm 
At t = 0, x = 0, y = + A. Hence equation should be a cos equation. Further wave is travelling in 
positive x-direction.  
Hence ?t and kx should have opposite signs. 
8.    From the figure, 
   
   = 3750 Hz  
  Particle velocity               .......(i) 
At t = 0, x - 0 Wave velocity v is positive. or slope of y - x graph is also positive.  
Therefore, from Eq. (i), particle velocity is  negative.  
Hence particle starts from means position (x = 0, t= 0) along negative y-direction.  
So at x = 0, equation should remain y = - A sin ?t. Further, wave velocity is positive.  
Hence ?t and kx should have opposite signs. 
    
   = 23562 rad/s 
 
  Now y = Asin (kx - ?t)  
9.   (a)