DC Pandey Solutions (JEE Advance): Wave Motion | DC Pandey Solutions for JEE Physics PDF Download

 Page 1


For JEE Advanced 
  Match the Columns 
Q 1.  For the wave equation, 
   y = a sin (bt - cx) Match the following two columns. 
Column I Column II 
(a) wave speed 
(p) 
b
2 ?
 
(b) maximum particle speed 
(q) 
c
2 ?
 
(c) wave frequency 
(r) 
b
c
 
(d) wavelength (s) None 
Q 2.  For the wave equation, 
   y = (4 cm) sin [ ?t + 2 ?x] 
  Here t is in second and x in meters. 
Column I Column II 
(a) at x = 0, particle velocity is maximum at t = (p) 0.5 s 
(b) at x = 0, particle acceleration is maximum at t = (q) 1.0 s 
(c) at x = 0.5 m, particle velocity is maximum at t = (r) zero 
(d) at x = 0.5 m, particle acceleration is maximum at t = (s) 1.5 s 
Q 3.  y-x graph of a transverse wave at a given instant is shown in figure. Match the following two 
columns. 
 
Column I Column II 
(a) velocity of particle A (p) positive 
(b) acceleration of particle A (q) negative 
(c) velocity of particle B (r) zero 
(d) acceleration of particle B (s) can't tell 
Q 4.  For a travelling wave match the following two columns. 
Column I Column II 
 (a) energy density (P) [ML
2
T
-3
] 
(b) power (q) 1/2 ? ?
2
A
2
Sv 
(c) intensity (r) [M
0
L
-1
T] 
(d) wave number (s) None 
Q 5.  Match following two columns. 
Page 2


For JEE Advanced 
  Match the Columns 
Q 1.  For the wave equation, 
   y = a sin (bt - cx) Match the following two columns. 
Column I Column II 
(a) wave speed 
(p) 
b
2 ?
 
(b) maximum particle speed 
(q) 
c
2 ?
 
(c) wave frequency 
(r) 
b
c
 
(d) wavelength (s) None 
Q 2.  For the wave equation, 
   y = (4 cm) sin [ ?t + 2 ?x] 
  Here t is in second and x in meters. 
Column I Column II 
(a) at x = 0, particle velocity is maximum at t = (p) 0.5 s 
(b) at x = 0, particle acceleration is maximum at t = (q) 1.0 s 
(c) at x = 0.5 m, particle velocity is maximum at t = (r) zero 
(d) at x = 0.5 m, particle acceleration is maximum at t = (s) 1.5 s 
Q 3.  y-x graph of a transverse wave at a given instant is shown in figure. Match the following two 
columns. 
 
Column I Column II 
(a) velocity of particle A (p) positive 
(b) acceleration of particle A (q) negative 
(c) velocity of particle B (r) zero 
(d) acceleration of particle B (s) can't tell 
Q 4.  For a travelling wave match the following two columns. 
Column I Column II 
 (a) energy density (P) [ML
2
T
-3
] 
(b) power (q) 1/2 ? ?
2
A
2
Sv 
(c) intensity (r) [M
0
L
-1
T] 
(d) wave number (s) None 
Q 5.  Match following two columns. 
Column I Column II 
(a) y = A sin ( ?t - kx) (p) travelling in positive x-direction 
(b) y= A sin (kx - ?t) (q) travelling in negative x-direction 
(c) y = - A cos ( ?t + kx) (r) at t = 0, velocity of particle is positive at x = 0 
(d) y=- A cos (kx - ?t) (s) at t = 0 acceleration of particle is positive at x = 0 
 
Solutions 
1.   (a) 
  (b) Maximum particle speed = ?A = (b) (a) 
   (c) 
  (d)   
2.        
    
  Now substitute value of t and x.      
3.   For velocity       
   = Slope of y- x graph. 
  Sign of v is not given in the questions. Hence direction of vp cannot be determined. 
  For particle acceleration. 
   ap ? - y 
  i.e., ap and y are away in opposite directions  
  If                           y = 0 
  then                      ap = 0 
4.  Energy density Energy per unit volume 
  Power = energy transfer per unit. 
    
  Intensity = energy transfer per unit per unit area 
    
  Wave number           
5.  
  It ?t and kx area of same sign, wave travels in negative x - direction. 
  If they area of opposite signs then wave travels in positive direction. 
 
Page 3


For JEE Advanced 
  Match the Columns 
Q 1.  For the wave equation, 
   y = a sin (bt - cx) Match the following two columns. 
Column I Column II 
(a) wave speed 
(p) 
b
2 ?
 
(b) maximum particle speed 
(q) 
c
2 ?
 
(c) wave frequency 
(r) 
b
c
 
(d) wavelength (s) None 
Q 2.  For the wave equation, 
   y = (4 cm) sin [ ?t + 2 ?x] 
  Here t is in second and x in meters. 
Column I Column II 
(a) at x = 0, particle velocity is maximum at t = (p) 0.5 s 
(b) at x = 0, particle acceleration is maximum at t = (q) 1.0 s 
(c) at x = 0.5 m, particle velocity is maximum at t = (r) zero 
(d) at x = 0.5 m, particle acceleration is maximum at t = (s) 1.5 s 
Q 3.  y-x graph of a transverse wave at a given instant is shown in figure. Match the following two 
columns. 
 
Column I Column II 
(a) velocity of particle A (p) positive 
(b) acceleration of particle A (q) negative 
(c) velocity of particle B (r) zero 
(d) acceleration of particle B (s) can't tell 
Q 4.  For a travelling wave match the following two columns. 
Column I Column II 
 (a) energy density (P) [ML
2
T
-3
] 
(b) power (q) 1/2 ? ?
2
A
2
Sv 
(c) intensity (r) [M
0
L
-1
T] 
(d) wave number (s) None 
Q 5.  Match following two columns. 
Column I Column II 
(a) y = A sin ( ?t - kx) (p) travelling in positive x-direction 
(b) y= A sin (kx - ?t) (q) travelling in negative x-direction 
(c) y = - A cos ( ?t + kx) (r) at t = 0, velocity of particle is positive at x = 0 
(d) y=- A cos (kx - ?t) (s) at t = 0 acceleration of particle is positive at x = 0 
 
Solutions 
1.   (a) 
  (b) Maximum particle speed = ?A = (b) (a) 
   (c) 
  (d)   
2.        
    
  Now substitute value of t and x.      
3.   For velocity       
   = Slope of y- x graph. 
  Sign of v is not given in the questions. Hence direction of vp cannot be determined. 
  For particle acceleration. 
   ap ? - y 
  i.e., ap and y are away in opposite directions  
  If                           y = 0 
  then                      ap = 0 
4.  Energy density Energy per unit volume 
  Power = energy transfer per unit. 
    
  Intensity = energy transfer per unit per unit area 
    
  Wave number           
5.  
  It ?t and kx area of same sign, wave travels in negative x - direction. 
  If they area of opposite signs then wave travels in positive direction. 
 
Subjective Questions 
Q 1.  The figure shows a snap photograph of a vibrating string at t = 0. The particle P is observed 
moving up with velocity 20 3 cm / s .
 
The tangent at P makes an angle 60° with x-axis. 
 
  (a) Find the direction in which the wave is moving. 
  (b) Write the equation of the wave. 
(c) The total energy carried by the wave per cycle of the string. Assuming that the mass per unit 
length of the string = 50g/m. 
Q 2.  A long string having a cross-sectional area 0.80 mm
2
 and density 12.5 g/cm
3
, is subjected to a 
tension of 64 N along the positive x-axis. One end of this string is attached to a vibrator at x = 0 
moving in transverse direction at a frequency of 20 Hz. At t = 0, the source is at a maximum 
displacement y = 1.0 cm. 
  (a) Find the speed of the wave travelling on the string. 
  (b) Write the equation for the wave. 
  (c) What is the displacement of the particle of the string at x = 50cm at time t = 0.05 s ? 
  (d) What is the velocity of this particle at this instant? 
Q 3.  One end of each of two identical springs, each of force-constant 0.5 N/mare attached on the 
opposite sides of a wooden block of mass 0.01kg. The other ends of the springs are connected to 
separate rigid supports such that the springs are unstretched and are collinear in a horizontal plane. 
To the wooden piece is fixed a pointer which
 
touches a vertically moving plane paper. The 
wooden piece, kept on a smooth horizontal table is now displaced by 0.02 m along the line of 
springs and released. If the speed of paper is 0.1 m/s, find the equation of the path traced by the 
pointer on the paper and the distance between two consecutive maximas on this path.  
 
Q 4.  A wave pulse is travelling on a string with a speed v towards the positive x-axis. The shape of the 
string at t = 0 is given by y(x) = A sin(x/a), where A and a are constants. 
  (a) What are the dimensions of A and a ? 
  (b) Write the equation of the wave for a general time t, if the wave speed is v. 
Q 5.  Figure shows a plot of the transverse displacement of the particle of a string at t = 0 through which 
a travelling wave is passing in the positive x-direction. The wave speed is 20 cm/s. Find  
  (a) the amplitude, (b) the wavelength, (c) the wave number and (d) the frequency of the wave. 
Page 4


For JEE Advanced 
  Match the Columns 
Q 1.  For the wave equation, 
   y = a sin (bt - cx) Match the following two columns. 
Column I Column II 
(a) wave speed 
(p) 
b
2 ?
 
(b) maximum particle speed 
(q) 
c
2 ?
 
(c) wave frequency 
(r) 
b
c
 
(d) wavelength (s) None 
Q 2.  For the wave equation, 
   y = (4 cm) sin [ ?t + 2 ?x] 
  Here t is in second and x in meters. 
Column I Column II 
(a) at x = 0, particle velocity is maximum at t = (p) 0.5 s 
(b) at x = 0, particle acceleration is maximum at t = (q) 1.0 s 
(c) at x = 0.5 m, particle velocity is maximum at t = (r) zero 
(d) at x = 0.5 m, particle acceleration is maximum at t = (s) 1.5 s 
Q 3.  y-x graph of a transverse wave at a given instant is shown in figure. Match the following two 
columns. 
 
Column I Column II 
(a) velocity of particle A (p) positive 
(b) acceleration of particle A (q) negative 
(c) velocity of particle B (r) zero 
(d) acceleration of particle B (s) can't tell 
Q 4.  For a travelling wave match the following two columns. 
Column I Column II 
 (a) energy density (P) [ML
2
T
-3
] 
(b) power (q) 1/2 ? ?
2
A
2
Sv 
(c) intensity (r) [M
0
L
-1
T] 
(d) wave number (s) None 
Q 5.  Match following two columns. 
Column I Column II 
(a) y = A sin ( ?t - kx) (p) travelling in positive x-direction 
(b) y= A sin (kx - ?t) (q) travelling in negative x-direction 
(c) y = - A cos ( ?t + kx) (r) at t = 0, velocity of particle is positive at x = 0 
(d) y=- A cos (kx - ?t) (s) at t = 0 acceleration of particle is positive at x = 0 
 
Solutions 
1.   (a) 
  (b) Maximum particle speed = ?A = (b) (a) 
   (c) 
  (d)   
2.        
    
  Now substitute value of t and x.      
3.   For velocity       
   = Slope of y- x graph. 
  Sign of v is not given in the questions. Hence direction of vp cannot be determined. 
  For particle acceleration. 
   ap ? - y 
  i.e., ap and y are away in opposite directions  
  If                           y = 0 
  then                      ap = 0 
4.  Energy density Energy per unit volume 
  Power = energy transfer per unit. 
    
  Intensity = energy transfer per unit per unit area 
    
  Wave number           
5.  
  It ?t and kx area of same sign, wave travels in negative x - direction. 
  If they area of opposite signs then wave travels in positive direction. 
 
Subjective Questions 
Q 1.  The figure shows a snap photograph of a vibrating string at t = 0. The particle P is observed 
moving up with velocity 20 3 cm / s .
 
The tangent at P makes an angle 60° with x-axis. 
 
  (a) Find the direction in which the wave is moving. 
  (b) Write the equation of the wave. 
(c) The total energy carried by the wave per cycle of the string. Assuming that the mass per unit 
length of the string = 50g/m. 
Q 2.  A long string having a cross-sectional area 0.80 mm
2
 and density 12.5 g/cm
3
, is subjected to a 
tension of 64 N along the positive x-axis. One end of this string is attached to a vibrator at x = 0 
moving in transverse direction at a frequency of 20 Hz. At t = 0, the source is at a maximum 
displacement y = 1.0 cm. 
  (a) Find the speed of the wave travelling on the string. 
  (b) Write the equation for the wave. 
  (c) What is the displacement of the particle of the string at x = 50cm at time t = 0.05 s ? 
  (d) What is the velocity of this particle at this instant? 
Q 3.  One end of each of two identical springs, each of force-constant 0.5 N/mare attached on the 
opposite sides of a wooden block of mass 0.01kg. The other ends of the springs are connected to 
separate rigid supports such that the springs are unstretched and are collinear in a horizontal plane. 
To the wooden piece is fixed a pointer which
 
touches a vertically moving plane paper. The 
wooden piece, kept on a smooth horizontal table is now displaced by 0.02 m along the line of 
springs and released. If the speed of paper is 0.1 m/s, find the equation of the path traced by the 
pointer on the paper and the distance between two consecutive maximas on this path.  
 
Q 4.  A wave pulse is travelling on a string with a speed v towards the positive x-axis. The shape of the 
string at t = 0 is given by y(x) = A sin(x/a), where A and a are constants. 
  (a) What are the dimensions of A and a ? 
  (b) Write the equation of the wave for a general time t, if the wave speed is v. 
Q 5.  Figure shows a plot of the transverse displacement of the particle of a string at t = 0 through which 
a travelling wave is passing in the positive x-direction. The wave speed is 20 cm/s. Find  
  (a) the amplitude, (b) the wavelength, (c) the wave number and (d) the frequency of the wave. 
 
Q 6.  Two wires of different densities but same area of cross-section are soldered together at one end 
and are stretched to a tension T. The velocity of a transverse wave in the first wire is double of that 
in the second wire. Find the ratio of the density of the first wire to that of the second wire. 
Q 7.  Two long strings A and B, each having linear mass density 1.2 × 10
-2
 kg/m are stretched by 
different tensions 4.8 N and 7.5 N respectively and are kept parallel to each other with their left 
ends at x = 0. Wave pulses are produced on the strings at the left ends at t = 0 on string A and at t 
= 20 ms on string B. When and where will the pulse on B overtake that on A ? 
Q 8.  A sinusoidal transverse wave travels on a string. The string has length 8.00 m and mass 6.00 g. 
The wave speed is 30.0 m/s and the wavelength is 0.200 m.  
  (a) If the wave is to have an average power of 50.0 W, what must be the amplitude of the wave ?  
  (b) For this same string, if the amplitude and wavelength are the same as in part (a) what is the 
average power for the wave if the tension is increased such that the wave speed is doubled ? 
Q 9.  A uniform rope with length L and mass m is held at one end and whirled in a horizontal circle 
with angular velocity ?. You can ignore the force of gravity on the rope. Find the time required 
for a transverse wave to travel from one end of the rope to the other. 
 Hint : 
1
22
dx x
sin
a
ax
?
??
?
??
??
?
?
 
Solutions 
1.   (a) 
  As vp and (slope)P are both positive, v must be negative.  
  Hence, the wave is moving in negative x-axis. 
  (b)                          ...(i) 
    
  A = 4 × 10
-3
 m = 0.4 cm  
At t = 0, x = 0, slope = + ve 
  vP = - v(slope) = + ve Further at t = 0, x = 0, y = + ve 
   
   Further, = - v tan 60° 
   v = - 20 cm/s 
    
    
Page 5


For JEE Advanced 
  Match the Columns 
Q 1.  For the wave equation, 
   y = a sin (bt - cx) Match the following two columns. 
Column I Column II 
(a) wave speed 
(p) 
b
2 ?
 
(b) maximum particle speed 
(q) 
c
2 ?
 
(c) wave frequency 
(r) 
b
c
 
(d) wavelength (s) None 
Q 2.  For the wave equation, 
   y = (4 cm) sin [ ?t + 2 ?x] 
  Here t is in second and x in meters. 
Column I Column II 
(a) at x = 0, particle velocity is maximum at t = (p) 0.5 s 
(b) at x = 0, particle acceleration is maximum at t = (q) 1.0 s 
(c) at x = 0.5 m, particle velocity is maximum at t = (r) zero 
(d) at x = 0.5 m, particle acceleration is maximum at t = (s) 1.5 s 
Q 3.  y-x graph of a transverse wave at a given instant is shown in figure. Match the following two 
columns. 
 
Column I Column II 
(a) velocity of particle A (p) positive 
(b) acceleration of particle A (q) negative 
(c) velocity of particle B (r) zero 
(d) acceleration of particle B (s) can't tell 
Q 4.  For a travelling wave match the following two columns. 
Column I Column II 
 (a) energy density (P) [ML
2
T
-3
] 
(b) power (q) 1/2 ? ?
2
A
2
Sv 
(c) intensity (r) [M
0
L
-1
T] 
(d) wave number (s) None 
Q 5.  Match following two columns. 
Column I Column II 
(a) y = A sin ( ?t - kx) (p) travelling in positive x-direction 
(b) y= A sin (kx - ?t) (q) travelling in negative x-direction 
(c) y = - A cos ( ?t + kx) (r) at t = 0, velocity of particle is positive at x = 0 
(d) y=- A cos (kx - ?t) (s) at t = 0 acceleration of particle is positive at x = 0 
 
Solutions 
1.   (a) 
  (b) Maximum particle speed = ?A = (b) (a) 
   (c) 
  (d)   
2.        
    
  Now substitute value of t and x.      
3.   For velocity       
   = Slope of y- x graph. 
  Sign of v is not given in the questions. Hence direction of vp cannot be determined. 
  For particle acceleration. 
   ap ? - y 
  i.e., ap and y are away in opposite directions  
  If                           y = 0 
  then                      ap = 0 
4.  Energy density Energy per unit volume 
  Power = energy transfer per unit. 
    
  Intensity = energy transfer per unit per unit area 
    
  Wave number           
5.  
  It ?t and kx area of same sign, wave travels in negative x - direction. 
  If they area of opposite signs then wave travels in positive direction. 
 
Subjective Questions 
Q 1.  The figure shows a snap photograph of a vibrating string at t = 0. The particle P is observed 
moving up with velocity 20 3 cm / s .
 
The tangent at P makes an angle 60° with x-axis. 
 
  (a) Find the direction in which the wave is moving. 
  (b) Write the equation of the wave. 
(c) The total energy carried by the wave per cycle of the string. Assuming that the mass per unit 
length of the string = 50g/m. 
Q 2.  A long string having a cross-sectional area 0.80 mm
2
 and density 12.5 g/cm
3
, is subjected to a 
tension of 64 N along the positive x-axis. One end of this string is attached to a vibrator at x = 0 
moving in transverse direction at a frequency of 20 Hz. At t = 0, the source is at a maximum 
displacement y = 1.0 cm. 
  (a) Find the speed of the wave travelling on the string. 
  (b) Write the equation for the wave. 
  (c) What is the displacement of the particle of the string at x = 50cm at time t = 0.05 s ? 
  (d) What is the velocity of this particle at this instant? 
Q 3.  One end of each of two identical springs, each of force-constant 0.5 N/mare attached on the 
opposite sides of a wooden block of mass 0.01kg. The other ends of the springs are connected to 
separate rigid supports such that the springs are unstretched and are collinear in a horizontal plane. 
To the wooden piece is fixed a pointer which
 
touches a vertically moving plane paper. The 
wooden piece, kept on a smooth horizontal table is now displaced by 0.02 m along the line of 
springs and released. If the speed of paper is 0.1 m/s, find the equation of the path traced by the 
pointer on the paper and the distance between two consecutive maximas on this path.  
 
Q 4.  A wave pulse is travelling on a string with a speed v towards the positive x-axis. The shape of the 
string at t = 0 is given by y(x) = A sin(x/a), where A and a are constants. 
  (a) What are the dimensions of A and a ? 
  (b) Write the equation of the wave for a general time t, if the wave speed is v. 
Q 5.  Figure shows a plot of the transverse displacement of the particle of a string at t = 0 through which 
a travelling wave is passing in the positive x-direction. The wave speed is 20 cm/s. Find  
  (a) the amplitude, (b) the wavelength, (c) the wave number and (d) the frequency of the wave. 
 
Q 6.  Two wires of different densities but same area of cross-section are soldered together at one end 
and are stretched to a tension T. The velocity of a transverse wave in the first wire is double of that 
in the second wire. Find the ratio of the density of the first wire to that of the second wire. 
Q 7.  Two long strings A and B, each having linear mass density 1.2 × 10
-2
 kg/m are stretched by 
different tensions 4.8 N and 7.5 N respectively and are kept parallel to each other with their left 
ends at x = 0. Wave pulses are produced on the strings at the left ends at t = 0 on string A and at t 
= 20 ms on string B. When and where will the pulse on B overtake that on A ? 
Q 8.  A sinusoidal transverse wave travels on a string. The string has length 8.00 m and mass 6.00 g. 
The wave speed is 30.0 m/s and the wavelength is 0.200 m.  
  (a) If the wave is to have an average power of 50.0 W, what must be the amplitude of the wave ?  
  (b) For this same string, if the amplitude and wavelength are the same as in part (a) what is the 
average power for the wave if the tension is increased such that the wave speed is doubled ? 
Q 9.  A uniform rope with length L and mass m is held at one end and whirled in a horizontal circle 
with angular velocity ?. You can ignore the force of gravity on the rope. Find the time required 
for a transverse wave to travel from one end of the rope to the other. 
 Hint : 
1
22
dx x
sin
a
ax
?
??
?
??
??
?
?
 
Solutions 
1.   (a) 
  As vp and (slope)P are both positive, v must be negative.  
  Hence, the wave is moving in negative x-axis. 
  (b)                          ...(i) 
    
  A = 4 × 10
-3
 m = 0.4 cm  
At t = 0, x = 0, slope = + ve 
  vP = - v(slope) = + ve Further at t = 0, x = 0, y = + ve 
   
   Further, = - v tan 60° 
   v = - 20 cm/s 
    
    
    
    
  ?  Energy carried per cycle 
   
  Substituting the values, we have 
   E = 1.6 × 10
-5
 J                       
2.   (a) 
    
   = 80 m/s 
  (b) 
     
  (c)  Substituting x = 0.5 m and t = 0.05 s, we get 
     
  (d) Particle velocity at time t. 
    
   
  Substituting x = 0.5 m and t = 0.05 s, we get 
   vP = 89 cm/s                                 
3.    keff =2k = 1.0 N/m 
    
   v = 0.1 m/s,   A = 0.02 m 
    
    
   = 0.02 cos 100(0. 1t - x) = 0.02 cos (10t - 100 x) m             
  The distance between two successive maxima 
     
4.    (a) Dimensions of A and Y are same.  
Similarly dimensions of a and x are same. (b) As the wave is travelling towards positive x-axis, 
there should be negative sign between term of x and term of t. Further, speed of wave