DC Pandey Solutions (JEE Main): Superposition of Waves | DC Pandey Solutions for JEE Physics PDF Download

 Page 1


Exercises 
For JEE Main 
  Subjective Questions 
  Reflection and Transmission of a Wave: Principle of Superposition and Interference 
Q 1.  Two waves are travelling in the same direction along a stretched string. The waves are 90° out of 
phase. Each wave has an amplitude of 4.0 cm Find the amplitude of the resultant wave. 
Q 2.  Two wires of different densities are soldered together end to end then stretched under tension T. 
The wave speed in the first wire is twice that in the second wire. 
  (a) If the amplitude of incident wave is A, what are amplitudes of reflected and transmitted waves? 
(b) Assuming no energy loss in the wire, find the fraction of the incident power that is reflected at 
the junction and fraction of the same that is transmitted. 
Q 3.  A wave is represented by y 1 =10cos(5x + 25t) where, x is measured in metres and t in seconds. A 
second wave for which 
   y 2 = 20 cos 5x 25t
3
? ??
??
??
??
 
  interferes with the first wave. Deduce the amplitude and phase of the resultant wave. 
Q 4.  Two waves passing through a region are represented by y 1 =(1.0cm)sin [(3.14cm
-1
 )x- (157s
-1
)t] 
and y 2 =(1.5 cm) sin [(1.57 cm
-1
 )x- (314 s
-1
) t] 
  Find the displacement of the particle at x = 4.5 cm at time t = 5.0 ms. 
Q 5.  A string of length 20 cm and linear mass density 0.4 g/cm is fixed at both ends and is kept under a 
tension of 16 N. A wave pulse is produced at t = 0 near an end as shown in figure which travels 
towards the other end. (a) When will the string have the shape shown in the figure again ? (b) 
Sketch the shape of the string at a time half of that found in part (a). 
 
Q 6.  A wave pulse on a string has the dimensions shown in figure. The wave speed is v = 1 cm/s. 
 
  (a) If point O is a fixed end, draw the resultant wave on the string at t = 3 s and t = 4 s. 
  (b) Repeat part (a) for the case in which O is a free end. 
Stationary Waves and Normal Modes of a String 
Q 7.  Two sinusoidal waves travelling in opposite directions interfere to produce a standing wave 
described by the equation 
   y = (1.5 m)sin (0.400 x)cos (200 t) 
Page 2


Exercises 
For JEE Main 
  Subjective Questions 
  Reflection and Transmission of a Wave: Principle of Superposition and Interference 
Q 1.  Two waves are travelling in the same direction along a stretched string. The waves are 90° out of 
phase. Each wave has an amplitude of 4.0 cm Find the amplitude of the resultant wave. 
Q 2.  Two wires of different densities are soldered together end to end then stretched under tension T. 
The wave speed in the first wire is twice that in the second wire. 
  (a) If the amplitude of incident wave is A, what are amplitudes of reflected and transmitted waves? 
(b) Assuming no energy loss in the wire, find the fraction of the incident power that is reflected at 
the junction and fraction of the same that is transmitted. 
Q 3.  A wave is represented by y 1 =10cos(5x + 25t) where, x is measured in metres and t in seconds. A 
second wave for which 
   y 2 = 20 cos 5x 25t
3
? ??
??
??
??
 
  interferes with the first wave. Deduce the amplitude and phase of the resultant wave. 
Q 4.  Two waves passing through a region are represented by y 1 =(1.0cm)sin [(3.14cm
-1
 )x- (157s
-1
)t] 
and y 2 =(1.5 cm) sin [(1.57 cm
-1
 )x- (314 s
-1
) t] 
  Find the displacement of the particle at x = 4.5 cm at time t = 5.0 ms. 
Q 5.  A string of length 20 cm and linear mass density 0.4 g/cm is fixed at both ends and is kept under a 
tension of 16 N. A wave pulse is produced at t = 0 near an end as shown in figure which travels 
towards the other end. (a) When will the string have the shape shown in the figure again ? (b) 
Sketch the shape of the string at a time half of that found in part (a). 
 
Q 6.  A wave pulse on a string has the dimensions shown in figure. The wave speed is v = 1 cm/s. 
 
  (a) If point O is a fixed end, draw the resultant wave on the string at t = 3 s and t = 4 s. 
  (b) Repeat part (a) for the case in which O is a free end. 
Stationary Waves and Normal Modes of a String 
Q 7.  Two sinusoidal waves travelling in opposite directions interfere to produce a standing wave 
described by the equation 
   y = (1.5 m)sin (0.400 x)cos (200 t) 
where, x is in metres and t is in seconds. Determine the wavelength, frequency and speed of the 
interfering waves. 
Q 8.  Two sinusoidal waves combining in a medium are described by the equations 
y1 = (3.0 cm) sin ? (x + 0.60 t) and y 2 = (3.0cm) sin ? (x - 0.60t) where, x is in centimetres and t is 
in seconds. Determine the maximum displacement of the medium at 
(a) x = 0.250 cm, (b) x= 0.500 cm and (c) x = 1.50 cm. (d) Find the three smallest values of x 
corresponding to antinodes. 
Q 9.  A standing wave is formed by the interference of two travelling waves, each of which has an 
amplitude A = ? cm, angular wave number k = ( ?/2)
 
per centimetre. 
  (a) Calculate the distance between two successive antinodes. 
  (b) What is the amplitude of the standing wave at x = 0.50 cm from a node ? 
Q 10.  Find the fundamental frequency and the next three frequencies that could cause a standing-wave 
pattern on a string that is 30.0 m long, has a mass per unit length of 9.00 × 10
-3
 kg /mand is 
stretched to a tension of 20.0 N. 
Q 11.  A string vibrates in its first normal mode with a frequency of 220 vibrations/s. The vibrating 
segment is 70.0 cm long and has a mass of 1.20 g. 
  (a) Find the tension in the string. 
  (b) Determine the frequency of vibration when the string vibrates in three segments. 
Q 12.  A 60.0cm guitar string under a tension of 50.0Nhas a mass per unit length of 0.100g/cm. What is 
the highest resonance frequency of the string that can be heard by a person able to hear 
frequencies upto 20,000 Hz? 
Q 13.  A wire having a linear density of 0.05 g/cm is stretched between two rigid supports with a tension 
of 450 N. It is observed that the wire resonates at a frequency of 420 Hz. The next higher 
frequency at which the same wire resonates is 490 Hz. Find the length of the wire. 
Q 14.  The vibrations from an 800 Hz tuning fork set up standing waves in a string clamped at both ends. 
The wave speed in the string is known to be 400 m/s for the tension used. The standing wave is 
observed to have four antinodes. How long is the string ? 
Q 15.  A string vibrates in 4 segments to a frequency of 400 Hz. 
  (a) What is its fundamental frequency ?  
  (b) What frequency will cause it to vibrate into 7 segments? 
Q 16.  A sonometer wire has a total length of 1 m between the fixed ends. Where should the two bridges 
be placed below the wire so that the three segments of the wire have their fundamental frequencies 
in the ratio 1:2:3? 
Q 17.  A guitar string is 90cmlong and has a fundamental frequency of 124 Hz. Where should it be 
pressed to produce a fundamental frequency of 186 Hz? 
Q 18.  Adjacent antinodes of a standing wave on a string are 15.0 cm apart. A particle at an antinode 
oscillates in simple harmonic motion with amplitude 0.850 cm and period 0.0750s. The string lies 
along the +x-axis and is fixed at x = 0. 
Page 3


Exercises 
For JEE Main 
  Subjective Questions 
  Reflection and Transmission of a Wave: Principle of Superposition and Interference 
Q 1.  Two waves are travelling in the same direction along a stretched string. The waves are 90° out of 
phase. Each wave has an amplitude of 4.0 cm Find the amplitude of the resultant wave. 
Q 2.  Two wires of different densities are soldered together end to end then stretched under tension T. 
The wave speed in the first wire is twice that in the second wire. 
  (a) If the amplitude of incident wave is A, what are amplitudes of reflected and transmitted waves? 
(b) Assuming no energy loss in the wire, find the fraction of the incident power that is reflected at 
the junction and fraction of the same that is transmitted. 
Q 3.  A wave is represented by y 1 =10cos(5x + 25t) where, x is measured in metres and t in seconds. A 
second wave for which 
   y 2 = 20 cos 5x 25t
3
? ??
??
??
??
 
  interferes with the first wave. Deduce the amplitude and phase of the resultant wave. 
Q 4.  Two waves passing through a region are represented by y 1 =(1.0cm)sin [(3.14cm
-1
 )x- (157s
-1
)t] 
and y 2 =(1.5 cm) sin [(1.57 cm
-1
 )x- (314 s
-1
) t] 
  Find the displacement of the particle at x = 4.5 cm at time t = 5.0 ms. 
Q 5.  A string of length 20 cm and linear mass density 0.4 g/cm is fixed at both ends and is kept under a 
tension of 16 N. A wave pulse is produced at t = 0 near an end as shown in figure which travels 
towards the other end. (a) When will the string have the shape shown in the figure again ? (b) 
Sketch the shape of the string at a time half of that found in part (a). 
 
Q 6.  A wave pulse on a string has the dimensions shown in figure. The wave speed is v = 1 cm/s. 
 
  (a) If point O is a fixed end, draw the resultant wave on the string at t = 3 s and t = 4 s. 
  (b) Repeat part (a) for the case in which O is a free end. 
Stationary Waves and Normal Modes of a String 
Q 7.  Two sinusoidal waves travelling in opposite directions interfere to produce a standing wave 
described by the equation 
   y = (1.5 m)sin (0.400 x)cos (200 t) 
where, x is in metres and t is in seconds. Determine the wavelength, frequency and speed of the 
interfering waves. 
Q 8.  Two sinusoidal waves combining in a medium are described by the equations 
y1 = (3.0 cm) sin ? (x + 0.60 t) and y 2 = (3.0cm) sin ? (x - 0.60t) where, x is in centimetres and t is 
in seconds. Determine the maximum displacement of the medium at 
(a) x = 0.250 cm, (b) x= 0.500 cm and (c) x = 1.50 cm. (d) Find the three smallest values of x 
corresponding to antinodes. 
Q 9.  A standing wave is formed by the interference of two travelling waves, each of which has an 
amplitude A = ? cm, angular wave number k = ( ?/2)
 
per centimetre. 
  (a) Calculate the distance between two successive antinodes. 
  (b) What is the amplitude of the standing wave at x = 0.50 cm from a node ? 
Q 10.  Find the fundamental frequency and the next three frequencies that could cause a standing-wave 
pattern on a string that is 30.0 m long, has a mass per unit length of 9.00 × 10
-3
 kg /mand is 
stretched to a tension of 20.0 N. 
Q 11.  A string vibrates in its first normal mode with a frequency of 220 vibrations/s. The vibrating 
segment is 70.0 cm long and has a mass of 1.20 g. 
  (a) Find the tension in the string. 
  (b) Determine the frequency of vibration when the string vibrates in three segments. 
Q 12.  A 60.0cm guitar string under a tension of 50.0Nhas a mass per unit length of 0.100g/cm. What is 
the highest resonance frequency of the string that can be heard by a person able to hear 
frequencies upto 20,000 Hz? 
Q 13.  A wire having a linear density of 0.05 g/cm is stretched between two rigid supports with a tension 
of 450 N. It is observed that the wire resonates at a frequency of 420 Hz. The next higher 
frequency at which the same wire resonates is 490 Hz. Find the length of the wire. 
Q 14.  The vibrations from an 800 Hz tuning fork set up standing waves in a string clamped at both ends. 
The wave speed in the string is known to be 400 m/s for the tension used. The standing wave is 
observed to have four antinodes. How long is the string ? 
Q 15.  A string vibrates in 4 segments to a frequency of 400 Hz. 
  (a) What is its fundamental frequency ?  
  (b) What frequency will cause it to vibrate into 7 segments? 
Q 16.  A sonometer wire has a total length of 1 m between the fixed ends. Where should the two bridges 
be placed below the wire so that the three segments of the wire have their fundamental frequencies 
in the ratio 1:2:3? 
Q 17.  A guitar string is 90cmlong and has a fundamental frequency of 124 Hz. Where should it be 
pressed to produce a fundamental frequency of 186 Hz? 
Q 18.  Adjacent antinodes of a standing wave on a string are 15.0 cm apart. A particle at an antinode 
oscillates in simple harmonic motion with amplitude 0.850 cm and period 0.0750s. The string lies 
along the +x-axis and is fixed at x = 0. 
  (a) Find the displacement of a point on the string as a function of position and time. 
  (b) Find the speed of propagation of a transverse wave in the string. 
  (c) Find the amplitude at a point 3.0 cm to the right of an antinode. 
Q 19.  A 1.50 m long rope is stretched between two supports with a tension that makes the speed of 
transverse waves 48.0 m/s. What are the wavelength and frequency of 
  (a) the fundamental ?  (b) the second overtone ?  (c) the fourth harmonic ? 
Q 20.  A thin, taut string tied at both ends and oscillating in its third harmonic has its shape described by 
the equation y(x,t)= (5.60 cm)sin[(0.0340 rad/cm)x] sin[(50.0 rad/s )t], where the origin is at the 
left end of the string, the x-axis is along the string and the y-axis is perpendicular to the string. 
  (a) Draw a sketch that shows the standing wave pattern. 
  (b) Find the amplitude of the two travelling waves that make up this standing wave. 
  (c) What is the length of the string? 
  (d) Find the wavelength, frequency, period and speed of the travelling wave.  
  (e) Find the maximum transverse speed of a point on the string. 
  (f) What would be the equation y(x, t) for this string if it were vibrating in its eighth harmonic ? 
Q 21.  A wire with mass 40.0 g is stretched so that its ends are tied down at points 80.0cm apart. The 
wire vibrates in its fundamental mode with frequency 60.0 Hz and with an amplitude at the 
antinodes of 0.300 cm. 
  (a) What is the speed of propagation of transverse wave in the wire ? 
  (b) Compute the tension in the wire. 
  (c) Find the maximum transverse velocity and acceleration of particles in the wire. 
Solutions 
1.  
   
   = 5.66 cm 
2.          
  (a) 
    
    
  (b)  Since reflected waves comes in the same medium, we can say that, 
Page 4


Exercises 
For JEE Main 
  Subjective Questions 
  Reflection and Transmission of a Wave: Principle of Superposition and Interference 
Q 1.  Two waves are travelling in the same direction along a stretched string. The waves are 90° out of 
phase. Each wave has an amplitude of 4.0 cm Find the amplitude of the resultant wave. 
Q 2.  Two wires of different densities are soldered together end to end then stretched under tension T. 
The wave speed in the first wire is twice that in the second wire. 
  (a) If the amplitude of incident wave is A, what are amplitudes of reflected and transmitted waves? 
(b) Assuming no energy loss in the wire, find the fraction of the incident power that is reflected at 
the junction and fraction of the same that is transmitted. 
Q 3.  A wave is represented by y 1 =10cos(5x + 25t) where, x is measured in metres and t in seconds. A 
second wave for which 
   y 2 = 20 cos 5x 25t
3
? ??
??
??
??
 
  interferes with the first wave. Deduce the amplitude and phase of the resultant wave. 
Q 4.  Two waves passing through a region are represented by y 1 =(1.0cm)sin [(3.14cm
-1
 )x- (157s
-1
)t] 
and y 2 =(1.5 cm) sin [(1.57 cm
-1
 )x- (314 s
-1
) t] 
  Find the displacement of the particle at x = 4.5 cm at time t = 5.0 ms. 
Q 5.  A string of length 20 cm and linear mass density 0.4 g/cm is fixed at both ends and is kept under a 
tension of 16 N. A wave pulse is produced at t = 0 near an end as shown in figure which travels 
towards the other end. (a) When will the string have the shape shown in the figure again ? (b) 
Sketch the shape of the string at a time half of that found in part (a). 
 
Q 6.  A wave pulse on a string has the dimensions shown in figure. The wave speed is v = 1 cm/s. 
 
  (a) If point O is a fixed end, draw the resultant wave on the string at t = 3 s and t = 4 s. 
  (b) Repeat part (a) for the case in which O is a free end. 
Stationary Waves and Normal Modes of a String 
Q 7.  Two sinusoidal waves travelling in opposite directions interfere to produce a standing wave 
described by the equation 
   y = (1.5 m)sin (0.400 x)cos (200 t) 
where, x is in metres and t is in seconds. Determine the wavelength, frequency and speed of the 
interfering waves. 
Q 8.  Two sinusoidal waves combining in a medium are described by the equations 
y1 = (3.0 cm) sin ? (x + 0.60 t) and y 2 = (3.0cm) sin ? (x - 0.60t) where, x is in centimetres and t is 
in seconds. Determine the maximum displacement of the medium at 
(a) x = 0.250 cm, (b) x= 0.500 cm and (c) x = 1.50 cm. (d) Find the three smallest values of x 
corresponding to antinodes. 
Q 9.  A standing wave is formed by the interference of two travelling waves, each of which has an 
amplitude A = ? cm, angular wave number k = ( ?/2)
 
per centimetre. 
  (a) Calculate the distance between two successive antinodes. 
  (b) What is the amplitude of the standing wave at x = 0.50 cm from a node ? 
Q 10.  Find the fundamental frequency and the next three frequencies that could cause a standing-wave 
pattern on a string that is 30.0 m long, has a mass per unit length of 9.00 × 10
-3
 kg /mand is 
stretched to a tension of 20.0 N. 
Q 11.  A string vibrates in its first normal mode with a frequency of 220 vibrations/s. The vibrating 
segment is 70.0 cm long and has a mass of 1.20 g. 
  (a) Find the tension in the string. 
  (b) Determine the frequency of vibration when the string vibrates in three segments. 
Q 12.  A 60.0cm guitar string under a tension of 50.0Nhas a mass per unit length of 0.100g/cm. What is 
the highest resonance frequency of the string that can be heard by a person able to hear 
frequencies upto 20,000 Hz? 
Q 13.  A wire having a linear density of 0.05 g/cm is stretched between two rigid supports with a tension 
of 450 N. It is observed that the wire resonates at a frequency of 420 Hz. The next higher 
frequency at which the same wire resonates is 490 Hz. Find the length of the wire. 
Q 14.  The vibrations from an 800 Hz tuning fork set up standing waves in a string clamped at both ends. 
The wave speed in the string is known to be 400 m/s for the tension used. The standing wave is 
observed to have four antinodes. How long is the string ? 
Q 15.  A string vibrates in 4 segments to a frequency of 400 Hz. 
  (a) What is its fundamental frequency ?  
  (b) What frequency will cause it to vibrate into 7 segments? 
Q 16.  A sonometer wire has a total length of 1 m between the fixed ends. Where should the two bridges 
be placed below the wire so that the three segments of the wire have their fundamental frequencies 
in the ratio 1:2:3? 
Q 17.  A guitar string is 90cmlong and has a fundamental frequency of 124 Hz. Where should it be 
pressed to produce a fundamental frequency of 186 Hz? 
Q 18.  Adjacent antinodes of a standing wave on a string are 15.0 cm apart. A particle at an antinode 
oscillates in simple harmonic motion with amplitude 0.850 cm and period 0.0750s. The string lies 
along the +x-axis and is fixed at x = 0. 
  (a) Find the displacement of a point on the string as a function of position and time. 
  (b) Find the speed of propagation of a transverse wave in the string. 
  (c) Find the amplitude at a point 3.0 cm to the right of an antinode. 
Q 19.  A 1.50 m long rope is stretched between two supports with a tension that makes the speed of 
transverse waves 48.0 m/s. What are the wavelength and frequency of 
  (a) the fundamental ?  (b) the second overtone ?  (c) the fourth harmonic ? 
Q 20.  A thin, taut string tied at both ends and oscillating in its third harmonic has its shape described by 
the equation y(x,t)= (5.60 cm)sin[(0.0340 rad/cm)x] sin[(50.0 rad/s )t], where the origin is at the 
left end of the string, the x-axis is along the string and the y-axis is perpendicular to the string. 
  (a) Draw a sketch that shows the standing wave pattern. 
  (b) Find the amplitude of the two travelling waves that make up this standing wave. 
  (c) What is the length of the string? 
  (d) Find the wavelength, frequency, period and speed of the travelling wave.  
  (e) Find the maximum transverse speed of a point on the string. 
  (f) What would be the equation y(x, t) for this string if it were vibrating in its eighth harmonic ? 
Q 21.  A wire with mass 40.0 g is stretched so that its ends are tied down at points 80.0cm apart. The 
wire vibrates in its fundamental mode with frequency 60.0 Hz and with an amplitude at the 
antinodes of 0.300 cm. 
  (a) What is the speed of propagation of transverse wave in the wire ? 
  (b) Compute the tension in the wire. 
  (c) Find the maximum transverse velocity and acceleration of particles in the wire. 
Solutions 
1.  
   
   = 5.66 cm 
2.          
  (a) 
    
    
  (b)  Since reflected waves comes in the same medium, we can say that, 
    
    
  ?  Fraction of power transmitted 
    
3.  
   = 26.46 cm 
    
   = 0.714 rad 
    
4.   Find y 1 and y 2 at given values of x and t and then simply add them to get net value of y. 
5.  
    
  (a) 
   = 0.02 s. 
  (b) At half the time pulse is at other end and it gets a phase difference of ?.  
  Hence the shape is as shown below. 
    
6.   (a) For fixed end. 
    
  By the superposition of these two pulses we will get the resultant.  
  But only to the left of point O, where string is actually present.  
  (b) For free end 
    
7.  
Page 5


Exercises 
For JEE Main 
  Subjective Questions 
  Reflection and Transmission of a Wave: Principle of Superposition and Interference 
Q 1.  Two waves are travelling in the same direction along a stretched string. The waves are 90° out of 
phase. Each wave has an amplitude of 4.0 cm Find the amplitude of the resultant wave. 
Q 2.  Two wires of different densities are soldered together end to end then stretched under tension T. 
The wave speed in the first wire is twice that in the second wire. 
  (a) If the amplitude of incident wave is A, what are amplitudes of reflected and transmitted waves? 
(b) Assuming no energy loss in the wire, find the fraction of the incident power that is reflected at 
the junction and fraction of the same that is transmitted. 
Q 3.  A wave is represented by y 1 =10cos(5x + 25t) where, x is measured in metres and t in seconds. A 
second wave for which 
   y 2 = 20 cos 5x 25t
3
? ??
??
??
??
 
  interferes with the first wave. Deduce the amplitude and phase of the resultant wave. 
Q 4.  Two waves passing through a region are represented by y 1 =(1.0cm)sin [(3.14cm
-1
 )x- (157s
-1
)t] 
and y 2 =(1.5 cm) sin [(1.57 cm
-1
 )x- (314 s
-1
) t] 
  Find the displacement of the particle at x = 4.5 cm at time t = 5.0 ms. 
Q 5.  A string of length 20 cm and linear mass density 0.4 g/cm is fixed at both ends and is kept under a 
tension of 16 N. A wave pulse is produced at t = 0 near an end as shown in figure which travels 
towards the other end. (a) When will the string have the shape shown in the figure again ? (b) 
Sketch the shape of the string at a time half of that found in part (a). 
 
Q 6.  A wave pulse on a string has the dimensions shown in figure. The wave speed is v = 1 cm/s. 
 
  (a) If point O is a fixed end, draw the resultant wave on the string at t = 3 s and t = 4 s. 
  (b) Repeat part (a) for the case in which O is a free end. 
Stationary Waves and Normal Modes of a String 
Q 7.  Two sinusoidal waves travelling in opposite directions interfere to produce a standing wave 
described by the equation 
   y = (1.5 m)sin (0.400 x)cos (200 t) 
where, x is in metres and t is in seconds. Determine the wavelength, frequency and speed of the 
interfering waves. 
Q 8.  Two sinusoidal waves combining in a medium are described by the equations 
y1 = (3.0 cm) sin ? (x + 0.60 t) and y 2 = (3.0cm) sin ? (x - 0.60t) where, x is in centimetres and t is 
in seconds. Determine the maximum displacement of the medium at 
(a) x = 0.250 cm, (b) x= 0.500 cm and (c) x = 1.50 cm. (d) Find the three smallest values of x 
corresponding to antinodes. 
Q 9.  A standing wave is formed by the interference of two travelling waves, each of which has an 
amplitude A = ? cm, angular wave number k = ( ?/2)
 
per centimetre. 
  (a) Calculate the distance between two successive antinodes. 
  (b) What is the amplitude of the standing wave at x = 0.50 cm from a node ? 
Q 10.  Find the fundamental frequency and the next three frequencies that could cause a standing-wave 
pattern on a string that is 30.0 m long, has a mass per unit length of 9.00 × 10
-3
 kg /mand is 
stretched to a tension of 20.0 N. 
Q 11.  A string vibrates in its first normal mode with a frequency of 220 vibrations/s. The vibrating 
segment is 70.0 cm long and has a mass of 1.20 g. 
  (a) Find the tension in the string. 
  (b) Determine the frequency of vibration when the string vibrates in three segments. 
Q 12.  A 60.0cm guitar string under a tension of 50.0Nhas a mass per unit length of 0.100g/cm. What is 
the highest resonance frequency of the string that can be heard by a person able to hear 
frequencies upto 20,000 Hz? 
Q 13.  A wire having a linear density of 0.05 g/cm is stretched between two rigid supports with a tension 
of 450 N. It is observed that the wire resonates at a frequency of 420 Hz. The next higher 
frequency at which the same wire resonates is 490 Hz. Find the length of the wire. 
Q 14.  The vibrations from an 800 Hz tuning fork set up standing waves in a string clamped at both ends. 
The wave speed in the string is known to be 400 m/s for the tension used. The standing wave is 
observed to have four antinodes. How long is the string ? 
Q 15.  A string vibrates in 4 segments to a frequency of 400 Hz. 
  (a) What is its fundamental frequency ?  
  (b) What frequency will cause it to vibrate into 7 segments? 
Q 16.  A sonometer wire has a total length of 1 m between the fixed ends. Where should the two bridges 
be placed below the wire so that the three segments of the wire have their fundamental frequencies 
in the ratio 1:2:3? 
Q 17.  A guitar string is 90cmlong and has a fundamental frequency of 124 Hz. Where should it be 
pressed to produce a fundamental frequency of 186 Hz? 
Q 18.  Adjacent antinodes of a standing wave on a string are 15.0 cm apart. A particle at an antinode 
oscillates in simple harmonic motion with amplitude 0.850 cm and period 0.0750s. The string lies 
along the +x-axis and is fixed at x = 0. 
  (a) Find the displacement of a point on the string as a function of position and time. 
  (b) Find the speed of propagation of a transverse wave in the string. 
  (c) Find the amplitude at a point 3.0 cm to the right of an antinode. 
Q 19.  A 1.50 m long rope is stretched between two supports with a tension that makes the speed of 
transverse waves 48.0 m/s. What are the wavelength and frequency of 
  (a) the fundamental ?  (b) the second overtone ?  (c) the fourth harmonic ? 
Q 20.  A thin, taut string tied at both ends and oscillating in its third harmonic has its shape described by 
the equation y(x,t)= (5.60 cm)sin[(0.0340 rad/cm)x] sin[(50.0 rad/s )t], where the origin is at the 
left end of the string, the x-axis is along the string and the y-axis is perpendicular to the string. 
  (a) Draw a sketch that shows the standing wave pattern. 
  (b) Find the amplitude of the two travelling waves that make up this standing wave. 
  (c) What is the length of the string? 
  (d) Find the wavelength, frequency, period and speed of the travelling wave.  
  (e) Find the maximum transverse speed of a point on the string. 
  (f) What would be the equation y(x, t) for this string if it were vibrating in its eighth harmonic ? 
Q 21.  A wire with mass 40.0 g is stretched so that its ends are tied down at points 80.0cm apart. The 
wire vibrates in its fundamental mode with frequency 60.0 Hz and with an amplitude at the 
antinodes of 0.300 cm. 
  (a) What is the speed of propagation of transverse wave in the wire ? 
  (b) Compute the tension in the wire. 
  (c) Find the maximum transverse velocity and acceleration of particles in the wire. 
Solutions 
1.  
   
   = 5.66 cm 
2.          
  (a) 
    
    
  (b)  Since reflected waves comes in the same medium, we can say that, 
    
    
  ?  Fraction of power transmitted 
    
3.  
   = 26.46 cm 
    
   = 0.714 rad 
    
4.   Find y 1 and y 2 at given values of x and t and then simply add them to get net value of y. 
5.  
    
  (a) 
   = 0.02 s. 
  (b) At half the time pulse is at other end and it gets a phase difference of ?.  
  Hence the shape is as shown below. 
    
6.   (a) For fixed end. 
    
  By the superposition of these two pulses we will get the resultant.  
  But only to the left of point O, where string is actually present.  
  (b) For free end 
    
7.  
    
8.    y - y 1 + y 2 = (6.0 cm) sin ( ?x) cos (0.6 ?t) 
   Ax = 6.0 cm sin ( ?x)              ...(i) 
  In parts (a), (b) and (c) substitute the given values of x and find the displacement amplitude at 
these locations.  
  (d) At antinodes 
   Ax = maximum = ?
 
6.0 cm 
   
  or              x = 0.5 cm, 1.5 cm, 2.5 cm 
9.    (a) Distance between successive antinodes 
    
  (b) Amax = 2A = (2 ?)
 
cm 
  If x = 0 is taken as node, we shall taken sin equation for Ax 
   Ax = Amax = sin kx  
   = (2 ? cm) sin kx 
   
10.  
    
  Next three frequencies are 2f1, 3f 1 and 4 f 1 
11.  (a)  
   T = 4l
2
f
2 
? 
    
  (b) f 3 = 3f 1 = 3 × 220 = 660 Hz 
12.    Fundamental frequency.                         
    
  Let n th harmonic is the highest frequency then, (58.93)n = 20,000 n =339.38 
  Hence 339 is the highest frequency  
  ?       fmax = (339) (58.93) Hz = 19977 Hz 
13.    Fundamental frequency = (490- 420) Hz 
   f 0 = 70 Hz