PHYS 102 Quiz Problems Chapter 16 : Waves I Dr. M. F. Al-Kuhaili
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1 PHYS 102 Quiz Problems Chapter 16 : Waves I Dr. M. F. Al-Kuhaili 1. (TERM 001) A sinusoidal wave traveling in the negative x direction has amplitude of 20.0 cm, a wavelength of 35.0 cm, and a frequency of 12.0 Hz. (a) Find the angular wave number of the wave. (b) Find the angular frequency of the wave. (c) Find the phase velocity of the wave. (d) Write an expression for the wave function y (x,t) {take φ = 0}. 2. (TERM 001) A string of linear density kg/m and length 3.0 m is stretched between two points. It vibrates at 200 Hz. What tension in the string will establish a three-loop standing-wave pattern along the string? 3. (TERM 001) A standing wave is formed by the interference of two traveling waves, each of which has an amplitude y m = π cm, angular wave number k = (π/2) cm -1, and angular frequency ω = 10π rad/s. (a) Calculate the distance between the first two antinodes. (b) What is the amplitude of the standing wave at x = 0.25 cm? 4. (TERM 001) The equation of a certain traveling wave on a string is given by: y (x,t) = (0.3 cm) sin (0.1x t +0.5), where x and y are in centimeters and t is in seconds. For this wave, find the following: (a) The wavelength. (b) The period. (c) The speed. (d) The tension in the string if the mass of the string is 100 g and its length is 250 cm. (e) The power transmitted by the wave. 5. (TERM 002) Write down the equation for a wave traveling in the negative direction along the x axis and having an amplitude of 0.01 m, a frequency of 550 Hz, and a speed of 330 m/s. (Take φ = 0) 6. (TERM 002) The equation of a transverse wave traveling along a string is: y (x,t) = 0.15 sin (0.79 x 13 t), in which x and y are measured in meters and t is in seconds. (a) What is the displacement y at x = 2.3 m, t = 0.16 s? (b) Write the equation of a wave that, when added to the given one, would produce standing waves on the string. (c) What is the displacement of the resultant standing wave at x = 2.3 m, t = 0.16 s? 7. (TERM 002) A string of length 125 cm has a mass of 2.00 g. It is stretched with a tension of 7.00 N between fixed supports. (a) What is the wave speed for this string? (b) What is the lowest resonant frequency for this string? 8. (TERM 002) A string of length 1.50 m has a mass of 8.70 g. It is stretched with a tension of 120 N between fixed supports. (a) What is the wave speed for this string? (b) Calculate the wavelengths of the waves that produce one-loop and two-loop standing waves on the string. (c) Calculate the frequencies of the waves that produce one-loop and two-loop standing waves on the string.
2 9. (TERM 012) A stretched string is 150 cm long and has a linear density of g/cm. What tension in the string will result in a second harmonic with a frequency of 450 Hz? 10. (TERM 012) (a) Write an equation describing a sinusoidal transverse wave traveling on a string in the + x direction with a wavelength of 12 cm, a frequency of 400 Hz, and amplitude of 1.5 cm. (b) What is the maximum speed of a point on the string? (c) What is the speed of the wave? (d) What power is transmitted by the wave (µ = 3.0 g/m)? 11. (TERM 012) A stretched string has a linear density of 7.2 g/m and is under a tension of 130 N. The fixed supports are 96.0 cm apart. The string is oscillating in the standing wave pattern shown in the figure. Calculate the speed, wavelength and frequency of the traveling waves whose interference gives this standing wave? 12. (TERM 021) A sinusoidal wave is traveling to the right along a stretched string of length 2.00 m and mass 10.0 g. The tension in the string is 4.50 N. The frequency of the wave is 50.0 Hz and its amplitude is 2.00 cm. (a) What is the speed of the wave? (c) Write down the equation for the wave. 13. (TERM 021) A string 3.0 long is oscillating as a three-loop standing wave whose amplitude is 1.0 cm. The wave speed is 100 m/s. (a) What is the frequency? (b) Write equations (with numbers) for two waves that, when combined, will result in this standing wave. 14. (TERM 021) Two waves traveling along a string are given by: y 1 (x,t) = 0.04 sin (3x 2t), y 2 (x,t) = 0.04 sin (3x + 2t), (a) What are the speed and wavelength of each wave? (b) Write down the equation for the resultant wave. (c) Find the maximum displacement of the resultant at x = 2.3 cm. 15. (TERM 022) A standing wave having a frequency of 60 Hz is set up on a string whose length is 96 cm and mass is 45 g. What is the tension in the string if the standing wave oscillates in four loops? 16. (TERM 022) A stretched string has a mass per unit length of 5.0 g/cm and is under a tension of 10 N. A sinusoidal wave on this string has amplitude of 0.15 mm and a frequency of 200 Hz and is traveling in the negative direction of x. Write an equation for this wave.
3 17. (TERM 022) A string, of mass 5.0 g and fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the string is given by: y = 0.10 (sin πx/2) (cos 15π t), where x and y are in meters. (a) What is the frequency of the wave? (b) What is the speed of waves on the string? (c) What is the length of the string? (d) What is the tension in the string? 18. (TERM 033) A string, of mass 5.0 g and length 2.5 m is fixed at both ends. It has two adjacent resonant frequencies at 112 and 140 Hz. (a) What is the fundamental frequency of the string? (b) What is the speed of waves on the string? (c) What is the wavelength of the 140-Hz resonance? (d) What is the tension in the string? 19. (TERM 042) A sinusoidal transverse wave is traveling to the left (negative x direction) along a stretched string. The wave has a frequency of 50.0 Hz and amplitude of 3.00 cm. The string has a linear mass density of 50.0 g/m and is stretched to a tension of 5.00 N. (a) Determine the speed of the wave. (c) Write an equation for the wave. (d) What is the average power carried by the wave? 20. (TERM 042) A stretched string has a length of 1.35 m and a mass of 3.38 g. A sinusoidal transverse wave is traveling on the string. It is represented by the following wave function: y (x,t) = sin (6.98 x t), where x and y are in meters, and t is in seconds. (a) Determine the frequency of the wave. (c) What is the wave speed? (d) What is the direction of travel of the wave? (e) What is the tension in the string? 21. (TERM 042) A stretched string, that is fixed at both ends, has a length of 4.00 m and a mass of 5.00 g. A standing wave causes the string to vibrate in six loops with a frequency of 216 Hz. (a) What is the fundamental frequency of the string? (b) What is the speed of waves on the string? (c) What is the tension in the string? (d) What is the distance between two adjacent nodes on the string? 22. (TERM 052) A stretched string is fixed at both ends. It has a mass of 5.80 grams. A standing wave with four loops is established on the string having a frequency of 320 Hz and an antinode vertical displacement of 3.75 cm. The length of each loop is m. a. What is the tension in the string? b. What is the equation of the standing wave? (Write the general form and then substitute the numerical values)
4 23. (TERM 052) A transverse sinusoidal wave is traveling to the right on a string that has a mass per unit length of kg/m and is under a tension of 20 N. The amplitude of the wave is 6.8 cm and its frequency is 100 Hz. a) Calculate the speed of the wave. b) What is the wavelength of the wave? c) Write down the equation of the wave, assuming that y = 0 at t = 0 and x = 0. d) What is the maximum transverse speed of any particle on the string? e) Suppose that another identical wave is traveling in the same direction as the original wave but phase-shifted from it by φ = 0.4π rad. What will be the maximum transverse speed of any particle on the string in this case? 24. (TERM 052) A sinusoidal transverse wave is traveling to the left along a string. The figure below shows the displacement as a function of position at time t = 0. The string tension is 3.6 N and its mass per unit length is kg/m. (a) What is the speed of the wave? (b) What are the amplitude and wavelength of the wave? (c) If the equation of the wave is written as: y( x, t) = ym sin( kx ± ω t + φ). Find k, ω, φ, and the correct sign. 25. (TERM 061) A transverse sinusoidal wave is traveling on a string that has a mass per unit length of kg/m. The equation for the wave is given by: y ( x, t) = 0.050sin(3.0x + 7.5t), where x and y are in meters, and t is in seconds. (a) What is the direction of travel of the wave? (c) What is the frequency of the wave? (d) What is the maximum transverse speed of any particle on the string? (e) What is the tension in the string? 26. (TERM 061) A stretched string that is fixed at both ends has a mass of 5.00 grams and a length of 2.00 m, and is under a tension of 15.7 N. A standing wave pattern is set up on the string such that the string vibrates in the fourth harmonic (n = 4). (a) What is the speed of waves on the string? (b) What is the wavelength of the waves whose interference resulted in the standing wave? (c) What is the angular frequency of the waves whose interference resulted in the standing wave? (d) What is the fundamental frequency of the string? 27. (TERM 061) Two sinusoidal waves traveling in opposite directions interfere to produce a standing wave described by: y ( x, t) = (1.50)(sin 0.400x)(cos200t), where x and y are in meters, and t is in seconds. (a) Determine the wavelength, frequency, and speed of the interfering waves. (b) What is the distance between two adjacent nodes? (c) What is the maximum displacement at the position x = 0.40 m?
5 28. (TERM 062) A transverse sinusoidal wave, of frequency 50 Hz and amplitude 1.0 cm, is traveling to the right on a stretched string whose linear density is kg/m and is under a tension of 1.7 N. At time t = 0, the particle at x = 0 had a displacement y = 0.40 cm. Assume that the general form of the wave is y (x,t) = y m sin (kx ± ωt + φ). a) What is the speed of waves on the string? b) What is the wavelength of the wave? c) What is the angular frequency of the wave? d) What is the angular wave number of the wave? e) What is the phase constant of the wave? f) Write down the equation of this wave. 29. (TERM 062) A string, fixed at both ends, oscillates in a third-harmonic standing wave pattern with a frequency f 3 = 46 Hz. The length of the string is 160 cm and it has a mass per unit length of kg/m. a) What is the speed of waves on the string? b) What is the wavelength of the standing wave? c) What is the tension in the string? 30. (TERM 062) Consider the following sinusoidal waves: y 1 (x,t) = 3.0 sin (2.0 x 4.0 t), y 2 (x,t) = 3.0 sin (2.0 x 4.0 t + π/3), y 3 (x,t) = 5.0 sin (3.0 x 2.0 t), y 4 (x,t) = 5.0 sin (3.0 x t), a) Will the wave resulting form the interference of y 1 and y 2 be a standing or traveling wave? b) Write down the equation for the wave that results from the interference of y 1 and y 2. c) Will the wave resulting form the interference of y 3 and y 4 be a standing or traveling wave? d) Write down the equation for the wave that results from the interference of y 3 and y 4. e) For the standing wave, what is the distance between two adjacent antinodes? 31. (TERM 063) A stretched string is fixed between two supports. It has a linear density µ = kg/m and a length of 0.60 m. The string is observed to form a standing wave with three anti-nodes when driven at a frequency of 420 Hz. a) What is the wavelength of this mode? b) What is the tension in the string? c) What is the frequency of the fifth harmonic of this string? 32. (TERM 063) A string fixed at both ends has a length of 2.5 m and a mass of kg. It has two adjacent resonances at frequencies 112 Hz and 140 Hz. (a) Determine the fundamental frequency of the string. (b) Determine the speed of the waves on the string. (c) What is the tension in the string? (d) Determine the wavelength of the 140 Hz resonance. (e) How far from either end of the string does the first node occur for the 140 Hz resonance?
6 33. (TERM 063) The displacement of a vibrating string versus position along the string is shown in the figure. The wave has a speed of 0.10 m/s. A and B are two points on the string. (a) What is the amplitude of the wave? (c) What is the frequency of the wave? (d) What is the phase difference (in radians) between points A and B? 4 B 2 y (mm) x (cm) 4 A 34. (TERM 071) A transverse sinusoidal wave on a string, with a linear density of µ = kg/m, is described by the following equation: y (x,t) = sin (21.0x 419t), (a) In what direction is the wave traveling? (c) What is frequency of the wave? (d) What is the speed of the wave? (e) What is tension in the string? 35. (TERM 071) Two identical sinusoidal waves with wavelengths of 3.0 m travel in the same direction along the same string. The two waves originate from two points that are a distance d apart, as shown in the figure below. The amplitude of each wave is 5.0 cm, and the amplitude of their resultant is 2.5 cm. (a) What is the phase difference (φ) (in radians) between the two waves? (b) What is the distance d? 36. (TERM 071) Two identical sinusoidal waves are traveling in opposite directions along a stretched string that is fixed at both ends and has a length of 1.0 m. The resulting standing wave is given by the following equation: y(x,t) = (0.15).sin(5π x).cos(315π t), (a) What is the speed of waves on the string? (b) What is the wavelength of the interfering waves? (c) What is the harmonic number (n) of the standing wave? (d) How many nodes are there? (e) What is the distance between two adjacent nodes?
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