16 - Wave Motion

0 ms at t 0 a particle on the string at problems x 0

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Unformatted text preview: at 515 Problems x 0 has a displacement of 2.00 cm and travels downward with a speed of 2.00 m/s. (a) What is the amplitude of the wave? (b) What is the initial phase angle? (c) What is the maximum transverse speed of the string? (d) Write the wave function for the wave. 33. A sinusoidal wave of wavelength 2.00 m and amplitude 0.100 m travels on a string with a speed of 1.00 m/s to the right. Initially, the left end of the string is at the origin. Find (a) the frequency and angular frequency, (b) the angular wave number, and (c) the wave function for this wave. Determine the equation of motion for (d) the left end of the string and (e) the point on the string at x 1.50 m to the right of the left end. (f) What is the maximum speed of any point on the string? 34. A sinusoidal wave on a string is described by the equation y (0.51 cm) sin(kx t) where k 3.10 rad/cm and 9.30 rad/s. How far does a wave crest move in 10.0 s? Does it move in the positive or negative x direction? 35. A wave is described by y (2.00 cm) sin(kx t ), where k 2.11 rad/m, 3.62 rad/s, x is in meters, and t is in seconds. Determine the amplitude, wavelength, frequency, and speed of the wave. 36. A transverse traveling wave on a taut wire has an amplitude of 0.200 mm and a frequency of 500 Hz. It travels with a speed of 196 m/s. (a) Write an equation in SI units of the form y A sin(kx t ) for this wave. (b) The mass per unit length of this wire is 4.10 g/m. Find the tension in the wire. 37. A wave on a string is described by the wave function y (0.100 m) sin(0.50x 20t) (a) Show that a particle in the string at x 2.00 m executes simple harmonic motion. (b) Determine the frequency of oscillation of this particular point. Section 16.8 Rate of Energy Transfer by Sinusoidal Waves on Strings 38. A taut rope has a mass of 0.180 kg and a length of 3.60 m. What power must be supplied to the rope to generate sinusoidal waves having an amplitude of 0.100 m and a wavelength of 0.500 m and traveling with a speed of 30.0 m/s? 39. A two-dimensional water wave spreads in circular wave fronts. Show that the amplitude A at a distance r from the initial disturbance is proportional to 1/√r. (Hint: Consider the energy carried by one outward-moving ripple.) 40. Transverse waves are being generated on a rope under constant tension. By what factor is the required power increased or decreased if (a) the length of the rope is doubled and the angular frequency remains constant, (b) the amplitude is doubled and the angular fre- WEB quency is halved, (c) both the wavelength and the amplitude are doubled, and (d) both the length of the rope and the wavelength are halved? 41. Sinusoidal waves 5.00 cm in amplitude are to be transmitted along a string that has a linear mass density of 4.00 10 2 kg/m. If the source can deliver a maximum power of 300 W and the string is under a tension of 100 N, what is the highest vibrational frequency at which the source can operate? 42. It is found that a 6.00-m segment of a long string contains four complete waves and has a mass of 180 g. The string is vibrating sinusoidally with a frequency of 50.0 Hz and a peak-to-valley displacement of 15.0 cm. (The “peak-to-valley” distance is the vertical distance from the farthest positive displacement to the farthest negative displacement.) (a) Write the function that describes this wave traveling in the positive x direction. (b) Determine the power being supplied to the string. 43. A sinusoidal wave on a string is described by the equation y (0.15 m) sin(0.80x 50t ) where x and y are in meters and t is in seconds. If the mass per unit length of this string is 12.0 g/m, determine (a) the speed of the wave, (b) the wavelength, (c) the frequency, and (d) the power transmitted to the wave. 44. A horizontal string can transmit a maximum power of (without breaking) if a wave with amplitude A and angular frequency is traveling along it. To increase this maximum power, a student folds the string and uses the “double string” as a transmitter. Determine the maximum power that can be transmitted along the “double string,” supposing that the tension is constant. (Optional) Section 16.9 The Linear Wave Equation 45. (a) Evaluate A in the scalar equality (7 3)4 A. (b) Evaluate A, B, and C in the vector equality 7.00 i 3.00 k A i B j C k. Explain how you arrive at your answers. (c) The functional equality or identity A B cos(Cx Dt E) (7.00 mm) cos(3x 4t 2) is true for all values of the variables x and t, which are measured in meters and in seconds, respectively. Evaluate the constants A, B, C, D, and E. Explain how you arrive at your answers. 46. Show that the wave function y e b(x vt ) is a solution of the wave equation (Eq. 16.26), where b is a constant. 47. Show that the wave function y ln[b(x vt )] is a solution to Equation 16.26, where b is a constant. 48. (a) Show that the function y(x, t ) x 2 v 2t 2 is a solution to the wave equation. (b) Show that the function above can be written as f (x vt ) g(x vt ), and determine the functional forms for f and g. (c) Repeat parts (a) and (b...
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This note was uploaded on 03/24/2010 for the course PHYSICS 2202 taught by Professor Mihalisin during the Spring '09 term at Temple.

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