Concept explainers
The string shown in Figure P13.5 is driven at a frequency of 5.00 Hz. The amplitude of the motion is A = 12.0 cm, and the wave speed is v = 20.0 m/s. Furthermore, the wave is such that y = 0 at x = 0 and t = 0. Determine (a) the angular frequency and (b) the wave number for this wave. (c) Write an expression for the wave function. Calculate (d) the maximum transverse speed and (e) the maximum transverse acceleration of an element of the string.
Figure P13.5
(a)
The angular frequency of the wave.
Answer to Problem 5P
The angular frequency of the wave is
Explanation of Solution
Write the expression for the frequency of the string.
Here,
Solve equation (I) for
Conclusion:
Substitute
Therefore, the angular frequency of the wave is
(b)
The wave number of the wave.
Answer to Problem 5P
The wave number of the wave is
Explanation of Solution
Write the expression for the wavelength of the wave.
Here,
Solve equation (III) for
Write the expression for the wavelength in terms of speed of the wave.
Conclusion:
Substitute
Substitute
Therefore, The wave number of the wave is
(c)
Expression for the wave function.
Answer to Problem 5P
Expression for the wave function is
Explanation of Solution
The general form of a wave function can be represented as,
Here,
Conclusion:
Using initial conditions, to make this fit,
In this case, taking initial conditions, substitute
Therefore, Expression for the wave function is
(d)
The maximum transverse speed of the wave.
Answer to Problem 5P
The maximum transverse speed of the wave is
Explanation of Solution
Write the expression for the transverse speed.
Differentiate equation (VI) in equation (VII),
Conclusion:
The maximum magnitude is given by,
Substitute
Therefore, the maximum transverse speed of the wave is
(e)
The maximum transverse acceleration of an element of the string.
Answer to Problem 5P
The maximum transverse acceleration of an element of the string.is
Explanation of Solution
Write the expression for the transverse acceleration.
Use equation (VIII) in equation (VII),
Conclusion:
The maximum magnitude is given by,
Substitute
Therefore, The maximum transverse acceleration of an element of the string.is
Want to see more full solutions like this?
Chapter 13 Solutions
Principles of Physics: A Calculus-Based Text
- A harmonic transverse wave function is given by y(x, t) = (0.850 m) sin (15.3x + 10.4t) where all values are in the appropriate SI units. a. What are the propagation speed and direction of the waves travel? b. What are the waves period and wavelength? c. What is the amplitude? d. If the amplitude is doubled, what happens to the speed of the wave?arrow_forwardAs in Figure P18.16, a simple harmonic oscillator is attached to a rope of linear mass density 5.4 102 kg/m, creating a standing transverse wave. There is a 3.6-kg block hanging from the other end of the rope over a pulley. The oscillator has an angular frequency of 43.2 rad/s and an amplitude of 24.6 cm. a. What is the distance between adjacent nodes? b. If the angular frequency of the oscillator doubles, what happens to the distance between adjacent nodes? c. If the mass of the block is doubled instead, what happens to the distance between adjacent nodes? d. If the amplitude of the oscillator is doubled, what happens to the distance between adjacent nodes? FIGURE P18.16arrow_forwardOne end of a clothesline is given a sinusoidal motion with a frequency of 5.0 Hz and an amplitude of 0.010m. At the time t=0, the end has a zero displacement and is moving in the +y direction. The speed of the wave is 10.0 m/s. What is the mave number, the wavelength and the angular frequency?arrow_forward
- A standing wave is the result of superposition of two harmonic waves given by the equations y1 (x, t) = A sin(wt – kæ) and y2(x, t) = A sin(wt + kæ). The angular frequency is w = 37 rad/s and the k = 27 rad/m is the wave number. (a) Give an expression for the amplitude of standing wave. (b) Determine the frequency. (c) Determine the wavelength of the wavearrow_forwardA piano wire with mass 3.50 g and length 75.0 cm is stretched with a tension of 27.0 N. A wave with frequency 100 Hz and amplitude 1.90 mm travels along the wire. Calculate the average power carried by the wave. Express your answer in watts. What happens to the average power if the wave amplitude is halved? Express your answer in watts.arrow_forwardA taut rope is tied to a machine that causes it to oscillate sinusoidally. You take a picture of the rope and see that at that moment there are four complete cycles along 10m. If the oscillator frequency is 20HZ, find: a) The wave number b) The angular frequency c) If at time t = Os, the height of the wave is Om when x = 0m, find the phase shift of the oscillation. d) If at time t = Os, the transverse velocity is 250m / s when x = Om, find the amplitude of the oscillation. e) Write the wave function that describes the behavior of this wave.arrow_forward
- A wave travels along the x-axis in the direction of decreasing x along a cable having a linear mass density of Mu = 50.0 g/cm and under a tension of T = 60.0 N. The frequency of the wave is f = 20.0 Hz. a) Find the wavelength and period of this wave. b) If the amplitude of the wave is 1.5 cm, find the equation for the wave as a function of x and time, t. c) Determine the average power of the wave.arrow_forwardProblem 4: A traveling wave along the x-axis is given by the following wave functionψ(x, t) = 3.6 cos(1.4x - 9.2t + 0.34),where x in meter, t in seconds, and ψ in meters. Read off the appropriate quantities for this wave function and find the following characteristics of this plane wave: Part (a) The amplitude in meters. Part (b) The frequency, in hertz. Part (c) The wavelength in meters. Part (d) The wave speed, in meters per second. Part (e) The phase constant in radians.arrow_forwardThe mathematical model for a wave on a tightly stretched wire is y(x, t) = 0.340 sin 12xt Злх + 4 where x and y are in meters, t is in seconds, and u of the wire is 86.0 g/m. (a) Calculate the average rate energy is conveyed along the wire. Enter a number. (b) What is the energy per cycle of the wave? Enter a number.arrow_forward
- A wave is represented by y=(3.00m)sin(2.5t - 3.14x). Find the (a) amplitude, (b) angular velocity, (c) frequency, (d) wave number, (e) wavelength, and (f) speed of the wave. (Okay, since i've submitted the question already and the letter a, b and c were already answered please help me answer letter d, e , and f) the computed (a) amplitude= 3m , (b) the angular velocity is 2.5 rad/s, (c) the frequency of the wave is 0.398 Hz.. please do check if it's correctarrow_forwardThe string shown in the figure is driven at a frequency of 5.00 Hz. The amplitude of the motion is A = 14.0 cm, and the wave speed is v= 18.0 m/s. Furthermore, the wave is such that y = 0 at x = 0 and t = 0. (a) Determine the angular frequency for this wave (in rad/s). 31.4 rad/s (b) Determine the wave number for this wave (in rad/m). 3.6 This is the wave length not the wave number. rad/m (c) Write an expression for the wave function. (Use the following as necessary: t, x. Let x be in meters and t be in seconds. Do not include units in your answer. Assume SI units.) y = sin (d) Calculate the maximum transverse speed (in m/s). m/sarrow_forwardA 1.88-m long rope has a mass of 0.139 kg. The tension is 55.4 N. An oscillator at one end sends a harmonic wave with an amplitude of 1.09 cm down the rope. The other end of the rope is terminated so all of the energy of the wave is absorbed and none is reflected. What is the frequency of the oscillator if the power transmitted is 118 W?arrow_forward
- Principles of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPhysics for Scientists and Engineers, Technology ...PhysicsISBN:9781305116399Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPhysics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage Learning