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A wave on a string is described by \(y(x, t) = A \mathrm{cos}(kx - \omega t)\). (a) Graph \(y, v_y\), and \(a_y\) as functions of \(x\) for time \(t = 0\). (b) Consider the following points on the string: (i) \(x =\) 0; (ii) \(x = \pi/4k\); (iii) \(x = \pi/2k\); (iv) \(x = 3\pi/4k\); (v) \(x = \pi k\); (vi) \(x = 5\pi/4k\); (vii) \(x = 3\pi/2k\); (viii) \(x = 7\pi/4k\). For a particle at each of these points at \(t = 0\), describe in words whether the particle is moving and in what direction, and whether the particle is speeding up, slowing down, or instantaneously not accelerating.

Short Answer

Expert verified
Graph at t=0, describe motion and acceleration of particles at specified x-values on a string wave.

Step by step solution

01

Understand the Wave Equation

The equation of the wave on the string is given by \( y(x, t) = A \cos(kx - \omega t) \). This is a sinusoidal wave with amplitude \( A \), wave number \( k \), and angular frequency \( \omega \). At time \( t = 0 \), the wave equation simplifies to \( y(x, 0) = A \cos(kx) \). This represents the displacement of particles on the string as a function of position \( x \).
02

Calculate Velocity and Acceleration

The velocity of a particle on the string is the time derivative of \( y(x, t) \). Thus, \( v_y(x, t) = \frac{\partial y}{\partial t} = A \omega \sin(kx - \omega t) \). At \( t = 0 \), \( v_y(x, 0) = A \omega \sin(kx) \).The acceleration is the time derivative of velocity: \( a_y(x, t) = \frac{\partial v_y}{\partial t} = -A \omega^2 \cos(kx - \omega t) \). At \( t = 0 \), \( a_y(x, 0) = -A \omega^2 \cos(kx) \).
03

Graph Displacement, Velocity, and Acceleration

Plot \( y(x, 0) = A \cos(kx) \) against \( x \) to display the displacement.Plot \( v_y(x, 0) = A \omega \sin(kx) \) against \( x \) for the velocity.Plot \( a_y(x, 0) = -A \omega^2 \cos(kx) \) against \( x \) for the acceleration.These graphs represent wave displacement, velocity, and acceleration as functions of position.
04

Analyze Specific Points on the String

For each of the specified points, substitute \( x \) into the simplified equations for \( y(x,0) \), \( v_y(x,0) \), and \( a_y(x,0) \):(i) \( x = 0 \) yields \( y = A \), \( v_y = 0 \), \( a_y = -A \omega^2 \).(ii) \( x = \pi/4k \) yields \( y = A/\sqrt{2} \), \( v_y = A \omega/\sqrt{2} \), \( a_y = -A \omega^2/\sqrt{2} \).(iii) \( x = \pi/2k \) yields \( y = 0 \), \( v_y = A \omega \), \( a_y = 0 \).(iv) \( x = 3\pi/4k \) yields \( y = -A/\sqrt{2} \), \( v_y = A \omega/\sqrt{2} \), \( a_y = A \omega^2/\sqrt{2} \).(v) \( x = \pi k \) yields \( y = -A \), \( v_y = 0 \), \( a_y = A \omega^2 \).(vi) \( x = 5\pi/4k \) yields \( y = -A/\sqrt{2} \), \( v_y = -A \omega/\sqrt{2} \), \( a_y = A \omega^2/\sqrt{2} \).(vii) \( x = 3\pi/2k \) yields \( y = 0 \), \( v_y = -A \omega \), \( a_y = 0 \).(viii) \( x = 7\pi/4k \) yields \( y = A/\sqrt{2} \), \( v_y = -A \omega/\sqrt{2} \), \( a_y = -A \omega^2/\sqrt{2} \).Evaluate the movement direction from \( v_y \) and acceleration changes from \( a_y \).
05

Interpret Results for Each Point

- If \( v_y = 0 \) and \( a_y eq 0 \), the particle is instantaneously not moving but accelerating toward or away from the equilibrium.- If \( v_y eq 0 \) and same sign as \( a_y \), the particle is speeding up.- If \( v_y eq 0 \) and opposite sign as \( a_y \), the particle is slowing down.At each \( x \):(i) Stationary, accelerates negatively.(ii) Moving in positive velocity, decreases as acceleration is negative.(iii) Maximum positive speed, no acceleration.(iv) Moving positive, increasing velocity as positive acceleration.(v) Stationary, positively accelerating.(vi) Moving negative, decelerates as acceleration positive.(vii) Maximum negative speed, no acceleration.(viii) Moving negatively, accelerates in negative direction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sinusoidal Wave
Sinusoidal waves are one of the most fundamental types of waves, characterized by smooth periodic oscillations. These waves are described by the equation, such as \( y(x, t) = A \cos(kx - \omega t) \), where:
  • \( A \) is the amplitude of the wave, indicating the maximum displacement from equilibrium.
  • \( k \) represents the wave number, which is related to the wavelength \( \lambda \) by \( k = \frac{2\pi}{\lambda} \).
  • \( \omega \) is the angular frequency, connected to the period \( T \) by \( \omega = \frac{2\pi}{T} \).
At time \( t = 0 \), this equation simplifies to \( y(x, 0) = A \cos(kx) \), showcasing the wave's displacement along the string across different positions \( x \). Sinusoidal waves are essential in understanding wave motion as they capture many real-world scenarios, including sound waves and electromagnetic waves. They exhibit oscillatory motion where particles move up and down in simple harmonic motion while the wave itself travels along the string.
Wave Equation
The wave equation is a mathematical representation of wave behavior, particularly how waves propagate through different mediums. The standard form of the wave equation on a string is \( y(x, t) = A \cos(kx - \omega t) \). It helps describe the displacement \( y \) of particles on the string at any given position \( x \) and time \( t \).
  • The wave equation incorporates key features like amplitude, frequency, and wavelength.
  • It reveals how the wave travels with velocity given by \( v = \frac{\omega}{k} \).
  • The term \( kx - \omega t \) denotes the phase of the wave, which is critical in defining wave interference and wave fronts.
The wave equation illustrates how each point on the string oscillates over time while maintaining a consistent wave shape. It's crucial for predicting how waves behave under various boundary conditions and how they interact when waves intersect.
Particle Motion on String
The motion of individual particles on a string as a wave passes can be analyzed through their position, velocity, and acceleration. For a wave given by \( y(x, t) = A \cos(kx - \omega t) \), the instantaneous velocity \( v_y(x, t) \) and acceleration \( a_y(x, t) \) of a particle are found by differentiating the wave equation with respect to time.
  • The velocity is \( v_y(x, t) = A \omega \sin(kx - \omega t) \) indicating how fast and in what direction a particle is moving at any point.
  • The acceleration is \( a_y(x, t) = -A \omega^2 \cos(kx - \omega t) \), showing how quickly the velocity of a particle is changing.
For each position \( x \), the particle could be moving or stationary, and accelerating positively or negatively, which changes over time. At peaks (where \( y \) is maximal), particles change direction, causing acceleration to be highest. At nodes (where \( y \) is zero), particles have maximum speed but zero acceleration. This understanding helps explain phenomena like standing waves and harmonics in instruments.

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Most popular questions from this chapter

A large rock that weighs 164.0 N is suspended from the lower end of a thin wire that is 3.00 m long. The density of the rock is 3200 kg/m\(^3\). The mass of the wire is small enough that its effect on the tension in the wire can be ignored. The upper end of the wire is held fixed. When the rock is in air, the fundamental frequency for transverse standing waves on the wire is 42.0 Hz. When the rock is totally submerged in a liquid, with the top of the rock just below the surface, the fundamental frequency for the wire is 28.0 Hz. What is the density of the liquid?

In your physics lab, an oscillator is attached to one end of a horizontal string. The other end of the string passes over a frictionless pulley. You suspend a mass \(M\) from the free end of the string, producing tension \(Mg\) in the string. The oscillator produces transverse waves of frequency \(f\) on the string. You don't vary this frequency during the experiment, but you try strings with three different linear mass densities \(\mu\). You also keep a fixed distance between the end of the string where the oscillator is attached and the point where the string is in contact with the pulley's rim. To produce standing waves on the string, you vary \(M\); then you measure the node-to-node distance \(d\) for each standingwave pattern and obtain the following data: (a) Explain why you obtain only certain values of \(d\). (b) Graph \(\mu d^2\) (in kg \(\cdot\) m) versus \(M\) (in kg). Explain why the data plotted this way should fall close to a straight line. (c) Use the slope of the best straight-line fit to the data to determine the frequency \(f\) of the waves produced on the string by the oscillator. Take \(g = 9.80 \, \mathrm{m/s}^2\). (d) For string A (\(\mu = 0.0260\) g/cm), what value of \(M\) (in grams) would be required to produce a standing wave with a node-to-node distance of 24.0 cm? Use the value of \(f\) that you calculated in part (c).

A water wave traveling in a straight line on a lake is described by the equation $$y(x, t) = (\mathrm{2.75 \, cm) cos(0.410 \mathrm{rad/cm} \, \textit{x}} + 6.20 \, \mathrm{rad}/s \; t)$$ where y is the displacement perpendicular to the undisturbed surface of the lake. (a) How much time does it take for one complete wave pattern to go past a fisherman in a boat at anchor, and what horizontal distance does the wave crest travel in that time? (b) What are the wave number and the number of waves per second that pass the fisherman? (c) How fast does a wave crest travel past the fisherman, and what is the maximum speed of his cork floater as the wave causes it to bob up and down?

At a distance of \(7.00 \times 10^{12}\) m from a star, the intensity of the radiation from the star is 15.4 W/m\(^2\). Assuming that the star radiates uniformly in all directions, what is the total power output of the star?

A piano wire with mass 3.00 g and length 80.0 cm is stretched with a tension of 25.0 N. A wave with frequency 120.0 Hz and amplitude 1.6 mm travels along the wire. (a) Calculate the average power carried by the wave. (b) What happens to the average power if the wave amplitude is halved?

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