Problem 14 - University Physics with Modern Physics

Question

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.

Step-by-Step Solution

Answer

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

Step 1: 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 \( x \).

Step 2: 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 \( t = 0 \), \( a_y(x, 0) = -A \omega^2 \cos(kx) \).

Step 3: 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.

Step 4: 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 \( a_y \).

Step 5: 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.

Key Concepts

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.