Problem 52 - University Physics with Modern Physics

Question

A transverse wave on a rope is given by $$y(x, t)=(0.750 \mathrm{cm}) \cos \pi\left[\left(0.400 \mathrm{cm}^{-1}\right) x+\left(250 \mathrm{s}^{-1}\right) t\right]$$ (a) Find the amplitude, period, frequency, wavelength, and speed of propagation. (b) Sketch the shape of the rope at these values of \(t : 0.0005\) s, 0.0010 s. (c) Is the wave wave traveling in the or -direction? (d) The mass per unit length of the rope is 0.0500 \(\mathrm{kg} / \mathrm{m} .\) Find the tension. (e) Find the average power of this wave.

Step-by-Step Solution

Answer

Amplitude: 0.750 cm, Period: 0.0251 s, Frequency: 39.89 Hz, Wavelength: 15.71 cm, Speed: 6.277 m/s. Wave: Negative x-direction. Tension: 1.97 N. Average Power: 11.05 W.

Step 1: Identify Given Equation Components

The wave equation is given as: \[ y(x, t)=(0.750 \, \text{cm}) \cos \pi\left[\left(0.400 \, \text{cm}^{-1}\right) x+\left(250\, \text{s}^{-1}\right) t\right] \]Identify the amplitude (A), wave number (k), and angular frequency (ω) from the equation:\[ A = 0.750 \, \text{cm} \]\[ k = 0.400 \, \text{cm}^{-1} \]\[ \omega = 250 \, \text{s}^{-1} \]

Step 2: Calculate Amplitude and Period

The amplitude (A) is 0.750 cm.The period (T) is calculated from angular frequency:\[ T = \frac{2\pi}{\omega} = \frac{2\pi}{250 \, \text{s}^{-1}} = \frac{2\pi}{250} \approx 0.0251 \, \text{s} \]

Step 3: Calculate Frequency and Wavelength

The frequency (f) is the reciprocal of the period:\[ f = \frac{1}{T} = \frac{1}{0.0251} \approx 39.89 \, \text{Hz} \]The wavelength (λ) is calculated from the wave number:\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{0.400 \, \text{cm}^{-1}} = 15.71 \, \text{cm} \]

Step 4: Determine Wave Speed

The speed (v) of the wave is the product of frequency and wavelength:\[ v = f \cdot \lambda = 39.89 \, \text{Hz} \times 15.71 \, \text{cm} = 627.7 \, \text{cm/s} \] or 6.277 m/s.

Step 5: Determine Wave Direction

The wave function is \( y(x, t)=A \cos(kx+\omega t) \).Since the equation has a positive angular frequency term \(+\omega t\), the wave is traveling in the negative x-direction.

Step 6: Sketch Wave Shape

Use the given wave function to sketch the wave at \( t = 0.0005 \, \text{s} \) and \( t = 0.0010 \, \text{s} \), noting changes in phase due to time.

Step 7: Calculate Tension in Rope

The speed of a wave on a string is given by:\[ v = \sqrt{\frac{T}{\mu}} \]Given \( v = 6.277 \, \text{m/s} \) and mass per unit length \( \mu = 0.0500 \, \text{kg/m} \), solve for tension (T):\[ T = \mu v^2 = 0.0500 \, \text{kg/m} \times (6.277 \, \text{m/s})^2 \approx 1.97 \, \text{N} \]

Step 8: Calculate Average Power

The average power (P) of the wave is given by:\[ P = \frac{1}{2} \mu \omega^2 A^2 v \]Substitute the values:\[ P = \frac{1}{2} (0.0500 \, \text{kg/m}) (250 \, \text{s}^{-1})^2 (0.0075 \, \text{m})^2 (6.277 \, \text{m/s}) \approx 11.05 \, \text{W} \]

Key Concepts

Wave Equation

The wave equation is a mathematical representation of a wave in motion. For a transverse wave on a rope, the equation typically takes the form: \[ y(x, t) = A \cos(kx + \omega t) \]where:

• \( y(x, t) \) represents the wave displacement at a point \( x \) and time \( t \).
• \( A \) is the amplitude of the wave, indicating the maximum displacement from the equilibrium position.
• \( k \) is the wave number, expressing the number of wave cycles per unit distance, defined as \( k = \frac{2\pi}{\lambda} \).
• \( \omega \) is the angular frequency, showing how many oscillations occur per unit time, given by \( \omega = 2\pi f \).

Together, these components describe both the spatial and temporal aspects of the wave’s motion.
The formula helps in understanding how waves propagate through space and time, serving as a foundational concept in physics.

Wave Parameters

Wave parameters are essential characteristics that define the behavior of a wave. These include amplitude, period, frequency, and wavelength.
**Amplitude** is the height of the wave peak from its rest position. It represents the energy of the wave, where higher amplitude means more energy. In the given problem, the amplitude is 0.750 cm.The **period** (\( T \)) is the time taken for one complete wave cycle to pass a given point. It's calculated as the reciprocal of frequency: \( T = \frac{1}{f} \).
**Frequency** (\( f \)) is how many wave cycles occur in a second, given in hertz (Hz). It is implicated in sound waves as pitch or color in light waves.
**Wavelength** (\( \lambda \)) is the distance between consecutive points of similar phase, such as crest to crest.
By identifying and calculating these parameters, we can gain deeper insights into the specific behavior and properties of the wave in various contexts.

Wave Speed

Wave speed describes how quickly a wave travels through a medium and is dictated by the formula:\[ v = f \cdot \lambda \]This indicates that the speed \( v \) of a wave is the product of its frequency \( f \) and wavelength \( \lambda \). In our exercise, the calculated speed is 6.277 m/s.
Wave speed is relevant in various applications, from sound traveling through air to light in optical fibers. Understanding wave speed helps us predict how quickly information or energy is transmitted in different environments.Moreover, the medium's characteristics, like elasticity and density, significantly affect the speed. For instance, waves typically travel faster in solids compared to liquids, which in turn are faster than gases.

Power of a Wave

The power of a wave is the rate at which energy is transferred by the wave through a medium. It can be quantified using:\[ P = \frac{1}{2} \mu \omega^2 A^2 v \]where:

• \( \mu \) is the mass per unit length of the rope.
• \( \omega \) is the angular frequency.
• \( A \) is the amplitude.
• \( v \) is the wave speed.

In practical terms, wave power is crucial in contexts such as harnessing energy from ocean waves or transmitting electrical signals.
In this exercise, the average power calculated is approximately 11.05 W.
Analyzing wave power allows us to design systems and applications that either maximize energy use or adequately handle the energy transmitted by waves. This concept is fundamental in fields like engineering and environmental science.