Problem 28 - University Physics with Modern Physics

Question

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is \(y(x, t)=\) 2.30 \(\mathrm{mm} \cos [(6.98 \mathrm{rad} / \mathrm{m}) x+(742 \mathrm{rad} / \mathrm{s}) t] .\) Being more practical, you measure the rope to have a length of 1.35 \(\mathrm{m}\) and a mass of 0.00338 kg. You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength; (d) wave speed; (e) direction the wave is traveling; (f) tension in the rope; (g) average power transmitted by the wave.

Step-by-Step Solution

Answer

(a) 2.30 mm, (b) 118.1 Hz, (c) 0.900 m, (d) 106.3 m/s, (e) negative x-direction, (f) 28.2 N, (g) 0.432 W.

Step 1: Determine the amplitude

The wave function is given in the form \(y(x, t) = A \, \cos(kx + \omega t)\). Here, \(A\) represents the amplitude.From the given wave function, \(y(x, t) = 2.30 \, \text{mm} \, \cos((6.98 \, \text{rad/m})x + (742 \, \text{rad/s})t)\).So, the amplitude \(A = 2.30 \text{ mm}\).

Step 2: Calculate the angular frequency

The angular frequency \(\omega\) is given in the wave function as \(742 \, \text{rad/s}\). To find the frequency \(f\), we use the relation:\[f = \frac{\omega}{2\pi}\]Substituting the known value of \(\omega\):\[f = \frac{742}{2\pi}\approx 118.1 \, \text{Hz}\].

Step 3: Calculate the wavelength

The wavenumber \(k\) is given as \(6.98 \, \text{rad/m}\). The relationship between the wavenumber and the wavelength \(\lambda\) is:\[\lambda = \frac{2\pi}{k}\]Substitute in the given value:\[\lambda = \frac{2\pi}{6.98} \approx 0.900 \, \text{m}\].

Step 4: Calculate the wave speed

Wave speed \(v\) can be found using the relationship between frequency \(f\) and wavelength \(\lambda\):\[v = f \cdot \lambda\]Substitute the known values:\[v = 118.1 \, \text{Hz} \cdot 0.900 \, \text{m} \approx 106.3 \, \text{m/s}\].

Step 5: Determine the direction of the wave

The wave function is \(y(x, t) = 2.30 \, \text{mm} \, \cos((6.98 \, \text{rad/m})x + (742 \, \text{rad/s})t)\). The positive sign in front of \(kx + \omega t\) shows that the wave is traveling in the negative x-direction.

Step 6: Calculate the tension in the rope

Using the formula for wave speed on a string, \(v = \sqrt{\frac{T}{\mu}}\), where \(T\) is the tension and \(\mu\) is the linear mass density.First, calculate linear mass density \(\mu\):\[\mu = \frac{\text{Mass}}{\text{Length}} = \frac{0.00338 \, \text{kg}}{1.35 \, \text{m}} = 0.00250 \, \text{kg/m}\]Now solve for \(T\):\[T = v^2 \cdot \mu = (106.3 \, \text{m/s})^2 \cdot 0.00250 \, \text{kg/m} \approx 28.2 \, \text{N}\].

Step 7: Calculate the average power transmitted by the wave

The average power \(P\) transmitted by a wave on a string is given by:\[P = \frac{1}{2} \mu \omega^2 A^2 v\]Substitute for \(\mu = 0.00250 \, \text{kg/m}\), \(\omega = 742 \, \text{rad/s}\), \(A = 2.30 \, \times 10^{-3} \, \text{m}\), and \(v = 106.3 \, \text{m/s}\):\[P = \frac{1}{2}(0.00250)(742)^2(2.30 \times 10^{-3})^2(106.3) \approx 0.432 \, \text{W}\].

Key Concepts

Wave Function

The wave function describes the oscillations of a wave as it travels through space. In the given problem, the wave function can be expressed as \( y(x, t) = 2.30 \, \text{mm} \, \cos((6.98 \, \text{rad/m})x + (742 \, \text{rad/s})t) \). This represents a traveling wave on a rope where:

• \(2.30\; \text{mm}\) is the amplitude, indicating the maximum displacement of the wave.
• \((6.98\, \text{rad/m})\) is the wavenumber, related directly to the wavelength.
• \((742\, \text{rad/s})\) is the angular frequency, connecting to the wave's frequency.

The wave function is pivotal in wave mechanics. It helps predict the position of wave particles at any given time \(t\) and position \(x\). Understanding each component of the wave function provides insight into the wave's characteristics and behavior.

Wave Speed

Wave speed describes how fast waves travel through a medium. It's calculated using the formula \( v = f \cdot \lambda \), where \(v\) represents wave speed, \(f\) is frequency, and \(\lambda\) is wavelength. In this exercise, the wave speed was calculated as 106.3 m/s. This means every second, the wave crests travel 106.3 meters along the rope. Wave speed can vary based on medium type and characteristics. For instance, tension in the rope and wave properties influence speed significantly. A crucial concept in both theoretical and practical aspects of wave mechanics.

Wave Amplitude

Amplitude refers to the wave's maximum displacement from its resting position. In this scenario, the amplitude \(A\) is \(2.30\, \text{mm}\). Amplitude helps to gauge the energy carried by the wave. Larger amplitudes mean higher energy transmission. For waves on a rope, amplitude signifies the extent of oscillation, affecting how much kinetic energy the particles of the rope have at any moment. At rest, the rope would lie flat, but as the wave passes, it oscillates up and down, reaching new crests and troughs defined by this amplitude value.

Frequency

Frequency \(f\) measures how many wave cycles pass through a point every second. It's expressed in hertz (Hz), where 1 Hz equals one cycle per second. In the exercise, the frequency was calculated using the angular frequency \(\omega\) with the formula \(f = \frac{\omega}{2\pi}\), resulting in approximately \(118.1 \, \text{Hz}\). Higher frequencies imply rapid oscillations, impacting how energy is perceived or transmitted through the wave. Frequency also influences sound waves heard by humans where higher frequencies produce higher pitches. In mechanical waves like the one on a rope, frequency remains constant, influenced by the source of the vibrations.

Wavelength

Wavelength \(\lambda\) is the physical length between two consecutive crests or troughs of a wave. It's inversely related to the wavenumber \(k\) by the equation \(\lambda = \frac{2\pi}{k}\). In this problem, the wavelength was calculated to be approximately \(0.900\, \text{m}\). Wavelength reflects the spatial period of a wave, influencing how it interacts with obstacles and boundaries. Longer wavelengths can bend around objects more effectively, leading to constructive or destructive interference. Understanding wavelength is crucial for analyzing wave behavior, including diffraction patterns and resonance on various mediums.

Tension in the Rope

Tension \(T\) in the rope affects wave propagation. It's a force that keeps the rope stretched, influencing the wave speed. Calculating tension involves the wave speed formula \(v = \sqrt{\frac{T}{\mu}}\). Solving for \(T\), after determining linear mass density \(\mu = \frac{\text{mass}}{\text{length}}\), we arrive at \(T = v^2 \cdot \mu\). In the provided exercise, the tension was found to be approximately \(28.2\, \text{N}\). This tension value ensures the wave remains consistent as it travels through the rope, affecting how the wave energy is transmitted without damping. Higher tension typically means faster wave speeds, translating into more robust and potent wave motion on the string.

Power Transmitted by Wave

The power transmitted by a wave indicates how much energy moves along the rope per unit time. The formula to find average power \(P\) is \(P = \frac{1}{2} \mu \omega^2 A^2 v\), considering factors like linear mass density, angular frequency, wave amplitude, and wave speed. For this exercise, the power was calculated to be around \(0.432\, \text{W}\). The average power reflects the capability of the wave to perform work, such as moving or deforming objects it interacts with. A higher average power suggests a greater energy transfer rate, which could be crucial in engineering applications and energy computations in physical systems. Understanding wave power is vital for managing energy resources in mechanical systems and ensuring energy efficiency.