Problem 36 - University Physics with Modern Physics

Question

A sinusoidal wave on a string is described by the equation \(y=(0.100 \mathrm{~m}) \sin (0.75 x-40 t),\) where \(x\) and \(y\) are in meters and \(t\) is in seconds. If the linear mass density of the string is \(10 \mathrm{~g} / \mathrm{m}\), determine (a) the phase constant, (b) the phase of the wave at \(x=2.00 \mathrm{~cm}\) and \(t=0.100 \mathrm{~s}\) (c) the speed of the wave, (d) the wavelength, (e) the frequency, and (f) the power transmitted by the wave.

Step-by-Step Solution

Answer

Answer: The phase constant is (a) ϕ = 0. The phase of the wave at x = 2.00 cm and t = 0.100 s is (b) Θ = -3.97 radians. The speed of the wave is (c) v = 53.33 m/s. The wavelength is (d) λ = 8.377 m. The frequency is (e) f = 6.366 Hz. The power transmitted by the wave is (f) P = 42.22 W.

Step 1: Identify the given parameters from the wave equation

The wave equation is given as \(y(x, t) = (0.100 m) \sin (0.75 x - 40 t)\). From this equation, we can identify the given parameters: Amplitude (A) = 0.100 m Wave number (k) = 0.75 rad/m Angular frequency (ω) = 40 rad/s

Step 2: Calculate the phase constant (a)

The phase constant (ϕ) is not explicitly given in the equation. However, we can observe that the equation is in the standard form \(y(x, t) = A\sin(kx - \omega t + \phi)\), and the phase constant is 0. Therefore, the phase constant is: \(\phi = 0\).

Step 3: Calculate the phase of the wave (b)

To find the phase of the wave at a given position (\(x = 2.00 cm\)) and time (\(t = 0.100 s\), we can use the equation \(\Theta = kx-\omega t+\phi\). Thus: \(\Theta = (0.75)(0.02) - (40)(0.100) + 0\) \(\Theta = -3.97\) radians

Step 4: Calculate the speed of the wave (c)

To find the speed of the wave (v), we can use the relationship \(v=\frac{\omega}{k}\). Thus: \(v=\frac{40 \text{ rad/s}}{0.75\text{ rad/m}}=53.33 \frac{\text{m}}{\text{s}}\)

Step 5: Calculate the wavelength (d)

We can find the wavelength (λ) by using the relationship \(\lambda = \frac{2\pi}{k}\). Thus: \(\lambda = \frac{2\pi}{0.75\text{ rad/m}}=8.377 \text{ m}\).

Step 6: Calculate the frequency (e)

We can find the frequency (f) by using the relationship \(f=\frac{\omega}{2\pi}\). Thus: \(f = \frac{40\text{ rad/s}}{2\pi}=6.366 \text{ Hz}\).

Step 7: Calculate the power transmitted by the wave (f)

Finally, we can find the power transmitted by the wave (P) using the relationship \(P=\frac{1}{2} \rho v \omega^2A^2\). The linear mass density of the string is given as \(\text{10 g/m}=\text{0.01 kg/m}\). Thus: \(P = \frac{1}{2}(0.01 \frac{\text{kg}}{\text{m}})(53.33 \frac{\text{m}}{\text{s}})(40 \text{ rad/s})^2(0.100 \text{ m})^2\) \(P = 42.22 \text{ W}\). So, the required values are: (a) \(\phi = 0\), (b) \(\Theta = -3.97\) radians, (c) \(v = 53.33 \frac{\text{m}}{\text{s}}\), (d) \(\lambda = 8.377 \text{ m}\), (e) \(f = 6.366 \text{ Hz}\), and (f) \(P = 42.22 \text{ W}\).

Key Concepts

Sinusoidal Wave

Sinusoidal waves are one of the most fundamental types of waveforms in physics, representing a smooth, periodic oscillation. Think of them akin to the waves seen in water or the oscillation of a swinging pendulum. Defined by a mathematical function resembling a sine wave, they help describe oscillatory behaviors.

These waves are often used to model periodic phenomena such as sound waves and light waves, making them crucial in fields like acoustics, optics, and electromagnetism.

• Their characteristic "hill" and "valley" shape makes their mathematical properties predictable and reliable.
• Understanding sinusoidal waves involves parameters like amplitude, frequency, and phase.
• Sine and cosine functions can interchangeably describe sinusoidal waves, depending on the phase shift.

By mastering sinusoidal waves, you'll gain a better understanding of how waves behave and interact in various media.

Wave Equation

The wave equation is a mathematical representation that describes the propagation of waves through a medium. It indicates how physical quantities related to the wave, like displacement of a wave at a particular point, change over time and space.

For sinusoidal waves, the equation is typically written as:\[ y(x, t) = A \, \sin(kx - \omega t + \phi) \]where:

• \(y\) is the displacement of the wave at position \(x\) and time \(t\).
• \(A\) is the amplitude, representing the wave's maximum displacement.
• \(k\) is the wave number, which relates to the number of oscillations per unit distance.
• \(\omega\) is the angular frequency, which relates to how many oscillations occur per unit time.
• \(\phi\) is the phase constant, allowing for horizontal shifts of the wave.

The wave equation provides the detailed description necessary to solve wave-related problems, such as determining wave speed or calculating energy transmission.

Linear Mass Density

Linear mass density is a measure of a string's mass per unit length, vital in understanding wave mechanics on strings. The relationship between the tension in a string and its linear mass density is crucial for calculating the wave speed.

Mathematically, linear mass density is expressed as:\[ \mu = \frac{m}{L} \]where:

• \(\mu\) is the linear mass density, measured in kg/m.
• \(m\) is the mass of the string.
• \(L\) is the length of the string.

Knowing the linear mass density helps in evaluating how massive a wave will be and provides insights into its energy transmission characteristics. It plays an essential role in systems like musical instruments where string tension and mass affect vibration.

Wave Speed

Wave speed is a crucial parameter in wave mechanics, representing how fast the wave propagates through the medium. Understanding wave speed clarifies how quickly wave patterns and associated energy are transferred from one point to another.

For a wave traveling through a medium like a string, the speed \(v\) can be calculated using the formula:\[ v = \frac{\omega}{k} \]This formula connects angular frequency \(\omega\) and wave number \(k\), allowing simplified calculation based on known wave properties.

• Wave speed is influenced by the medium's properties, such as tension and linear mass density in strings.
• In non-dispersive media, wave speed remains constant across wavelengths, enhancing predictability.
• Different physical settings may alter the wave speed, requiring recalculations based on the environment.

Understanding wave speed is foundational for applications such as signal transmission, where delays or speed changes impact performance.

Wavelength

Wavelength is the spatial period of a sine wave—the distance over which the wave's shape repeats. It's an essential concept in describing waves, determining how far wave crests (or troughs) are apart.

For sinusoidal waves, wavelength \(\lambda\) is inversely related to the wave number \(k\):\[ \lambda = \frac{2\pi}{k} \]Wavelength provides insight into various wave properties and interactions:

• Shorter wavelengths typically associate with higher energy and frequency waves.
• Wavelength influences phenomena like diffraction, interference, and resonance.
• Knowing the wavelength helps in comprehending wave propagation in different media.

By understanding wavelength, you can better grasp how different frequencies and energies interact in physical and applied sciences.