Problem 84 - Fundamentals of Physics

Question

Oscillation of a \(600 \mathrm{~Hz}\) tuning fork sets up standing waves in a string clamped at both ends. The wave speed for the string is \(400 \mathrm{~m} / \mathrm{s}\). The standing wave has four loops and an amplitude of \(2.0 \mathrm{~mm}\). (a) What is the length of the string? (b) Write an equation for the displacement of the string as a function of position and time.

Step-by-Step Solution

Answer

(a) The length of the string is 1.333 m. (b) Displacement equation: \(u(x,t) = 2.0 \times 10^{-3} \sin(3\pi x) \cos(1200\pi t)\).

Step 1: Calculate the wavelength

A standing wave with four loops indicates there are two complete wavelengths along the string, since each loop represents a half-wavelength: \ \[ n = 4 \text{ loops} = \frac{4 \times \lambda}{2} = 2\lambda \] \ The frequency of oscillation \( f \) is given as \(600\, \mathrm{Hz}\) and the wave speed \( v \) is \(400\, \mathrm{m/s}\). The wavelength \( \lambda \) can be calculated using the formula \( v = f \cdot \lambda \): \ \[ \lambda = \frac{v}{f} = \frac{400}{600} = \frac{2}{3} \mathrm{~m}\]

Step 2: Determine the length of the string

As we derived in Step 1, the length of the string is \(2\lambda\) since there are four half-wavelength loops making two full wavelengths. So, \ \[ L = 2\lambda = 2 \times \frac{2}{3} = \frac{4}{3} \mathrm{~m} = 1.333 \mathrm{~m} \] \ The length \( L \) of the string is \(1.333 \, \mathrm{m}\).

Step 3: Write the displacement equation for the string

The displacement equation of a standing wave can be written as: \ \[ u(x,t) = A \sin\left(\frac{n\pi}{L}x\right)\cos(\omega t) \] \ where \(A = 2.0 \times 10^{-3} \mathrm{~m}\) is the amplitude, \(n = 4\) is the number of loops, \(L\) is the length of the string, and \(\omega\) is the angular frequency. First, we calculate the angular frequency \(\omega\): \ \[ \omega = 2\pi f = 2\pi \times 600 = 1200\pi \mathrm{~rad/s} \] \ Substituting these values into the equation: \ \[ u(x,t) = 2.0 \times 10^{-3} \sin\left(\frac{4 \pi x}{1.333}\right)\cos(1200\pi t) \] \(u(x,t) = 2.0 \times 10^{-3} \sin\left(3\pi x\right) \cos(1200\pi t)\)

Key Concepts

Wave Speed

Wave speed is a crucial concept when dealing with standing waves. It defines how fast the wave propagates along the medium. In this exercise, the wave speed is given as \(400\, \mathrm{m/s}\). It is the rate at which the wave travels through the string, thus determining how quickly points on the wave move.Wave speed \(v\) can generally be determined using the formula:

• \(v = f \cdot \lambda\)

where \(f\) is the frequency of the wave, and \(\lambda\) is its wavelength. This relationship shows how frequency and wavelength interplay to define the wave speed. If either the frequency or the wavelength changes, the speed is adjusted accordingly.In situations involving standing waves, like in this exercise, analyzing the wave speed helps us solve for other parameters like wavelength and string length, as these waves result in specific patterns due to fixed endpoints.

Wavelength Calculation

To understand the behavior of standing waves, calculating the wavelength is vital. From the original problem, the standing wave formed on the string has four loops, indicating a specific wavelength characteristic.Each loop in a standing wave represents a half-wavelength \((\frac{\lambda}{2})\). So, with four loops:

• \(n = 4 \text{ loops} = 2\lambda\)

Using the wave speed \(v = 400 \mathrm{~m/s}\) and the frequency \(f = 600\, \mathrm{Hz}\), we can find the wavelength using the formula \(v = f \cdot \lambda\):\[ \lambda = \frac{v}{f} = \frac{400}{600} = \frac{2}{3} \mathrm{~m} \]This calculated wavelength allows us to find more characteristics about the standing waves, such as the length of the string. Each time there's a change in frequency or wave speed, the wavelength will adjust, affecting the entire setup of the standing wave.

Displacement Equation

A displacement equation offers a mathematical expression of how each point on a wave moves over time. For standing waves, where nodes and antinodes form fixed points on a string, this equation can provide detailed insights.The displacement equation for this standing wave is:\[ u(x,t) = A \sin\left(\frac{n\pi}{L}x\right)\cos(\omega t) \]Here's a breakdown of each component:

• \(A = 2.0 \times 10^{-3} \mathrm{~m}\): Amplitude, the maximum displacement.
• \(n = 4\): Number of loops or antinodes in this problem.
• \(L\): The length of the string.
• \(\omega\): Angular frequency.

This equation provides a clear depiction of the wave's behavior across the string's length \(x\) and time \(t\). It's essential for predicting how the wave will move and at what points it reaches maximum displacement. As seen, changing any parameter will directly influence how the wave propagates.

Angular Frequency

Angular frequency is a measure of how fast the oscillations occur in a wave, relating to the number of cycles the wave completes per unit time. It's crucial for defining the wave's dynamics as it determines how rapidly points on a wave undergo their oscillations.It is denoted by \(\omega\) and calculated using:\[ \omega = 2\pi f \]For the tuning fork in our exercise, with a frequency \(f = 600 \mathrm{~Hz}\), the angular frequency \(\omega\) is:\[ \omega = 2\pi \times 600 = 1200\pi \mathrm{~rad/s} \]This calculation integrates into the displacement equation, influencing the wave pattern over time. Angular frequency links seamlessly with the concepts of frequency and period of waves, thus offering an angle-based perspective on wave oscillations. It plays a vital role in understanding the time dynamics of standing waves and their inherent oscillatory nature.