Problem 52 - Fundamentals of Physics

Question

A rope, under a tension of \(200 \mathrm{~N}\) and fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the rope is given by $$ y=(0.10 \mathrm{~m})(\sin \pi x / 2) \sin 12 \pi t $$ where \(x=0\) at one end of the rope, \(x\) is in meters, and \(t\) is in seconds. What are (a) the length of the rope, (b) the speed of the waves on the rope, and (c) the mass of the rope? (d) If the rope oscillates in a third-harmonic standing wave pattern, what will be the period of oscillation?

Step-by-Step Solution

Answer

(a) 4 m, (b) 24 m/s, (c) 1.39 kg, (d) 0.056 s

Step 1: Find the Length of the Rope

The displacement equation is given by \(y=(0.10 \, \text{m})(\sin \pi x / 2) \sin 12 \pi t\). This is a second-harmonic standing wave, meaning there are two complete half-wavelengths along the length of the rope. Thus, \(L = 2(\lambda / 2) = \lambda\). The wave number \(k = \frac{\pi}{2}\) relates to the wavelength by \(k = \frac{2\pi}{\lambda}\). Hence, \(\lambda = \frac{2\pi}{k} = 4\). Therefore, the length of the rope is 4 m.

Step 2: Calculate the Speed of the Waves

The angular frequency \(\omega = 12\pi\), and the wave number \(k = \frac{\pi}{2}\). The wave speed \(v\) is given by \(v = \frac{\omega}{k} = \frac{12\pi}{\pi/2} = 24\, \text{m/s}\).

Step 3: Calculate the Mass of the Rope

The speed of the wave is also expressed in terms of tension \(T\) and linear mass density \(\mu\) by \(v = \sqrt{\frac{T}{\mu}}\). Solving for \(\mu\), we have \(\mu = \frac{T}{v^2} = \frac{200}{24^2} = \frac{200}{576} \approx 0.3472\, \text{kg/m}\). The mass \(m\) of the rope is \(m = \mu \times L = 0.3472 \times 4 \approx 1.3888\, \text{kg}\).

Step 4: Determine the Period of Third-Harmonic Oscillation

In the third-harmonic pattern, the frequency becomes 3 times the fundamental frequency. Since the period \(T\) is the inverse of frequency, \(T_{3rd} = \frac{T_{2nd}}{3}\). Given \(T_{2nd} = \frac{1}{6}\) as \(\omega = 12\pi\) gives \(f_{2nd} = \frac{\omega}{2\pi} = 6\), thus \(T_{3rd} = \frac{1}{18}\, \text{s} \approx 0.0556\, \text{s}\).

Key Concepts

Wave Speed

Wave speed is a crucial concept in understanding how waves travel through different mediums. In general, wave speed can be determined by the relationship between the angular frequency and the wave number. The formula for wave speed is given by the equation:

• \( v = \frac{\omega}{k} \)

Here, \( \omega \) is the angular frequency and \( k \) is the wave number. In the given exercise, the angular frequency \( \omega \) is 12π, and the wave number \( k \) is \( \frac{\pi}{2} \). By substituting these values into the formula, students can calculate the wave speed as \( v = \frac{12\pi}{\pi/2} = 24 \text{ m/s} \).
This speed reflects how quickly the wave cycles travel along the rope under the given tension.

Linear Mass Density

Linear mass density is essential for understanding how mass is distributed along a given length of rope. It is often denoted by the symbol \( \mu \) and is measured in kilograms per meter (kg/m).

The relationship between wave speed, tension, and linear mass density is given by the following formula:

• \( v = \sqrt{\frac{T}{\mu}} \)

Where \( T \) is the tension in the rope and \( \mu \) is the linear mass density. To find \( \mu \), rearrange the formula to:

• \( \mu = \frac{T}{v^2} \)

Using the wave speed calculated earlier (24 m/s) and the tension provided (200 N), we can determine \( \mu = \frac{200}{24^2} = \frac{200}{576} \approx 0.3472 \text{ kg/m} \).
This tells us how much mass per meter is distributed along the rope.

Standing Wave

Standing waves occur when two waves of the same frequency and amplitude travel through a medium in opposite directions, creating nodes and antinodes along the medium's length. In our exercise, the rope exhibits a second-harmonic standing wave pattern.

This means the rope has been divided exactly into two complete wavelengths (or a full sine wave and its reflection).

• Nodes are points of no displacement, while antinodes experience maximum displacement.

The number of nodes and antinodes determines the harmonic of the wave, with the second harmonic indicating one additional node and antinode compared to the fundamental frequency.
It's an important concept in understanding how wave patterns behave differently depending on factors like tension and length.

Second Harmonic

The second harmonic, especially in the context of a standing wave, refers to a state where the length of the medium (in this case, the rope) contains exactly two half-wavelengths.
In our exercise problem, this is represented by the displacement equation \( y = (0.10 \text{ m})(\sin \frac{\pi x}{2}) \sin 12\pi t \), which shows the_wave pattern of the rope.
This implies that the rope's length corresponds to exactly one wavelength, as the wavelength \( \lambda \) is determined by:

• \( \lambda = \frac{2\pi}{k} = 4 \text{ m} \)

Where \( k \) is the wave number.
The second harmonic is fundamental for analyzing how the medium vibrates with more complex patterns compared to the first harmonic, contributing to a deeper understanding of wave physics.