Problem 29 - Physics

Question

A string has a linear density of \(8.5 \times 10^{-3} \mathrm{~kg} / \mathrm{m}\) and is under a tension of \(280 \mathrm{~N}\). The string is \(1.8 \mathrm{~m}\) long, is fixed at both ends, and is vibrating in the standing wave pattern shown in the drawing. Determine the (a) speed, (b) wavelength, and (c) frequency of the traveling waves that make up the standing wave.

Step-by-Step Solution

Answer

The wave speed is approximately 181.44 m/s, the wavelength is 3.6 m, and the frequency is about 50.4 Hz.

Step 1: Calculate the Speed of the Wave on the String

The speed of a wave traveling on a string can be found using the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension in the string and \( \mu \) is the linear density. Here, \( T = 280 \ \text{N} \) and \( \mu = 8.5 \times 10^{-3} \ \text{kg/m} \). Substitute these values into the formula:\[v = \sqrt{\frac{280}{8.5 \times 10^{-3}}} = \sqrt{32941.18} \approx 181.44 \ \text{m/s}.\]

Step 2: Determine the Wavelength of the Standing Wave

The length of the string and the mode number determine the wavelength. For a string fixed at both ends, the wavelength depends on the number of antinodes (or nodes) in the standing wave pattern. If the pattern shows a half-wavelength fitting into the string length, the wavelength is twice the length of the string. Here, \( \lambda = 2 \times 1.8 \text{ m} = 3.6 \text{ m}. \)

Step 3: Calculate the Frequency of the Wave

Frequency can be found using the relationship \( f = \frac{v}{\lambda} \). Using the values obtained previously, \( v = 181.44 \ \text{m/s} \) and \( \lambda = 3.6 \ \text{m} \), substitute these values into the formula:\[f = \frac{181.44}{3.6} \approx 50.4 \ \text{Hz}.\]

Key Concepts

Standing Wave

Standing waves are a fascinating phenomenon that occur when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. On a string fixed at both ends, standing waves appear as patterns that seem to stand still. They are characterized by nodes and antinodes. Nodes are points where the string does not move, while antinodes are points where the string moves with maximum amplitude.

For a string to form a standing wave, it must vibrate at certain frequencies, which are called harmonic frequencies. These frequencies depend on the length of the string and the speed of the wave traveling along it. Only specific wavelengths that fit into the length of the string create standing waves, leading to specific frequencies that produce vibrant patterns.

Wavelength

The wavelength is a crucial parameter in understanding waves. It's defined as the distance over which the wave's shape repeats. In standing waves on a string, the wavelength can be determined by examining the number of nodes and antinodes.

• For a string fixed at both ends, only those wavelengths that create whole numbers of half-wavelengths fit into the string's length will produce standing waves.
• In our specific problem, if the string's length fits one-half of the wavelength, then the entire wavelength is twice the string's length.
• So for a string that is 1.8 meters long, the wavelength would be 3.6 meters if there's a single antinode in the middle of the string.

This understanding of wavelength helps in predicting the pattern of the standing wave.

Wave Frequency

Wave frequency refers to how often the particles of a medium oscillate as a wave passes through it. It's typically measured in hertz (Hz), which is the number of cycles per second.

• When it comes to standing waves on strings, frequency is pivotal because it dictates the wave pattern that forms.
• It is derived from the wave speed and the wavelength using the formula: \( f = \frac{v}{\lambda} \).
• In our exercise, with a wave speed of approximately 181.44 m/s and a wavelength of 3.6 meters, the wave frequency is calculated to be approximately 50.4 Hz.

This relationship highlights how the speed and the wavelength directly influence the frequency of a wave.

Tension in String

Tension in the string is a force that is exerted along the length of the string and plays a significant role in wave properties. It heavily influences the speed of waves traveling through the string.

• The greater the tension, the faster the wave can travel.
• In mathematical terms, this relationship is captured by the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) represents tension and \( \mu \) stands for the linear mass density of the string.
• For instance, in our example where the tension in the string is 280 N and the linear density is \(8.5 \times 10^{-3} \text{kg/m}\), the wave speed is calculated as approximately 181.44 m/s.

This formula emphasizes tension as a controlling factor in how quickly signals travel down a string.