Question

A string fixed at both ends is \(8.36 \mathrm{~m}\) long and has a mass of \(122 \mathrm{~g} .\) It is subjected to a tension of \(96.7 \mathrm{~N}\) and set vibrating. (a) What is the speed of the waves in the string? (b) What is the wavelength of the longest possible standing wave? (c) Give the frequency of that wave.

Short answer

a) The speed of the waves in the string is 250 m/s. b) The wavelength of the longest possible standing wave is 16.72 m. c) The frequency of that wave is 14.96 Hz

Step-by-step solution

Calculate the mass per unit length

\(\mu = \frac{m}{l} = \frac{0.122 \mathrm{~kg}}{8.36 \mathrm{~m}} = 0.0146 \mathrm{~kg/m}\)

Calculate the wave speed

\(v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{96.7 \mathrm{~N}}{0.0146 \mathrm{~kg/m}}} = 250 \mathrm{~m/s}\)

Calculate the longest possible standing wave wavelength

\(\lambda = 2L = 2 * 8.36 \mathrm{~m} = 16.72 \mathrm{~m}\)

Calculate the frequency

f = \(\frac{v}{\lambda}\) = \(\frac{250 \mathrm{~m/s}}{16.72 \mathrm{~m}}\) = 14.96 Hz

Key concepts

Wave Speed

Understanding the concept of wave speed is fundamental in physics, especially when it comes to standing waves in strings. Wave speed is defined as the distance a wave travels per unit time, typically measured in meters per second (m/s). It's key to note that the wave speed on a string depends on two critical factors: the tension applied to the string and the linear mass density (mass per unit length) of the string.

In our exercise, we are provided with the mass of the string and its length. By dividing the mass by the length, we derive the linear mass density, denoted by the Greek letter \(\mu\). Once we have this value, we can use the formula \(v = \sqrt{\frac{T}{\mu}}\) to calculate the wave speed \(v\). This formula reveals that the wave speed increases with greater tension and decreases as the string's mass density increases.

For example, the application of a tension force of 96.7 Newtons to a string with a mass density of 0.0146 kg/m results in a high wave speed. Here, this results in a wave speed of approximately 250 m/s, which is indicative of how fast wave disturbances will propagate through the string. Such information can be crucial for a variety of applications, from musical instruments to engineering designs that incorporate vibrating strings.

Wavelength Calculation

The calculation of wavelength is central to understanding the behavior of standing waves in strings. In a standing wave, nodes and antinodes appear at certain points along the string, representing points of no motion and maximum motion, respectively. The wavelength (usually represented by the Greek letter \(\lambda\)) is the distance between two consecutive points that are in phase – for example, from crest to crest or trough to trough.

In the context of standing waves on a string that is fixed at both ends, the longest possible wavelength occurs when there is just one antinode and two nodes (one at each end). This is referred to as the fundamental frequency or the first harmonic. The length of the string is half the wavelength of the fundamental, so to compute the wavelength we use the equation \(\lambda = 2L\), where \(L\) is the length of the string.

For example, an 8.36-meter-long string would have its longest possible standing wave with a wavelength of 16.72 meters, as calculated from doubling the string's length. Knowing this wavelength is crucial when we look into properties such as the standing wave's frequency and the harmonics being produced in the string.

Frequency of Standing Wave

The frequency of a standing wave relates to how often the wave oscillates back and forth over a particular point. For strings, like the one in our example, the frequency also determines the pitch of the sound produced when the string is vibrated. Frequency is inversely related to the wavelength and directly proportional to the wave speed - when one increases, the other decreases, and vice versa.

The basic formula to find the frequency (\(f\)) of a wave is \(f = \frac{v}{\lambda}\), where \(v\) is the wave speed and \(\lambda\) is the wavelength. Following this formula, if we have a wave speed of 250 m/s and a wavelength of 16.72 meters, as in our example, the frequency would be approximately 14.96 Hertz (Hz).

This frequency represents the first harmonic or fundamental frequency, which is the lowest frequency at which the string can support a standing wave. Higher harmonics, or overtones, can be calculated as integer multiples of the fundamental frequency, each corresponding to different standing wave patterns with more nodes and antinodes. Thus, understanding how to calculate the frequency of the fundamental helps in grasping the complex patterns of vibration that can occur in strings with fixed ends.