Question

A \(50.0-\mathrm{cm}\) -long wire with a mass of \(10.0 \mathrm{~g}\) is under a tension of \(50.0 \mathrm{~N}\) Both ends of the wire are held rigidly while it is plucked. a) What is the speed of the waves on the wire? b) What is the fundamental frequency of the standing wave? c) What is the frequency of the third harmonic?

Short answer

The speed of the waves on the wire is 158.1 m/s, the fundamental frequency of the standing wave is 158.1 Hz, and the frequency of the third harmonic is 474.3 Hz.

Step-by-step solution

Calculate the linear mass density of the wire.

To find the speed of the waves on the wire and the fundamental frequency, we first need to calculate the linear mass density of the wire. The formula for linear mass density is: \(\mu = \frac{m}{L}\) Given the mass \(m = 10.0 \mathrm{~g}\) and length \(L = 50.0\mathrm{~cm}\), we can find the linear mass density: \(\mu = \frac{10.0\mathrm{~g}}{50.0\mathrm{~cm}} = 0.2 \mathrm{~g/cm}\) We can convert the linear mass density to SI units, i.e., \(\mathrm{kg/m}\): \(\mu = 0.2 \mathrm{~g/cm} \times \frac{1 \mathrm{~kg}}{1000 \mathrm{~g}} \times \frac{100 \mathrm{~cm}}{1 \mathrm{~m}} = 0.002\mathrm{~kg/m}\)

Calculate the wave speed.

Now we can calculate the wave speed using the formula: \(v = \sqrt{\frac{T}{\mu}}\) Given the tension \(T = 50.0\mathrm{~N}\), we can find the wave speed: \(v = \sqrt{\frac{50.0\mathrm{~N}}{0.002\mathrm{~kg/m}}} = \sqrt{25000\mathrm{~m^2/s^2}} = 158.1\mathrm{~m/s}\)

Calculate the fundamental frequency.

We can now calculate the fundamental frequency using the formula: \(f_1 = \frac{v}{2L}\) To use SI units, we must convert the length \(L\) to meters: \(L = 50.0 \mathrm{~cm} \times \frac{1 \mathrm{~m}}{100 \mathrm{~cm}} = 0.5\mathrm{~m}\). Thus, the fundamental frequency is: \(f_1 = \frac{158.1\mathrm{~m/s}}{2\times 0.5\mathrm{~m}} = 158.1\mathrm{~Hz}\)

Calculate the frequency of the third harmonic.

We can find the frequency of the third harmonic using the formula: \(f_n = nf_1\) For the third harmonic, \(n = 3\). Thus, the frequency of the third harmonic is: \(f_3 = 3 \times 158.1\mathrm{~Hz} = 474.3\mathrm{~Hz}\) The results are: a) The speed of the waves on the wire is \(158.1\mathrm{~m/s}\). b) The fundamental frequency of the standing wave is \(158.1\mathrm{~Hz}\). c) The frequency of the third harmonic is \(474.3\mathrm{~Hz}\).

Key concepts

Linear Mass Density

In the study of wave mechanics, particularly when analyzing strings and wires that transmit vibrations, linear mass density is a fundamental property.

It is the measure of mass per unit length of a given string or wire and is represented by the symbol \(\mu\). To find the linear mass density, one divides the mass \(m\) of the object by its total length \(L\), as shown in the equation \(\mu = \frac{m}{L}\). Understanding linear mass density is essential because it directly affects the speed at which waves can travel through the material. In practice, units of grams per centimeter (g/cm) are often converted to kilograms per meter (kg/m) to adhere to the International System of Units (SI).

Higher linear mass density indicates a more massive string for the same length, which generally leads to slower wave propagation due to the greater inertia that must be overcome for the string to vibrate.

Wave Speed

The speed at which waves travel along a string, known as the wave speed, depends on the tension in the string and its linear mass density. The relationship is given by the equation \(v = \sqrt{\frac{T}{\mu}}\), where \(v\) is the wave speed, \(T\) is the tension, and \(\mu\) is the linear mass density.

Tension, measured in Newtons (N), contributes to the stiffness of the string, and a higher tension usually means higher wave speed since the string is more resistant to deformation. Conversely, heavier strings, with greater linear mass density, tend to be slower in transmitting vibrations. It's important to understand that the wave speed is not influenced by the wave's amplitude or frequency but is fundamentally set by the physical properties of the string or wire.

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a string or wire can vibrate to form a standing wave. It represents the first harmonic and is given by the formula \(f_1 = \frac{v}{2L}\), where \(f_1\) is the fundamental frequency, \(v\) is the wave speed, and \(L\) is the length of the string.

This frequency is significant because it establishes the base tone of the stringed instrument or any vibrating system bounded at two ends and is associated with the simplest pattern of standing waves, i.e., one antinode and two nodes at the fixed ends. The fundamental frequency is foundational in music and acoustics because all other vibrational modes (harmonics) are based on it, creating the unique timbre of instruments.

Harmonics

In the context of wave mechanics, harmonics refer to the various vibrational modes of a string or a column of air at frequencies that are multiples of the fundamental frequency. The fundamental frequency is known as the first harmonic, the second harmonic is twice the frequency of the fundamental, the third harmonic is three times the frequency, and so on.

Each harmonic corresponds to a unique pattern of nodes (points of no displacement) and antinodes (points of maximum displacement) along the medium. For example, the third harmonic on a string will have three antinodes and four nodes. Mathematically, the frequency of the nth harmonic is described by the equation \(f_n = nf_1\), where \(n\) is an integer representing the harmonic number, and \(f_1\) is the fundamental frequency. Understanding harmonics is key in music theory and the design of musical instruments where different notes are produced by different harmonic vibrations.