Question

A \(1.48\) -m-long wire has a mass of \(8.62 \mathrm{~g}\) and is held under a tension of \(122 \mathrm{~N}\). The wire is held rigidly at both ends and set into vibration. Calculate \((a)\) the speed of waves on the wire, (b) the wavelengths of the waves that produce one- and twoloop standing waves on the wire, and ( \(c\) ) the frequencies of the waves in \((b)\).

Short answer

The speed of waves on the wire is approximately 486.5 m/s. The wavelengths that produce one- and two-loop standing waves on the wire are approximately 2.96m and 1.48m respectively. The frequencies of the waves that correspond to these wavelengths are approximately 164.2 Hz and 328.4 Hz respectively.

Step-by-step solution

Calculate the speed of waves on the wire

First, calculate the linear density \(\mu\) of the wire by dividing its mass \(m\) by its length \(L\), both in SI units. The mass should be converted from grams to kilograms. Then, substitute the values of \(T\) and \(\mu\) into the equation \(v = \sqrt{\frac{T}{\mu}}\) to find the speed \(v\).

Calculate the wavelengths that produce one- and two-loop standing waves

The wavelength of a standing wave can be calculated with \(\lambda = \frac{2L}{n}\), where \(L\) is the length and \(n\) is the number of loops. For one-loop and two-loop standing waves, \(n=1\) and \(n=2\) respectively.

Calculate the frequencies of the waves

The frequency of a wave can be calculated with \(f = \frac{v}{\lambda}\), where \(v\) is the speed and \(\lambda\) is the wavelength. Substitute the values obtained for \(v\) and \(\lambda\) into the equation to find \(f\). Do it twice, once for \(n=1\) and once for \(n=2\)

Review your answer

Verify that your answers are consistent with the given conditions and they make sense in the context of the problem. Return to any previous steps if necessary.

Key concepts

Linear Density

Linear density, often denoted by the Greek letter \( \mu \), is a fundamental property in the study of waves on strings or wires. It describes how much mass is contained per unit length of the wire or string. To calculate the linear density of a wire, you divide its total mass \( m \) by its length \( L \). In mathematical terms, it's given as:\[ \mu = \frac{m}{L} \]

This allows us to relate the mass of the wire to its physical properties influencing wave speed. For instance, in order to use weights in kilograms and the length in meters, converting the mass from grams is often required. Understanding linear density is crucial for predicting how fast waves travel along the wire, using the formula for wave speed, \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension in the wire.

Standing Waves

Standing waves are a unique phenomenon where certain frequencies of vibration lead to the formation of stationary patterns of waves on a medium, like a wire. When a wire is rigidly held at both ends and vibrated, specific patterns can emerge, characterized by nodes and antinodes. Nodes are points of zero amplitude, while antinodes are points of maximum amplitude. These patterns depend largely on the length of the wire and the frequency of vibration.

For a wire, standing waves form when the wire's length is a multiple of half the wavelength, resulting in what are called loops or harmonics. The simplest forms are a one-loop (also known as the fundamental mode) and a two-loop. Such standing waves can be explained by the formula for wavelength \( \lambda = \frac{2L}{n} \), where \( n \) denotes the number of loops. So for a one-loop standing wave, \( n=1 \), and for a two-loop, \( n=2 \).

Wave Frequency

Wave frequency is the number of cycles a wave completes in a second and is measured in hertz (Hz). The frequency of standing waves on a wire can be calculated if the speed of the wave \( v \) and the wavelength \( \lambda \) are known. This is given by the formula:\[ f = \frac{v}{\lambda} \]

This equation highlights the inverse relationship between wavelength and frequency, meaning that a longer wavelength corresponds to a lower frequency and vice versa. Once the speed of the wave and the wavelengths are determined by their harmonic order, calculating frequency becomes straightforward. This is essential for applications like musical instruments, where specific frequencies are necessary to produce desired notes.

Wavelength Calculation

Wavelength calculation is pivotal in understanding wave behavior across different harmonics. It involves determining the length of one complete wave cycle along the wire. The formula \( \lambda = \frac{2L}{n} \) encapsulates how the wire's length and its vibrational mode affect this calculation. Here, \( L \) represents the entire length of the wire, while \( n \) is the number of loops or antinodes formed.

For instance, when \( n=1 \) (one-loop mode), the wavelength equals twice the wire's length. When \( n=2 \) (two-loop mode), the wavelength is the same as the wire's length. Understanding these relationships is essential because they define how the wave is set to resonate within the constraints of the wire, influencing both the frequency and vibrational patterns observed.