Question

A \(3.00-\mathrm{m}\) -long string, fixed at both ends, has a mass of \(6.00 \mathrm{~g}\). If you want to set up a standing wave in this string having a frequency of \(300 . \mathrm{Hz}\) and three antinodes, what tension should you put the string under?

Short answer

The required tension on the string is 2880 N.

Step-by-step solution

Calculate the linear mass density of the string

To find the linear mass density, we need to divide the total mass of the string by its length. The mass of the string is given as \(6.00\,\text{g}\), and the length is \(3.00\,\text{m}\). The linear mass density, denoted as \(\mu\), can be calculated as: $$\mu = \frac{m}{L}$$

Find the wave speed and wavelength

The standing wave pattern has three antinodes, so this is the 3rd harmonic. The distance between two consecutive nodes or two consecutive antinodes is half the wavelength. Therefore, with three antinodes, we have: $$\lambda = 2\cdot \frac{2 L}{3} = \frac{4 L}{3}$$ We know the frequency (f) of the wave is \(300\,\text{Hz}\). Now we can find the wave speed (v) by multiplying the frequency with the wavelength: $$v = f \cdot \lambda$$

Find the required tension

With the linear mass density and wave speed, we can now find the required tension (T) on the string, using the formula: $$v^2 = \frac{T}{\mu}$$ Solve for the tension: $$T = \mu \cdot v^2$$ Now combine and solve all the equations in order: Step 1: $$\mu = \frac{6.00\,\text{g}}{3.00\,\text{m}} = \frac{6.00\times 10^{-3}\,\text{kg}}{3.00\,\text{m}} = 2.00\times 10^{-3}\,\mathrm{\frac{kg}{m}}$$ Step 2: $$\lambda = \frac{4 \cdot 3.00\,\text{m}}{3} = 4.00\,\text{m}$$ $$v = 300\,\text{Hz} \cdot 4.00\,\text{m} = 1200\,\frac{\text{m}}{\text{s}}$$ Step 3: $$T = (2.00\times 10^{-3}\,\mathrm{\frac{kg}{m}}) \cdot (1200\,\frac{\text{m}}{\text{s}})^2 = 2880\,\text{N}$$ So the string should be under a tension of \(2880\,\text{N}\) to set up a standing wave with a frequency of \(300\,\text{Hz}\) and three antinodes.

Key concepts

Linear Mass Density

Linear mass density is a crucial concept in understanding how waves behave on a string. It is defined as the mass per unit length of the string, symbolized by the Greek letter \(\mu\). In the context of a vibrating string, linear mass density affects the wave speed, which is essential for calculating the tension required for a particular standing wave pattern.

To calculate the linear mass density, you divide the total mass of the string by its length. The lower the mass per unit length, the less inertia the string elements have, making it easier to move and change direction. This results in higher wave speeds and frequencies. On the other hand, a string with high linear mass density moves more sluggishly, reducing the wave speed and frequency.

In the example given, the string had a mass of \(6.00\text{g}\) and a length of \(3.00\text{m}\), resulting in a linear mass density of \(2.00\times 10^{-3}\frac{kg}{m}\). This figure is foundational for further calculations related to the standing wave and tension.

Harmonic Wave Frequency

Harmonic wave frequency refers to the frequency at which standing waves are generated on a string when it vibrates in segments forming nodes and antinodes. The frequency must match one of the natural frequencies of the string, which correspond to the harmonics of the string.

A system has multiple harmonics, each associated with a different standing wave pattern. The fundamental frequency (first harmonic) is the lowest frequency at which the string will naturally vibrate. Each subsequent harmonic creates a more complex standing wave pattern with an increased number of nodes and antinodes. For instance, the third harmonic will have three distinct loops when vibrating.

In our exercise, a frequency of \(300\text{Hz}\) is needed to produce three antinodes (the third harmonic). The frequency plays a pivotal role in determining the standing wave's speed, which, combined with the linear mass density, reveals the required tension on the string.

String Wave Speed

Wave speed on a string is a vital aspect when studying wave phenomena. It's the speed at which the wave travels along the string, depending on its tension and linear mass density. Mathematically, the wave speed \(v\) is determined by the square root of the tension \(T\) divided by the linear mass density \(\mu\), represented by the equation \(v = \sqrt{\frac{T}{\mu}}\).

For a given harmonic frequency, the wave speed is calculated by multiplying the frequency by the wavelength, as seen in our example \(v = 300\text{Hz} \times 4.00\text{m} = 1200\frac{\text{m}}{\text{s}}\). High tension and low mass density lead to higher wave speed, essential for transferring waves quickly along the string.

Understanding this relationship is key when calculating the tension needed for particular wave frequencies and harmonics, providing insight into the dynamics of the string's vibrations.