Question

A wave traveling on a string has the equation of motion \(y(x, t)=0.02 \sin (5.00 x-8.00 t)\) a) Calculate the wavelength and the frequency of the wave. b) Calculate its velocity. c) If the linear mass density of the string is \(\mu=0.10 \mathrm{~kg} / \mathrm{m}\), what is the tension on the string?

Short answer

Wavelength (λ): 1.26 m Frequency (f): 1.27 Hz Wave velocity (v): 1.60 m/s Tension (T): 0.256 N

Step-by-step solution

Find the wave number and the angular frequency

We are given the wave equation: $$y(x, t) = 0.02 \sin(5.00 x - 8.00 t)$$ The wave number \(k\) and the angular frequency \(\omega\) in the given equation are: $$k = 5.00 \, m^{-1}$$ $$\omega = 8.00 \, s^{-1}$$

Calculate the wavelength and frequency

To find the wavelength \(\lambda\), use the formula: $$\lambda = \frac{2\pi}{k}$$ $$\lambda = \frac{2\pi}{5.00}$$ $$\lambda = 1.26\,\mathrm{m}$$ To find the frequency \(f\), use the formula: $$f = \frac{\omega}{2\pi}$$ $$f = \frac{8.00}{2\pi}$$ $$f = 1.27\,\mathrm{Hz}$$

Calculate the velocity of the wave

To calculate the velocity \(v\), use the formula \(v=\frac{\omega}{k}\): $$v = \frac{8.00}{5.00}$$ $$v = 1.60\, \mathrm{m/s}$$

Calculate the tension on the string

We are given the linear mass density of the string, \(\mu = 0.10 \,\mathrm{kg/m}\). Use the wave speed equation to find the tension \(T\): $$v = \sqrt{\frac{T}{\mu}}$$ Rearrange the formula to solve for \(T\): $$T = \mu v^2$$ Substitute the values for \(\mu\) and \(v\): $$T = 0.10\,\mathrm{kg/m} \times (1.60\,\mathrm{m/s})^2$$ $$T = 0.256 \,\mathrm{N}$$ The solutions to the given exercise are: a) Wavelength, \(\lambda = 1.26\, \mathrm{m}\), and frequency, \(f = 1.27\, \mathrm{Hz}\). b) Wave velocity, \(v = 1.60\, \mathrm{m/s}\). c) Tension on the string, \(T = 0.256\, \mathrm{N}\).

Key concepts

Wavelength Calculation

Wavelength represents the physical distance between consecutive crests or troughs in a wave pattern. In physics, the importance of calculating wavelength lies in understanding the physical and geometrical properties of waves.

To calculate wavelength of a sinusoidal wave, the formula \(\lambda = \frac{2\pi}{k}\) is used, where \(\lambda\) is the wavelength and \(k\) is the wave number. The wave number is obtained from the spatial component of the wave equation, and it is inversely proportional to the wavelength. For instance, a wave number of \(5.00 \, m^{-1}\) leads to a wavelength of \(1.26\, \mathrm{m}\), indicating the wave repeats its pattern every 1.26 meters along the medium.

Frequency Calculation

Frequency is a crucial concept in wave motion physics as it measures how often the particles of a medium vibrate when a wave passes through it. Defined in Hertz (Hz), the frequency represents the number of oscillations per unit time.

To find a wave's frequency, use the formula \(f = \frac{\omega}{2\pi}\), where \(f\) is the frequency and \(\omega\) is the angular frequency of the wave. The angular frequency reflects the rate of change of the wave's phase and is related to the time component of the wave equation. A higher frequency means more cycles or oscillations are occurring each second. For the given wave with an angular frequency of \(8.00 \, s^{-1}\), the calculated frequency is \(1.27\,\mathrm{Hz}\), describing the rate of oscillation.

Wave Velocity

Wave velocity, often denoted as \(v\), describes the speed at which a wave travels through a medium. It is a fundamental concept, as it affects how quickly energy or information is transmitted by the wave.

The formula \(v = \frac{\omega}{k}\) is instrumental for calculating the velocity of a wave. Where \(\omega\) is the angular frequency and \(k\) is the wave number, their ratio gives the wave's speed in units of meters per second (\(\mathrm{m/s}\)). This velocity depends on both time and spatial factors of the wave. From our example, dividing the angular frequency \(8.00 \,\mathrm{s^{-1}}\) by the wave number \(5.00 \,\mathrm{m^{-1}}\) results in a wave velocity of \(1.60\,\mathrm{m/s}\), indicating how fast the wave propagates along the string.

String Tension Physics

In the context of waves on strings, tension is the force that keeps the string taut as it vibrates. Tension is a vital element because it influences the speed of wave propagation along the string.

The relationship between wave speed \(v\), string tension \(T\), and linear mass density \(\mu\) of a string is described by the equation \(v = \sqrt{\frac{T}{\mu}}\). To solve for the tension on the string, rearrange this formula to \(T = \mu v^2\). Applying the provided values of the linear mass density \(\mu = 0.10 \,\mathrm{kg/m}\) and the wave velocity \(1.60 \, \mathrm{m/s}\), you can calculate the required tension that must be applied to the string. In the example given, the calculated tension comes out to be \(0.256\, \mathrm{N}\), which is the force that keeps the string vibrating at the determined frequency and wavelength.