Question

A wave traveling on a string has the equation of motion \(y(x, t)=0.0200 \sin (5.00 x-8.00 t),\) where \(x\) and \(y\) are in meters and \(t\) is in seconds. a) Calculate the wavelength and the frequency of the wave. b) Calculate its velocity. c) If the linear mass density of the string is \(\mu=0.100 \mathrm{~kg} / \mathrm{m},\) what is the tension on the string?

Short answer

Question: Calculate the wavelength, frequency, velocity of the wave, and the tension on the string given its linear mass density using the wave equation: \(y(x, t)=0.0200 \sin(5.00x - 8.00t)\), and linear mass density \(\mu = 0.100\,\text{kg/m}\). Answer: The wavelength is \(\dfrac{2\pi}{5}\) meters, the frequency is approximately \(1.273\) Hz, the velocity is \(1.60\, \text{m/s}\), and the tension on the string is \(0.256\) N.

Step-by-step solution

a) Calculate the wavelength and the frequency of the wave

First, we need to recognize the given wave equation: \(y(x,t) = 0.0200 \sin(5.00x - 8.00t)\). From this equation, we can identify the wave number \(k = 5.00\) and the angular frequency \(\omega = 8.00\). To find the wavelength, we can use the relationship \(k = \dfrac{2\pi}{\lambda}\). Solving for \(\lambda\), we get: \(\lambda = \dfrac{2\pi}{k} = \dfrac{2\pi}{5.00} = \dfrac{2\pi}{5}\) meters. To find the frequency, we can use the relationship \(\omega = 2\pi f\). Solving for \(f\), we get: \(f = \dfrac{\omega}{2\pi} = \dfrac{8.00}{2\pi} \approx 1.273\) Hz.

b) Calculate its velocity

Next, we need to find the wave speed, \(v\). We can use the relationship \(v = \dfrac{\omega}{k}\): \(v = \dfrac{8.00}{5.00} = 1.60 \, \text{m/s}\).

c) Calculate the tension on the string

We are given the linear mass density \(\mu = 0.100\,\text{kg/m}\), and we found the wave speed, \(v = 1.60\, \text{m/s}\). We can find the tension on the string using the formula \(T = \mu v^2\): \(T = 0.100 \cdot (1.60)^2 = 0.100 \cdot 2.56 = 0.256\, \text{N}\). So, the tension on the string is \(0.256\) N.

Key concepts

Wavelength Calculation

Understanding how to calculate the wavelength of a wave is essential for anyone studying wave motion in physics. The wavelength, represented by the Greek letter lambda \(\lambda\), is the distance between two consecutive points that are in phase on a wave, such as two consecutive crests or troughs.

In the given exercise, the wave number \(k\) helps us determine the wavelength. The wave number and wavelength are inversely related, as shown by the equation \(k = \dfrac{2\pi}{\lambda}\). By rearranging this equation, we can calculate the wavelength \(\lambda\). It's important to have a clear understanding of this equation, as it helps us connect the spatial configuration of a wave to its physical properties.

Wave Frequency

The frequency of a wave, denoted as \(f\), is the number of cycles the wave completes per unit of time, typically measured in hertz (Hz). A wave's frequency and its angular frequency, \(\omega\), are directly related through the equation \(\omega = 2\pi f\).

When solving problems in wave motion, it is crucial to interpret the angular frequency correctly to determine the standard frequency of the wave. Learning to convert between angular frequency and standard frequency will enable students to better understand the wave's temporal characteristics and how often the wave oscillates.

String Wave Velocity

Wave velocity, or the speed of a wave \(v\), on a string describes how fast the wave is traveling along the string. It can be determined from the wave's angular frequency \(\omega\) and wave number \(k\) via the straightforward equation \(v = \dfrac{\omega}{k}\).

This formula is invaluable for predicting how quickly a wave propagates through a medium, in this case, a string. Students should remember that the velocity is not always constant; it can depend on the medium's properties, such as tension and linear mass density—topics we'll explore in further sections.

Linear Mass Density

Linear mass density is the measure of a mass per unit length, symbolized by \(\mu\) and typically expressed in kilograms per meter (kg/m). It quantifies the distribution of mass along the length of an object such as a string.

In wave motion physics, it plays a key role in understanding how the physical characteristics of a string affect wave propagation. The linear mass density, along with the wave tension, contributes to the velocity at which waves travel along the string. Recognizing how to apply this concept is an essential skill when analyzing the behavior of waves on strings.

Wave Tension

The tension in a string, indicated by the letter \(T\), affects the velocity and the power of a wave traveling through the string. Tension can be thought of as the force that is applied along the string's length, causing it to stretch and influencing how waves behave on it.

In the context of the problem at hand, tension is determined by the equation \(T = \mu v^2\), linking it to both the linear mass density and the velocity of the wave. Recognizing this relationship helps students appreciate how physical forces and material properties interact to dictate wave motion.