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Chapter 16: Problem 28

A transverse wave is traveling on a string. The displacement y of a particle from its equilibrium position is given by \(y=(0.021 \mathrm{m}) \sin (25 t-2.0 x) .\) Note that the phase angle \(25 t-2.0 x\) is in radians, \(,\) tis in seconds, and \(x\) is in meters. The linear density of the string is \(1.6 \times 10^{-2} \mathrm{kg} / \mathrm{m}\) . What is the tension in the string?

Short Answer

Expert verified
The tension in the string is 2.5 N.

Step by step solution

01

Identify Characteristics of the Wave Equation

The given wave equation is:\[ y = (0.021 \, \text{m}) \sin(25t - 2.0x) \]where the general form is:\[ y = A \sin(kx - \omega t) \] From this, we can identify the angular frequency \( \omega = 25 \, \text{rad/s} \) and the wave number \( k = 2.0 \, \text{rad/m} \).
02

Calculate the Wave Speed

The wave speed \( v \) can be calculated using the relation between wave speed, angular frequency, and wave number:\[ v = \frac{\omega}{k} \]Substitute the given values:\[ v = \frac{25}{2.0} = 12.5 \, \text{m/s} \]
03

Use the Formula for Tension in a Wave

With the wave speed calculated, we use the formula for wave speed in terms of tension \( T \) and linear density \( \mu \) to find the tension in the string:\[ v = \sqrt{\frac{T}{\mu}} \]We rearrange the equation to find the tension:\[ T = \mu v^2 \]
04

Calculate Linear Density

Given that the linear density \( \mu = 1.6 \times 10^{-2} \, \text{kg/m} \), we use this in the tension formula.
05

Calculate Tension in the String

Substitute \( \mu = 1.6 \times 10^{-2} \, \text{kg/m} \) and \( v = 12.5 \, \text{m/s} \) into the tension formula:\[ T = (1.6 \times 10^{-2}) (12.5)^2 \]\[ T = (1.6 \times 10^{-2}) (156.25) \]\[ T = 2.5 \, \text{N} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transverse Waves
Transverse waves are a type of wave where the particles of the medium move perpendicular to the direction of wave propagation. In simpler terms, if the wave moves horizontally, the medium's particles move up and down or vice versa.
A common example of transverse waves is waves on a string or water waves. Inside the exercise, the given wave equation represents a transverse wave on a string, described mathematically to show how each point on the string moves up and down as the wave travels.
  • Displacement: The displacement of particles (y) is perpendicular to the direction of the wave.
  • Equilibrium Position: This is the rest position of the string, and the sine function describes the deviation from this position.
  • Wave Equation Form: Generally follows the structure: \(y = A \sin(kx - \omega t)\)
Wave Speed
Wave speed is a crucial factor that defines how fast a wave travels through a medium. It is calculated by the relationship between angular frequency \( \omega \) and wave number \( k \). Mathematically, it is expressed as:
\[v = \frac{\omega}{k}\\]
For the problem at hand, the speed of the wave is calculated by plugging in the given values for angular frequency (25 rad/s) and wave number (2.0 rad/m), which gives a wave speed of 12.5 m/s.
  • Wave Number \( k \): Represents the number of waves per unit distance and is given in radians per meter (rad/m).
  • Angular Frequency \( \omega \): Represents how fast the wave oscillates in time and is given in radians per second (rad/s).
  • Importance of Wave Speed: Helps determine how quickly energy or information travels through the medium.
Tension in Strings
Tension in a string is an essential parameter for understanding wave behavior on it. It is the force that is applied to the string that helps maintain its tightness and impacts both the speed and characteristics of the wave on the string.
In this exercise, the formula to calculate tension when wave speed and linear density are known is:
\[T = \mu v^2\]
Where \(T\) is the tension, \(\mu\) is the linear density of the string, and \(v\) is the wave speed. Here, the provided values of \(\mu = 1.6 \times 10^{-2} \, \text{kg/m}\) and \(v = 12.5 \, \text{m/s}\) yield a tension of 2.5 N.
  • Linear Density \( \mu \): The mass per unit length of the string, crucial for finding tension.
  • How Tension Affects Waves: Higher tension results in faster wave speeds.
  • Application: Used widely in musical instruments and engineering.
Angular Frequency
Angular frequency is a measure of how quickly a wave oscillates in terms of radians per second. It is a fundamental concept in wave mechanics, particularly when analyzing waves that involve oscillatory motion.
For the exercise, the angular frequency, \( \omega \), is given as 25 rad/s. This means that the wave oscillates 25 radians in each second, indicating how rapid these oscillations are.
  • Relation to Wave Speed: Part of the calculation for wave speed through the formula \( v = \frac{\omega}{k} \).
  • Use in Equations: Often seen in equations describing sinusoidal motion, akin to those describing alternating current (AC) in physics.
  • Importance in Physics: Offers insights into the energy and frequency of the wave.

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Most popular questions from this chapter

To measure the acceleration due to gravity on a distant planet, an astronaut hangs a 0.055-kg ball from the end of a wire. The wire has a length of 0.95 \(\mathrm{m}\) and a linear density of \(1.2 \times 10^{-4} \mathrm{kg} / \mathrm{m}\) . Using electronic equipment, the astronaut measures the time for a transverse pulse to travel the length of the wire and obtains a value of 0.016 \(\mathrm{s}\) . The mass of the wire is negligible compared to the mass of the ball. Determine the acceleration due to gravity.

A jetskier is moving at 8.4 m/s in the direction in which the waves on a lake are moving. Each time he passes over a crest, he feels a bump. The bumping frequency is 1.2 Hz, and the crests are separated by 5.8 m. What is the wave speed?

A woman is standing in the ocean, and she notices that after a wave crest passes, five more crests pass in a time of 50.0 s. The distance between two successive crests is 32 m. Determine, if possible, the wave鈥檚 (a) period, (b) frequency, (c) wavelength, (d) speed, and (e) amplitude. If it is not possible to determine any of these quantities, Then so state.

A steel cable has a cross-sectional area \(2.83 \times 10^{-3} \mathrm{m}^{2}\) and is kept under a tension of \(1.00 \times 10^{4} \mathrm{N}\) . The density of steel is 7860 \(\mathrm{kg} / \mathrm{m}^{3}\) . Note that this value is not the linear density of the cable. At what speed does a transverse wave move along the cable?

Hearing damage may occur when a person is exposed to a sound intensity level of 90.0 dB (relative to the threshold of hearing) for a period of 9.0 hours. One particular eardrum has an area of \(2.0 \times 10^{-4} \mathrm{m}^{2} .\) How much sound energy is incident on this eardrum during this time?
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