Chapter 15 ( Medium )

Podcast Beta

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the linear mass density of a copper wire with a density of 8920 kg/m³ and a diameter of 2.40 mm?

0.04 kg/m (correct)

0.02 kg/m

0.10 kg/m

0.08 kg/m

If a tension of 10.00 N is applied to a copper wire, what is the speed of transverse waves through the wire with a linear mass density of 0.04 kg/m?

15.8 m/s (correct)

10.0 m/s

20.0 m/s

5.0 m/s

How far away is a cliff if an echo is heard 4 seconds after a horn sounds, with the speed of sound at 340 m/s?

1360 m

170 m

340 m

680 m (correct)

Given a tension of 10.00 N in a copper wire, which of these determines the wave speed the most significantly?

The tension in the wire Signup and view all the answers

What formula represents the relationship between wave speed, tension, and linear mass density in a wire?

$v = \frac{T}{\mu}$ Signup and view all the answers

What is a wave primarily defined as?

An excitation that propagates through space or medium over time Signup and view all the answers

Which of the following is an example of an electromagnetic wave?

X-ray Signup and view all the answers

Mechanical waves require what for propagation?

A medium Signup and view all the answers

Which type of wave moves in a direction perpendicular to the oscillation direction?

Transverse waves Signup and view all the answers

What distinguishes longitudinal waves from transverse waves?

Longitudinal waves compress and rarefy the medium. Signup and view all the answers

What is not a type of wave based on medium?

Transverse waves Signup and view all the answers

Which of the following describes the motion of particles in transverse waves?

Particles oscillate perpendicular to wave travel. Signup and view all the answers

In which context are mechanical waves primarily studied?

In the context of sound and water waves Signup and view all the answers

What is the amplitude of the wave given by the wave function?

0.002 m Signup and view all the answers

What is the angular frequency (ω) of the wave?

346 rad/s Signup and view all the answers

Which expression correctly calculates the wave number (k) of the wave?

k = 78.7 rad/m Signup and view all the answers

What is the velocity (v) of the wave?

4.4 m/s Signup and view all the answers

How is the wavelength (λ) calculated from the wave number (k)?

λ = 2π/k Signup and view all the answers

What is the period (T) of the wave derived from the angular frequency?

T = 2π/346 Signup and view all the answers

What would be the phase (∅) of the wave at x = 2.3 × 10⁻³ m and t = 2.1 s?

726.8 rad Signup and view all the answers

Which condition indicates that a wave is traveling in the negative x-direction?

k is negative and ω is positive Signup and view all the answers

Which of the following wave equations represents a wave traveling in the positive x-direction?

y(x,t) = (0.002 m) sin[(78.8 m⁻¹)x + (346 s⁻¹)t − 1] Signup and view all the answers

What is the primary characteristic of longitudinal waves?

They propagate along the direction in which the oscillators move. Signup and view all the answers

How is the wave number (k) mathematically defined?

The number of wavelengths in a distance of $2\pi$. Signup and view all the answers

What does the amplitude (A) of a wave signify?

The maximum displacement from the wave's equilibrium position. Signup and view all the answers

Which parameter describes the time for one complete oscillation of a wave?

Period Signup and view all the answers

If a wave has a period (T) of 0.5 seconds, what is its frequency (f)?

$2 Hz$ Signup and view all the answers

How is the angular frequency (ω) defined in terms of oscillations?

The number of oscillations in each time interval of $2\pi$. Signup and view all the answers

What measurement indicates the length of one complete wave oscillation?

Wavelength Signup and view all the answers

Which statement about frequency (f) is correct?

It is inversely proportional to the period (T). Signup and view all the answers

In the wave equation $y(x, t) = A \sin(kx - \omega t + \phi_0)$, what does k represent?

The wave number, related to the wavelength. Signup and view all the answers

What does the variable $\mu$ represent in the wave equation for a string?

Linear mass density in kg/m Signup and view all the answers

How is wave velocity ($v$) on a string related to tension ($T$) and linear mass density ($\mu$)?

$v = \sqrt{T \cdot \mu}$ Signup and view all the answers

Which factor, when increased, will lead to a decrease in wave speed on a string?

Increased linear mass density Signup and view all the answers

Given the wave function $y(x,t) = (0.002 m) \sin[(78.8 m^{-1})x + (346 s^{-1})t - 1]$, what does the term $346 s^{-1}$ represent?

Frequency of the wave Signup and view all the answers

What is the amplitude of the wave represented by $y(x,t) = (0.002 m) \sin[(78.8 m^{-1})x + (346 s^{-1})t - 1]$?

0.002 m Signup and view all the answers

If the mass of the steel cable is 38 kg and its length is 61 m, what is the linear mass density ($\mu$) of the cable?

0.623 kg/m Signup and view all the answers

How would increasing the tension in the elevator cable affect the velocity of the wave pulse?

Increase the velocity of the wave pulse Signup and view all the answers

Which statement about the function $D.y(x,t) = (0.002 m) \sin[(-78.8 m^{-1}) + (346 s^{-1})]$ is true?

It describes a wave moving to the left. Signup and view all the answers

If an elevator cable of mass 38 kg generates a wave pulse after a hammer tap, what factor determines how long it takes for the pulse to travel the entire length of 61 m?

Velocity of the wave in the cable Signup and view all the answers

Study Notes

Waves

A wave is an excitation traveling through space or a medium but does not transport matter.
Waves can be classified by medium (Electromagnetic & Mechanical) or propagation (Transverse and Longitudinal).
Electromagnetic waves don't require a medium for propagation (e.g., light, radio waves, x-rays).
Mechanical waves require a medium for propagation (e.g., sound waves, water waves).

Coupled Oscillators

Transverse waves move perpendicular to the direction of the oscillators, like a wave moving up and down on a rope, forming crests and troughs.
Longitudinal waves move parallel to the direction of the oscillators, like sound waves compressing and rarefying the air.

Mathematical Description of Waves

A wave can be described by a function of position (x) and time (t).
Sinusoidal waves have a mathematical description: 𝑦 𝑥, 𝑡 = 𝐴𝑠𝑖𝑛(𝑘𝑥 − 𝜔𝑡 + ∅0 ) where:
- A is the amplitude, the maximum displacement of the wave.
- λ is the wavelength, the distance between two identical points in a wave (e.g., crests or troughs).
- k is the wave number, the number of wavelengths in 2π.
- T is the period, the time for one complete oscillation.
- f is the frequency, the number of oscillations per second.
- ω is the angular frequency, the number of oscillations in 2π time.

Derivation of the Wave Equation

Waves on a string have a velocity dependent on the tension (T) and linear mass density (μ) of the string.
- The greater the tension, the faster the wave.
- The greater the linear density, the slower the wave.
The wave equation can be derived from the force balance on a small segment of the string.
The velocity of the wave on a string is given by 𝑣= 𝑇/𝜇

Example 15.1: Elevator Cable

An elevator repairman sends a signal with a hammer tap on an elevator cable.
The wave's speed is calculated using the tension in the cable (due to the weight of the elevator and man) and the cable's linear mass density.
The time for the wave pulse to travel the length of the cable is easily calculated from the distance and the wave's speed.

Extra Exercise: Copper Wire

A copper wire under tension has transverse waves sent down it.
The linear mass density of the wire is calculated using its density and cross-sectional area.
The speed of the wave in the wire is determined by the tension and linear mass density.

Extra Exercise: Ship Horn Echo

A ship horn's echo is used to determine the distance to a cliff.
The time it takes for the sound to travel to the cliff and back is used, along with the speed of sound, to calculate the distance.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Test your knowledge on waves and their classifications, including mechanical and electromagnetic types. Explore the mathematical descriptions of waves, including sinusoidal functions and their parameters. This quiz will enhance your understanding of wave dynamics and behaviors.