Module 6: Periodic Motion, Waves, and Sound Study Guide

Periodic Motion

• Periodic motion is defined as any motion that repeats itself as a function of time.
• Period ($T$): The time required for an object to complete exactly one repetition or cycle.
  • The unit of measurement for period is seconds ($s$).
  • Formula for Period: T = \frac{\Delta t}{\text{# of cycles}}
• Frequency ($f$): The rate of repetition of the motion.
  • The unit of measurement is Hertz ($Hz$), where one Hertz is defined as one cycle per second ($1\,Hz = 1\,\text{cycle/s}$).
  • Formula for Frequency: f = \frac{\text{# of cycles}}{\Delta t}
• Relationship between Period and Frequency:
  • The period and frequency are reciprocals of each other.
  • $T = \frac{1}{f}$
  • $f = \frac{1}{T}$

Simple Harmonic Motion (SHM)

• Equilibrium Position: The specific location of an object when it reaches equilibrium, characterized by a net force of zero ($F_{net} = 0$) acting on the object.
• Simple Harmonic Motion (SHM): A specific type of periodic motion where both the restoring force and the acceleration of the object are directly proportional to the distance of the object from its equilibrium position.
  • The restoring force and acceleration are always directed toward the equilibrium position.
• Amplitude ($A$): The maximum distance an object in simple harmonic motion travels from its equilibrium position.
  • Amplitude is a measurement of distance and is measured in meters ($m$).

Simple Harmonic Oscillators

• Pendulum: An apparatus consisting of a mass ($m$) attached to the end of a string or rod of length ($L$), which is pivoted at the opposite end to allow back-and-forth swinging.
• Period of a Pendulum ($T$): The time required for the pendulum to repeat its motion exactly once.
  • Formula: $T = \frac{1}{f} = 2\pi\sqrt{\frac{L}{g}}$
  • In this formula, $L$ is the length of the pendulum in meters ($m$) and $g$ is the acceleration due to gravity.
  • The value of $g$ is typically $9.80\,m/s^2$ when the pendulum is located on or near Earth's surface. This value changes if the pendulum is located high in the atmosphere or on a different planet.
• Mass Bouncing on a Spring: The period of an object oscillating on a spring.
  • Formula: $T = \frac{1}{f} = 2\pi\sqrt{\frac{m}{k}}$
  • In this formula, $m$ is the mass of the object in kilograms ($kg$) and $k$ is the spring constant of the spring.
  • Relationship: As the mass ($m$) increases, the period ($T$) of the oscillation also increases.

Damped Oscillations and Resonance

• Damped Oscillations: Occur when an external force removes energy from an oscillating system. This causes the amplitude of the oscillations to decrease over time until the motion eventually stops.
• Dampening Force: Friction is the force usually responsible for the dampening of oscillations.
• Resonance: A phenomenon that occurs when the frequency of one oscillation matches the natural frequency of another object, causing that second object to begin oscillating as well.

Waves

• Wave Definition: A periodic disturbance of a medium that facilitates the transfer of energy from one location to another without the transfer of matter.
  • Common examples include sound waves, light waves, and surf (water) waves.
• Medium: The material through which a wave travels and which is disturbed by the wave's passage.
• Wave Types by Medium Requirement:
  • Mechanical Waves: Require a physical medium to travel through (e.g., sound).
  • Electromagnetic Waves: Do not require a medium and can travel through a vacuum (e.g., light).
• Wave Types by Direction of Disturbance:
  • Transverse waves.
  • Longitudinal waves.
  • Surface waves.
• Properties of Waves:
  • Period ($T$): The time interval required to go from one crest to the next consecutive crest.
  • Frequency ($f$): The measure of how often a wave repeats itself as a function of time.
  • Amplitude ($A$): The maximum displacement of the medium from its rest position.
  • Velocity ($v$): The speed at which the wave travels through the medium.
  • Wavelength ($\lambda$): The physical distance between two consecutive identical points on a wave, such as from crest to crest or from compression to compression. It is measured in meters ($m$).
• Wave Velocity Formulas:
  • $v = \frac{\Delta d}{\Delta t} = \frac{\lambda}{T}$
  • $v = f\lambda$

Properties of Waves: Reflection

• Reflection: The process of a wave bouncing off a boundary between two different mediums.
• Normal to a Surface: An imaginary line drawn perpendicular to the surface at the point where a light ray strikes.
• Angle of Incidence: The angle measured between the incoming (incident) ray and the normal line.
• Angle of Reflection: The angle measured between the normal line and the reflected ray.
• The Law of Reflection: States that the incident ray, the reflected ray, and the normal line all exist within the same plane, and the angle of incidence is equal to the angle of reflection.
• Specular Reflection: Reflection occurring off a smooth surface, which allows a clear image of the object to be seen.
• Diffuse Reflection: Reflection occurring when light bounces off a rough surface, scattering the light.

Properties of Waves: Refraction

• Refraction: The change in direction of a wave as it crosses from one medium into another, caused by a change in the medium itself.
• Index of Refraction ($n$): The unitless ratio of the speed of light in a vacuum ($c$) to the speed of light in a specific medium ($v$).
  • Formula: $n = \frac{c}{v}$
• Snell's Law: Also known as the Law of Reflection in some contexts of refraction study, it relates the indices of refraction and the angles of the ray.
  • Formula: $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$

Properties of Waves: Superposition and Interference

• Superposition: A principle defined by three core parts:
  1. The ability of waves to overlap within the same medium, at the same location, and at the same time.
  2. The resulting wave is determined by the algebraic sum of the individual waves.
  3. Individual waves pass through one another and emerge undisturbed.
• Interference: A phenomenon derived from the principle of superposition.
  • Constructive Interference: Occurs when the algebraic sum of overlapping waves produces a resultant wave with a larger amplitude than the individual constituent waves.
  • Destructive Interference: Occurs when the resulting wave has a smaller amplitude than the individual constituent waves.
  • Complete Destructive Interference: Occurs when two waves perfectly match up and offset each other, resulting in a total cancellation of the wave (no wave at all).

Properties of Waves: Diffraction and Huygen's Principle

• Thin Film Interference: Interference resulting from light shining on a thin film, such as a soap film, as seen from the viewer's perspective.
• Diffraction: The phenomenon where a wave spreads out as it passes beyond a barrier or through an opening in a barrier.
• Huygen's Principle: A two-part principle used to explain diffraction for any type of wave:
  1. Every point on a wavefront acts as a source of a spherical wavelet that spreads outward at the speed of the wave.
  2. The shape of the wavefront at any later time is the curve that is tangent to all of these individual wavelets.

Sound

• Sound is categorized as a mechanical wave that commonly travels through air.
• Sound is created when a source vibrates back and forth, producing a longitudinal wave that is perceived as sound.

Electromagnetic Waves

• Electromagnetic (EM) waves are unique because they can travel through a vacuum and do not require a medium to exist, distinguishing them from mechanical waves.
• Electromagnetic Spectrum: The total collection of all frequencies of electromagnetic waves.
• The Three-Part Model of an Electromagnetic Wave:
  1. EM waves are sinusoidal transverse waves.
  2. They consist of an electric component and a magnetic component. These components are perpendicular to each other and perpendicular to the direction the wave is traveling.
  3. The electric and magnetic field strengths are always in phase; they reach maximum and minimum strengths at the exact same time.