Periodic Motion and Waves Notes

Periodic Motion

• Motion that repeats over the same path in equal intervals of time
• Orbiting satellites, planets, moons are periodic
• Other examples: a mass bouncing on spring, a pendulum swinging back and forth
• Includes oscillations, waves, vibrations

Simple Harmonic Motion

• Periodic motion through an equilibrium position
• Always a force trying to return object to eq. position
• Force (and resulting acceleration) is prop. to displacement from eq. position
• Always has inertial and "springy" components
• Amplitude is maximum displacement from eq. position
• Frequency ( f ) is rate of oscillation, vibrations or cycles per second
• Period ( T ) is time duration of one cycle
• Frequency and period are reciprocals T = 1/f
• Unit of frequency is hertz (Hz) eq. to sec.^-1

SHM and Circular Motion

• If image of particle moving in a circle is projected on diameter of circle, the image is seen to move in SHM
• Angular variables of circ. motion are related to linear variables of SHM
• Amplitude = radius; displacement = R·cos θ
• Graph of displacement vs. time gives cos graph

The Spring Oscillator

• Restoring force due to spring force F = -kx
• Inertial component is mass attached to spring

The Pendulum

• Object suspended so that it can swing back and forth about an axis
• Simple pendulum consists of a mass (the bob) and a cord
• "Springy" component is due to gravity
• Inertial component is due to mass of bob, but depends on length of cord

The Period of a Pendulum

• Period is independent of mass of bob
• Period is independent of amplitude for small arcs
• Period is directly proportional to square root of length, inversely prop. to square root of g

Physical Pendulum

• Real object suspended and swinging as pendulum
• All mass not concentrated in bob
• Has point called center of oscillation that corresponds to bob position in simple pendulum of same period

SHM and Energy

• At max displacement, energy is all potential, gravitational for pendulum, elastic for spring oscillator
• At equilibrium position, energy is all kinetic, max speed here also
• If dissipative forces present (friction), motion is damped, amplitude decreases, motion stops when all energy is lost

Measuring g

• Since period of pendulum depends on g, its value can be calculated by measuring length and period of pendulum

Vocabulary

• Periodic motion
• Simple harmonic motion
• simple pendulum
• physical pendulum

Wave: definition

• A quantity or disturbance that changes in magnitude with respect to time at a given location
• Also changes in magnitude from place to place at a given time
• Propagates through a medium or space
• Transfers energy, not matter
• water wave, sound, light, x-ray, earthquake

Wave Classification Schemes

• Whether medium is needed for propagation.
  • Mechanical
  • Electromagnetic
• How particles move compared to motion of wavefront.
  • Longitudinal
  • Transverse

Mechanical Waves

• Require elastic medium for propagation
• Energy source vibrates particles of medium about an equilibrium position
• Each particle exerts force on adjacent particles passing energy along
• Inertia of particles slows propagation of wave: wave speed depends on medium
• Particles move in SHM if wave train is generated by periodic motion
• Examples: water waves, sound waves, earthquake waves, vibrating strings

• Self-propagating--need no medium for propagation
• Can travel through vacuum of space
• Examples: light waves, microwaves, x-rays, radio & TV broadcasts

• Displacement of particles is perpendicular to direction of wave travel crest: point of max. positive displacement
• trough: point of max. negative displacement
• examples: water, light, string, all electromagnetic waves

• Displacement of particles is parallel to direction of wave travel
• Series of high and low pressure areas in medium
• Compression: high pressure area (like crest)
• Rarefaction: low pressure area (like trough)
• Ex: sound, Slinky, some earthquake waves

• Energy: waves transport energy
• Phase: relative position between waves
• Frequency: how many waves per second
• Period: how long a time for one wave
• Wavelength: distance between waves
• Speed: how fast wave travels
• Amplitude: how big the wave is

• Depends on amplitude, frequency, and density of medium
• Power (energy/time) is proportional to square of amplitude and/or frequency
• With no losses to system, each wave has same energy as source
• As wave moves outward, energy spreads over larger area, reducing amplitude

• In phase: waves (or particles) are moving together, peaks line up with peaks and troughs line up with troughs
• Out of phase: waves (or particles) are not aligned; totally out of phase, peaks line up with troughs, troughs with peaks
• Phase relationship can be expressed in degrees, related to circular motion

Frequency

• Number of wave pulses passing a point in a given time
• Measured from identical points on successive waves
• Symbol is f or Greek letter nu (ν)
• Unit is hertz (Hz), has SI units of sec^-1
• Old unit is cycles per second

Period

• Time for one wave cycle Symbol is T
• Reciprocal of frequency T= 1/f

Wavelength

• Distance between identical points on successive waves
• Also distance wave travels in one period
• Measured in meters (or parts of meters)
• Symbol is Greek letter lambda (λ)

Wave Speed

• Same as any speed: distance/time, symbol v, units m/s
• Depends on medium and often on wavelength
• When speed depends on l in medium, medium is called dispersive
• Causes dispersion, or spreading of wave according to wavelength; ex: rainbow
• v = λ/ T = fλ

Amplitude

• In transverse wave, equals maximum displacement from equilibrium position
• In longitudinal wave, equals maximum pressure change from normal pressure
• Damping by dissipative forces reduces amplitude as wave travels

Wave Properties

• Rectilinear Propagation Reflection
• Impedance
• Refraction
• Diffraction
• Interference

Rectilinear Propagation

• In uniform medium, waves travel in straight lines, perpendicular to wavefront
• Wave velocity direction also perpendicular to wavefront

Reflection

• Occurs at boundary between two media
• Wave is returned to original medium
• Can be partial or complete depending on how new media transmits wave energy
• The more wave speed changes at media boundary, the more wave is reflected
• Law of reflection: angle of incidence equals angle of reflection

Impedance

• A measure of how easily a wave can be produced in a medium
• Equals ratio of applied force producing wave to resulting displacement velocity
• If impedances of two media match, wave is not reflected and is transmitted with no loss
• Impedance matching using transformers important for energy transmission systems

Impedance and Reflection

• If wave can't create displacement in particles of new media, impedance is infinite, wave is reflected out of phase: fixed end reflection
• If wave producing force can't be transferred to new media, impedance is zero and wave is reflected in phase: free end reflection

Refraction

• Bending of wave path at boundary between media
• Due to different wave speed in new medium
• Wave must strike boundary obliquely
• Since v = fλ , change in speed changes wavelength
• If v in new media < v in old media, wave bends towards normal of boundary & vice versa

Diffraction

• Spreading of wave beyond edges of barrier or past small opening
• Causes bending of wavefront
• Opening must be approximately same size as wavelength to diffract
• Example: sound waves diffracted by doorways, light waves aren't

Superposition Principle

• When two or more waves travel through the same space (medium) at the same time.
• Each wave proceeds independently as though no other waves were present
• The resultant displacement of any particle is the vector sum of displacements each wave would give it alone.
• Produces complex waveforms

Interference

• Effects due to two or more superposed waves
• If 2 waves of same type and frequency are in phase, displacements add, creating greater amplitude -- constructive interference
• Same waves out of phase, resultant displacement is now difference, decreasing amplitude -- destructive interference
• Often destructive and constructive interference happens in different places at same time, creates interference pattern
• Points of zero displacement, complete cancellation are called nodes
• Points of max displacement called antinodes
• Total wave energy doesn't change, just rearranged
• Standing wave: produced by interference of 2 periodic waves of same amplitude and wavelength traveling in opposite directions
• Usually wave reflected onto itself
• Nodes remain stationary, energy remains standing at antinodes
• Basis for all string and wind instruments