Exam 4: Waves and Optics

SHM OF SPRINGS AND PENDULUMS

Oscillations of a Spring

• If an object vibrates back and forth over the same path, each cycle taking the same amount of time, the motion is called ——

• If the spring is hung vertically, the only change is in the —— position, which is at the point where the spring force equals the —— ——

Terminology

• Displacement is measured from the —- point

• Amplitude is the —— displacement from the equilibrium

  • If the equilibrium is not 0, you can use (max-min / 2)

• Wavelength is the distance between two consecutive corresponding points of the same phase on the wave (crest, trough)

  • Crest to crest

  • trough to trough

  • Equilibrium point to SECOND adjacent equilibrium point

• Wave number: the number of wave cycles per unit distance (1/λ)

• A cycle is a full to-and-fro motion

• Period is the —- required to complete —- —-

• Frequency: is the —- of cycles completed per —-

• Angular Frequency: measures the rate of phase change in radians per second

  • ω = 2πf = √k/m so f = 1/2π√k/m

  • Pendulum has a different equation, one that relates length and gravity b/c it’s independent of mass

periodic

equilibrium, gravitational force

equilibrium

maximum

time one cycle

number second

Spring Force and Hooke’s Law

• The force exerted by a spring depends on it's displacement, and if given by Hooke’s Law

  • Fs = -kx = ma

  • Negative sign means the restoring force acts in the direction opposite to the displacement

  • Force is —— constant so the acceleration is —- constant either, as both rely on ——-

not, not, displacement

Period and Frequency

• The time between pulses/cycles is called the ——, and the number of pulses/cycles per unit time is the ——

• period and frequency are ——

  • T = 1/f and f = 1/T

  • T unit = s; f unit = Hz

• For a mass spring system, in terms of angular frequency we can also say that f = 1/2π√k/m, and therefore T = 2π√m/k

period frequency

reciprocal

Elastic (Spring) Potential Energy

• Elastic force results from stretching or compressing an object

• Elastic PE is the energy gained when —- is done to stretch a spring to a distance (x)

  • PEs = 1/2kx²

• Simple harmonic motion occurs when the energy of the system repeatedly changes from —— energy to —— energy and back again

  • Energy added by doing work to stretch the spring is transformed back and forth between PE and KE

  • Total energy remains constant at all times

work

potential kinetic

Energy in the simple harmonic oscillator

• If the mass is at its limits of its motion (maximum amplitude), the energy is all ——-

• If the mass is at the equilibrium point, the energy is all ——-

• We know what the PE is at each turning point/max amplitude, and this represents the system’s total energy:

  • E = 1/2kA²

• Since E = PE + KE, we can say 1/2kx² + 1/2mv² = 1/2kA² and we can use this to solve for the velocity

  • Remember, the total energy is constant, and the system is switching between all PE at max amplitude and all KE at equilibrium position

potential, kinetic

Simple Pendulum

• As long as the cord is considered massless and the amplitude is small, the period does NOT depend on ——

• T = 2π√L/g

  • Remember, period of a mass-spring system is T = 2π√m/k, but since pendulum period is NOT dependent on mass we replace with:

  • L = analog to mass

  • g = analog to k

  • Since T is reciprocal to f, we can say that f = 1/2π√g/L

• Energy of a pendulum

  • At max amplitude, E = PE = mgh and KE = 0

  • At equilibrium position, E = KE = 1/2mv² and PE = 0

mass

Damped Harmonic Motion

• Harmonic motion with a —- or —- force

  • Amplitude ——- over time due to dissipative forces like friction or air resistance

• If the dampening is small, we can treat is as an envelope that modifies the undamped oscillation

  • cos graph with — amplitude = dampening

frictional, drag, decreases, decreased

Forced Oscillations and Resonance

• Forced vibrations occur when there is a period driving force

  • This force may or may not have the same period as the natural frequency of the system

• However, if the frequency is the —— as the natural frequency, the amplitude can become very ——, this is called resonance

  • System will begin to oscillate on its own without a driving force in the presence of its resonant frequency

same, large

Summary: SHM

• SHM is —— pattern

• The restoring force is —- to the displacement for (for small displacements it obeys Hooke’s law)

• The total energy is continually changing from —- to —- and back

• Period is independent of ——- (lab topic)

• Period is independent of —— in a pendulum (L/g), but NOT in mass-spring systems (m/k)

• In SHM, acceleration is —- ——- because force is not constant (depends on displacement, so acceleration also depends on displacement)

sinusoidal
Potential

potential kinetic

amplitude

mass

not constant

Longitudinal Waves

• Need a —- to propagate

• Region of compression moves —-/—- to the medium

• The displacement of the medium is along/parallel to the direction of the propagation of the wave

• — waves are longitudinal waves

  • Consist of —— variations in the air

  • The diaphragm of a speaker oscillates back and forth producing regions of higher and lower pressure, and these regions propagate through the air as variations in air pressure and density, forming longitudinal sound waves

Transverse Waves

• Displacement is —— to the direction of travel

• Example: light waves, pulling a string, ripple over water

medium, along parallel sound

pressure

perpendicular

Intensity

• The amplitude is air vibrations, what we hear is the intensity

  • The loudness of a sound is much more closely related to the log of the intensity

  • The intensity of a wave is the —— transported per unit time across a unit area with units of W/m²

• Sound level is measured in decibels and defined as

  • dB = 10log(I/Io) where Io is the threshold of hearing, the quietest intensity a human ear can detect

• Intensity of sound follows the —— —- law

  • I = 1/r²

  • So if you double the distance from the source of the sound, the intensity will be decreased by ¼

• Intensity is equal to the —- ——

• Log review

  • You can rearrange x = logy into y = 10^x

energy

inverse square

squared amplitude

Velocity

• Velocity can be related to wavelength and frequency with the equation v = λf

• The velocity of a wave depends on properties of the medium where the wave propagates

  • The stiffer the medium the —- the sound travels through the medium

faster

Interference

• A —- property of light, the process in which two or more waves combine

  • When two waves pass through the same region of space, they will interfere, and it can be destructive or constructive

• Principle of superposition

  • When two or more waves combine the resulting disturbance or displacement is equal to the —— of the individual disturbances

  • Addition depends on the PHASE of the 2 waves

• Types of interference

  • When two waves are exactly in-phase (0º) there is —— —- interference and the resulting amplitude is the sum of the two amplitudes (large)

  • When two waves are exactly out of phase (180º) there is —- —- interference and the two aves are cancelled out

  • When the two waves are partially out of phase there will be some —- interference and the resulting amplitude will be the difference between the two amplitudes (lower)

wave

sum

total constructive

total destructive

destructive

Standing Waves

• Standing waves can be produced on a string with both ends fixed , and it’s a wave pattern that appears to stand still rather than traveling

• Formed by the —- interference of 2 waves with the same —— and —— traveling in —- directions

• It oscillates in place with formation of —- and ——

  • Nodes = result of total —— interference, —- amplitude

  • Antinodes = result of total —- interference, —- amplitude

• The length of the medium that supports the standing wave must be an integer multiple of λ/2

  • The distance from one node to an adjacent node (or antinode to adjacent antinode) is EXACTLY —

  • Since the distance between nodes is exactly half a wavelength, the λ must be —— the length of the string

  • Depending on the number of λ/2 multiples that can be trapped in the length relates to the fundamental frequency of the 1st harmonic

    • Fundamental frequency is the frequency that creates — nodes and —- antinode

    • f = v/2L

• The pitch of the sound depends on its ——-, and the loudness of the sound depends on its —— (—- —-) and ear sensitivity

constructive, frequency amplitude, opposite, nodes antinodes, destructive no, constructive maximum, λ/2, twice, 2, 1

frequency, intensity, squared amplitude

Doppler Effect

• Occurs when a —— of sound is moving with respect to an ——

  • If the observer is moving with respect to the source, the λ remains the same, but the wave speed is different for the observer

• A source moving toward an observer appears to have a —— frequency and a —— wavelength

• A source moving away from an observer appears to have a —- frequency and a —— wavelength

source observer

higher, shorter

lower, longer

Beats and Beat Frequency

• Waves can also interfere in time, causing a phenomenon called beats

  • Periodic fluctuation in sound loudness

• When two sound waves of —— frequencies are played together, caused by —- and —— interference

  • Beats are the slow envelope around two waves that are relatively close in frequency

• Beat frequency is the rate at which this pattern ——, calculated as the absolute difference between the two frequencies

  • f_beat = |f₁ - f₂|

• Understanding the graph

  • The beat sum will be greatest/loudest when they are close to in-phase, and will be quietest when out of phase

  • When the peaks of both waves align, they add together to form a very tall wave (large amplitude)

  • When the peak of one wave aligns with the trough (low point) of the other, they cancel each other out, making a flat line (zero amplitude)

close, constructive destructive

repeats

Subtractive Color Mixing

• The selective —— of light is a form of subtractive color mixing, some pigments absorb some wavelengths of light more than others

  • Process of creating colors by absorbing/subtracting specific wavelengths of light from white light, dyes, pigments, or filters, allowing only the ——, —- light to reach the eye

  • In color printing, the three primary pigments are cyan, yellow, and magenta

• Examples

  • White light passed through a magenta filter will absorbs green but transmits and reflects red and blue

  • White light passed through a yellow filter will absorb blue but transmits and reflects green and red

  • White light passed through a cyan filter will absorb red but transmits and reflects green and blue

• When light strikes an object some of the light undergoes specular reflection where all of the light is reflected as if by a ——, and the rest of the light will undergoes diffuse reflection and is reflected in —- ——

absorption, unabsorbed reflected

mirror, all directions

Additive Color Mixing

• The process of —- different wavelengths of light in order to produce a response interpreted as a —— color

• Combining three primary colors: blue, green, and red in different amounts can produce responses in our brains corresponding to the colors we are used to identifying

  • Red + green = yellow

  • Blue + green = cyan

  • Blue + red = magenta

  • Combing all three primary colors = white

mixing

different

Law of Reflection

• The angle of reflection that the ray makes with the normal to the surface of incidence —— the angle of incidence

  • Ø1 = Ø2

equals

Plane Mirrors

• Diffuse Reflection

  • When light reflects from a rough surface, the law of reflection still holds, but the angle of ——- varies

  • With diffuse reflection, your eyes see reflected light at —- angles

• Specular reflection (from a plane/flat mirror)

  • All light is reflected in the —— direction

  • Your eyes must be in the correct position to see the reflected image

• Virtual Image

  • When we look in a plane mirror, the reflected image appears —— the mirror

  • As a result of the reflection from the mirror, we see a —— image, which we construct from the extension of the rays

• The image is —-, —-, and the —- —- as the object

  • When doing flat mirror problems, if you are asked for angle of reflection or refraction you are ALWAYS comparing to the normal, NOT the mirror plane itself

incidence all

same

behind, virtual

virtual upright same size

Mirrors

• Spherical mirrors can be either reflective on the inside surface (concave) or the outside surface (convex)

  • Concave

    • Reflects light from the —- surface

    • Parallel rays —- as they leave the mirror (opposite of lens)

    • Can magnify an image like a lens

  • Convex

    • Reflect light from —- surface

    • Parallel light rays —— as they leave the mirror (opposite of lens)

    • The image is ALWAYS ——-, ——, and —— (incident parallel rays seem to be coming from focal point behind the mirror)

inside, converge

outside, diverge, smaller, upright, virtual

Index of Refraction

• Says how —— light travels through a —— with respect to its speed in a ——/—-

  • n = c/v where c = speed of light in a vacuum and v = speed of light in the medium

• Smaller index of refraction = light travels ——

• Part of Snell’s law, impacts the amount of bending of light when it moves from one medium to another, the amount of bending depends on the change of IOR between mediums

• Index of refraction depends on ——

• The air has the lowest IOR possible at ——-

  • This is when speed of light in a vacuum (c) is equal to the speed of light in the media … this is just air

  • There is no upper limit, but must be greater than 1.00

fast, medium, vacuum air

fast

wavelength

1.00

Dispersion

• The index of refraction varies slightly depending on the —— of light

• This is how prisms and water droplets create a —— from sunlight

• Splitting of —— light into its constituent colors (like a rainbow) as they travel at different —— through a medium due to differences in IOR due to their differing wavelengths

wavelength

rainbows

white, speed

Law of Refraction

• Light changes —— (—-) when crossing a boundary from one medium to another, this is called refraction

• The angle the outgoing ray makes with the normal is called the angle of refraction

  • Remember, the normal is always perpendicular to the surface of the medium

  • SO angle of refraction is between the —- and the ——, NOT the —— boundary

direction bends

ray, normal , medium

Snell’s Law

• Relates the indices of refraction of two media to the directions of ray propagation in terms of the angles to the —— surface

• The medium with a higher IOR (—- speed of light) bends the ray —- to the normal plane (—- angle)

• n1Ø1 = n2Ø2

normal

slower, closer, smaller

Total Internal Reflection

• If light hits the plane of incidence at an angle higher than the —— angle when moving from a medium of high to low IOR, it can be totally internally reflected

  • Only happens when light travels from a medium of —— IOR to one of —— IOR

  • Øi > Øc = TIR

  • Øi = Øc then light travels at 90º along the medium boundary

  • Øc = sin^-1(n1/n2)

• This is how fiber optic cables work

critical, smaller

high low

Lenses

• Convex/Converging

  • Light rays will ——- toward the optical axis

  • Incident parallel rays all —— pass through the —- ——

  • The distance from the center of the lens to the focal point is the —- ——

    • Focal length determines where the light rays will be focused and the image is formed

  • Has a —— focal length value

• Concave/Diverging

  • Light rays will —— from the optical axis

  • Incident parallel rays all —— to be coming from the focal point

  • Has a —— focal length value

• Thin lens equation

  • 1/f = 1/di + 1/do

  • m = hi/ho = -di/do

• Refractive/Optical Power

  • Measured in ——

  • Optical power is —— to focal length (P = 1/f)

converge, actually focal point, focal length, positive

diverge, appear, negative

diopters, reciprocal

Diffraction

• Waves bend/diffract when passing through a small opening/—— or around small ——

  • The wave then spreads after passing through the slit

  • The wave properties of light are demonstrated by phenomena where light bends, spreads, or interferes with itself = diffraction

• Slit properties

  • If the slit is small enough that it is comparable to the size of the wavelength of light, we will see diffraction/spreading

  • If the slit is too big, there will be no diffraction just simple parallel rays

• Diffraction effect

  • Depends on the ratio of —— to —— ——- —— (λ/a)

  • Large ratio (small slit) = —- wave pattern

  • Small ratio (wide slit) = —- wave pattern

  • Basically, narrow slit = more spreading/diffraction

• Diffraction Limit/Rayleigh Criterion

  • the wave nature of light limits ability to see —— objects

  • Criterion for the minimum resolvable detail

  • Imaging process if diffraction limited when the first diffraction minimum of the image of one source point coincides with the diffraction maximum of another

  • Why we can see bacteria but not viruses under optical microscope

slit, objects

wavelength, slit opening size

wide narrow

small

Single Slit Diffraction

• The parallel rays will interfere at some points as they travel through the slit and onto the screen

  • Creates a broad central maxima bright band, with smaller bright spots alternating with dark spots

• Where crest meets crest exactly in phase = —- interference = —— spot

• Where crest meets trough exactly out of phase (shifting λ/2) = —— interference = —— spot

• This results in a pattern of light and dark spots across the screen, with the

• Equation: sinØ_dark = m(λ/a)

  • Calculates angle of a DARK SPOT

  • Where m = integer for each multiple of λ (central maxima m = 0, first dark spot m = 1)

  • Remember, sinø = x/L

constructive bright, destructive dark,

Double Slit Diffraction

• When light passes through two slits, it makes two lights waves —- —- with each other

  • These two waves will interfere constructively or destructively depending on a phase difference (λ, λ/2, etc)

  • If two waves travel equal optical distances to the screen, they interfere ——- and a —- spot is seen

  • If two waves travel distances differing by one full λ, the two waves will ALSO interfere —— and a —- spot is seen

  • But if two waves trvel distances differing by λ/2 then the two waves interfere —— and a —- spot is produced

  • This results in an —— pattern of bright and dark spots = fringe pattern

    • NO central maxima!! Evenly sized and spaced dark and bright spots

• Equation: sinØ = (mλ/d)

  • Calculates angle of a BRIGHT SPOT

  • d = distance between the two fringes, NOT the width of the slit like in the single slit equation

in phase

constructively bright

constructively bright

destructively dark

alternating

Spectroscopy

• Study and interpretation of colors of the —— spectra

• Atoms and molecules can be identified when they are in a thin gas through they characteristic emission lines

• Analyzes a set of color —— or —— by a compound to identify it

electromagnetic

emitted absorbed

Scattering of Light

• Redirection of light rays in random, different directions when hitting particles in a medium, causing effects like blue skies or red sunsets

• The sky is blue due to scattering

  • Rayleigh Scattering: Intensity of incident light = 1/λ⁴

    • Ex. the scatter of λ = 400nm (blue) is 9x stronger than that of λ = 700nm (red)

  • So shorter wavelengths will be scattered ——, so blue light will scatter more than red light, and this is why the sky is seen as blue

  • It’s the atmosphere and molecules that allow the sky it look blue, without atmosphere there would be nothing to scatter the light and it would just be black like space

  • In the evening, the sun is so low on the horizon the blue light scatters so much it has be removed, so the only the —- scattered wavelengths/colors appear (longer λ: red, orange, yellow)

more

least

Phenomena that represent wave properties of light include ——, ——-, ——-, —— and ——

interference, diffraction, polarization dispersion, scattering

Reflection: bouncing back of light after it hits a surface, the angle of reflection that the ray makes with the normal to the surface of incidence equals the angle of incidence

Refraction: Light changes direction (bends) when crossing a boundary from one medium to another

Dispersion: change of IOR with wavelength of light, causes prisms and rain rainbows (splitting of white light into its constituent colors (like a rainbow) as they travel at different speeds through a medium)

Diffraction: bending, spreading of light waves when they encounter an obstacle, edge, or pass through a narrow opening

Scattering: redirection of light rays in different directions when hitting particles in a medium, causing effects like blue skies or red sunsets

ok