Physics 7C, Lecture 1Physics 7C, Lecture 1

Winter Quarter -- 2007

Professor Robin Erbacher

343 Phy/Geo

erbacher@physics.ucdavis.edu

AnnouncementsAnnouncements

• Course syllabus (policy, philosophy) on the web: http://physics7.ucdavis.edu

• Quizzes every other lecture, ~20 minutes each, average of 4 best = 45% (or 20)% of grade.

• Final on Monday, March 19 10:30 am. If you cannot make this we suggest 7C in a different quarter.

• Quiz #1 next Wednesday, see calendar on web.

• Turn off cell phones and pagers during lecture.

Short Review of Physics 7BShort Review of Physics 7B

Simple Harmonic Oscillators

We will see how oscillatory motion learned in 7B that we see in all vibrational systems gives us

waves, important to much of the world around us.

Two basic motions in nature:•Linear or curvilinear (7B)•Periodic motion (7B & 7C)

Simple Harmonic Oscillator (SHO)

Simple Harmonic Oscillator (SHO)

Equlibrium Position The position at which all forces acting on an object sum to zero.

Displacement Change in the position of an object with respect to the equilibrium position.

Restoring Force Force that acts on the object that tends to make it move towards the Eq. position.

Simple Harmonic Motion When Restoring force proportional to the Displacement

First, the ideas of SH Motion (SHM): The Constructs

xF k x=−uur r

SHO: Simple 2-d PendulumSHO: Simple 2-d PendulumSimple Harmonic Oscillators:

•Need restoring force F = -kx•Restoring force proportional to displacement(for small enough displacements most oscillators obey SHM)

Modeling Simple Harmonic OscillatorsBegins with Newton’s 2nd Law:

F = ma

-g= d2 l dt2

€

F = m a

-mgsinθ = mld2θ

dt 2

- gsinθ = ld2θ

dt 2

sinθ ≈ θ (θ small)

-g

lθ =

d2θ

dt 2

T

mg

mgcos

mgsin

Periodic FunctionsPeriodic FunctionsWhat kind of function gives back thesame function when differentiated twice?

€

d

dt Asinbt = bAcosbt

d2

dt 2Asinbt =

d

dt[bAcosbt] = - b2Asinbt

€

Try θ(t) = Asinbt :

2

T = 2/b

Simple Harmonic Oscillators

Simple Harmonic Oscillators

So, do we have the solution?

€

-b2Asinbt = d2θ

dt 2

€

-g

lθ =

d2θ

dt 2

We have:

We want:

€

∴ b = g

l

€

T = 2πl

g… and …

€

y(t) = Asin(2π

Tt +φ)A generalized SHO

solution is of the form:

€

Asinbt = θ(t)and:

Example: A pendulum in the playground…

€

-k

my =

d2y

dt 2

€

-g

lθ =

d2θ

dt 2

Another SHO: Mass on a Spring

Another SHO: Mass on a Spring

Simple Harmonic Motion can be usedto describe many phenomena

€

F = ma = -ky

F = md2y

dt 2

-ky = md2y

dt 2

€

y(t) = Asin(2π

Tt +φ) + BA generalized solution

is of the form:

Hooke’s Law:Restoring force spring constant k(determines pull)

Displacement fromUnstretched length

€

∴T = 2πm

k… and …

Period Versus FrequencyPeriod Versus Frequency

2

€

T = 2πm

k€

T = 2πl

gPendulum:

Mass/Spring:

How is the frequency fof the oscillations relatedto the period T?

1) f is proportional to T 2) f,T are inversely proportional3) f is always twice T4) f is not related to T5) I have no clue6) I’d like to buy a vowel

€

T = 1f = 1

1 sec = secPeriod:

To Summarize…To Summarize…

y(t) = A sin (2 t/T + )y(t) = A sin (2 t/T + )

y=Displacement

A=AmplitudeT=Period =phase

Introduction to WavesIntroduction to Waves

Wave Phenomena

We will see how waves are responsible for sound, light, propagation of information, and all of

matter (when we get to quantum mechanics).

A wave is a disturbance: a type of internal motion of a medium, in which the displaced portion returns to equilibrium. This disturbance propagates in space.

Waves in NatureWaves in Nature

A surfer braves the monster waves that form in an area on the north shore of Maui called “Jaws”, where about 12 times a year, the conditions are just right to produce some of the largest waves in the world: the shape of the

beach sculpts the swells that originate from as far as Alaska into 40- to 70-foot [12- to 21-meter] walls of water.

Steven Kornreich www.beachlook.com

Thanks to Prof. Calderonwho found this

photo

Waves: Energy and Amplitude

Waves: Energy and Amplitude

Simple WavesSimple WavesWhat is a wave?Particles of the medium oscillate about their equil- ibrium positions in both a spatial and a temporal way.

What other kinds of waves are there?

Transverse Waves

Longitudinal Waves

Combo Waves

(circular)

WaterWaves

We will focus on these

What is the simplest type of wave?

A single wave due to a non-periodic disturbance: a traveling pulse wave.

Wave ParametersWave ParametersCertain independent parameters characterize all waves:1) Amplitude: Controlled by the magnitude of the forces that

started the wave.2) Speed: Determined by the properties of the medium.3) Direction: Determined by the direction of the forces starting

the wave. 1) Longitudinal: Oscillations in direction of wave velocity v 2) Transverse: Oscillations are perpendicular to v

4) Frequency f of oscillations: Controlled by forces starting the wave.

Wave: disturbancepropagates in x…

Snapshot v. MovieSnapshot v. MovieSome waves are simply a pulse, and some are repetitive. These are harmonic (or sinusoidal), generated by SHOs. Harmonic waves have a dependent variable, wavelength , the distance at which the oscillation repeats.

wavelength:vwave/f

2

y(x)

2

y(t)

Snapshot: Hold time constant, see where we are in space.

Movie: Go forward in time, see how spatial points move.

The Wave RepresentationThe Wave RepresentationDescribing the behavior of harmonic (sinusoidal) waves is extremely important in our physical world.

€

y(x,t) - y0 = A sin(2π

λx ±

2π

Tt +φ) + BThe most general

solution is of the form:

Because there is both a time-dependence and a translation of the wave in space, we need to represent the wave using both t and x.

€

What are all these parameters?x: location in the medium (spatial)t: time (temporal)T,f, period, freq., wavelengthA: amplitude phase

€

Δy(x,t) = Asin[Φ(x,t)]

Too complicated? Think of the sin argument as one big phase (or angle)

Total phase

Period, Frequency, Wavelength, Wave Speed

Period, Frequency, Wavelength, Wave Speed

€

T = 1f (sec)period:

€

f = v λ (1/sec)frequency:

wavelength:

€

= vwave f = vwaveT (meters)

What’s the wave velocity?

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

€

Φ = 2π

λx ±

2π

Tt +φ = wave phase

Ride the wave: constant

€

± x = λ

2πΦ m

λ

Tt -

λ

2πφ

€

v = dx

dt= m

λ

T

If we choose + in the wave function, the velocity is negative.

Particle VelocityParticle Velocity

€

vwave = dxdt = mλ T

So, the velocity of the wave, or propagating disturbance, can be found by riding along the wave at constant :

What is the velocity of a particle (or length of string) on the wave?

As always:

€

v = dy(x,t)

dt Why y?

Transverse Waves:Particle moves as SHO!

€

vparticle = A2π

Tcos(

2π

Tt ±

2π

λx +φ)

€

y(x,t) - y0 = Δy(x,t) = A sin(2π

Tt ±

2π

λx +φ) + B

Note: I swapped x and t term. Block notes differ from DL expression. Both ok. Use DL version

Mexican Wave!

Two Types of Wave PropagationTwo Types of Wave PropagationWaves transfer energy without bringing along the mass. Particles get disturbed, collide, but stay oscillating about the equilibrium, they don’t move with the wave.

But do all wave disturbances move perpendicular to direction of propagation?

Transverse wave: particle motion is perpendicular to direction of propagation.

Longitudinal wave: particle direction is same as (aLONG the) wave propagation

Longitudinal Waves

Sound WavesSound Waves

€

P(x,t) - Patm = ΔP(x,t) = A sin(2π

Tt ±

2π

λx) ⇒ AsinΦ(x,t)

The sound vibrations in 1-Dimension, such as long, narrow tubes, trombone, flute, trumpet, follows harmonic oscillations. But how does one describe the vibrations of the air?

It’s all about pressure (density) fluctuations!

Equilibrium = Atmospheric (or surrounding) pressure

Power and IntensityPower and IntensitySound is a pressure fluctuation in a medium. Sound energy is transported through the medium via these fluctuations.

Power: sound energy time

emitted by a source

Intensity: Psource

area(area of wavefront)

How About Light?How About Light?What kind of wave is a light wave? It’s a transverse excitation, perpendicular to the direction of wave propagation.

What’s the medium that’s displaced as the wave propagates?

Nothing!

Light propagates via oscillating electric and magnetic fields (more on this later in the course!)

The Enigmatic Ether!

Light: Visible, and Invisible Light: Visible, and InvisibleThe light we see is a small portion of the radiation that exists!

Visible Light:4.3-7.5 x 1014 Hz

Ultra Violet (UV)X-rays/rays

Infra Red IRwave, AM/FM, TV

frequency

wavelength

Light: Visible, and Invisible Light: Visible, and InvisibleThe light we see is a small portion of the radiation that exists! The light we see is a small portion of the radiation that exists!

Visible Light:4.3-7.5 x 1014 Hz

Ultra Violet (UV)X-rays/rays

Infra Red IRwave, AM/FM, TV

frequency

wavelength