View
250
Download
1
Embed Size (px)
Citation preview
Physics 7C, Lecture 1Physics 7C, Lecture 1
Winter Quarter -- 2007
Professor Robin Erbacher
343 Phy/Geo
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