Physics 41: Waves, Optics, Thermolwillia2/41/41ch15_s13.pdf · It is possible to compensate for the...
Transcript of Physics 41: Waves, Optics, Thermolwillia2/41/41ch15_s13.pdf · It is possible to compensate for the...
Physics 41: Waves, Optics, Thermo
•Spread Out in Space: NONLOCAL
•Superposition: Waves add in space
and show interference.
•Do not have mass or Momentum
•Waves transmit energy.
•Bound waves have discreet energy
states – they are quantized.
•Localized in Space: LOCAL
•Have Mass & Momentum
•No Superposition: Two particles
cannot occupy the same space at the
same time!
•Particles have energy.
•Particles can have any energy.
Particles & Waves
is a What you Hear
The Pressure Wave sets the Ear Drum into Vibration.
The ear converts sound energy to mechanical energy to a nerve impulse which is transmitted to the brain.
electroencephalogram
= Mach #Sv
v
A wave packet in a square well (an electron in a box)
changing with time.
Electron Waves Probability Waves in an Ocean of Uncertainty
Superposition: Waves ADD in Space
Interference: Waves interfere with other waves and with themselves without any permanent damage!
Contour Map of Interference Pattern
Natural Frequency & Resonance
All objects have a natural frequency of
vibration or oscillation. Bells, tuning
forks, bridges, swings and atoms all
have a natural frequency that is related
to their size, shape and composition.
A system being driven at its natural
frequency will resonate and produce
maximum amplitude and energy.
Some Sytems have only ONE natural frequency: springs, pendulums, tuning forks, satellites orbits
Some Systems have more than one
frequency they osciallate with: Harmonics.
When the driving vibration matches
thenatural frequency of an object, it
produces a Sympathetic Vibration -
it Resonates!
Natural Frequency & Resonance
Sound Waves: Mechanical Vibrations
The Ear: An Acoustic Tuner
Cilia: Acoustic Tuning Forks
Eyes: Optical Tuner Optical Antennae: Rods & Cones
Rods: Intensity Cones: Color
Light Waves: EM Vibrations
E = Emax cos (kx – ωt)
B = Bmax cos (kx – ωt)
0 0
1Ec
BSpeed of Light in a vacuum:
186,000 miles per second
300,000 kilometers per second
3 x 10^8 m/s /v c n
Coupled Oscillators Molecules, atoms and particles are modeled as coupled oscillators.
Waves Transmit Energy through coupled oscillators.
Forces are transmitted between the oscillators like springs
Coupled oscillators make the medium.
Atoms are EM Tuning Forks
They are ‘tuned’ to particular
frequencies of light energy.
The possible frequency and energy states of an electron in an
atomic orbit or of a wave on a string are quantized.
2
vf n
l
Strings & Atoms are Quantized
34
, n= 0,1,2,3,...
6.626 10
nE nhf
h x Js
Super Strings: Particles are string vibrations in resonance
DARK ENERGY The Vibration of Nothing
THE GAME
We want to describe the motion of oscillating systems and find the natural frequency of objects and systems.
If you know the natural frequency of an object, the frequency it can oscillate or vibrate with, then you know everything about it, most importantly it’s ENERGY and the MUSIC it makes!
Use Hooke’s Law!
Review: Hooke’s Law
An elastic system displaced from equilibrium oscillates in a
simple way about its equilibrium position with
Simple Harmonic Motion.
Hooke’s Law describes the elastic response to an applied force.
Elasticity is the property of an object or material which causes
it to be restored to its original shape after distortion.
Ut tensio, sic vis - as the extension, so is the force
Hooke’s Law It takes twice as much force to stretch a spring twice as far.
The linear dependence of displacement upon stretching force:
appliedF kx
Hooke’s Law Stress is directly proportional to strain.
( ) ( )appliedF stress kx strain
+
Hooke’s Law: F = - k x
+
Hooke’s Law: F = - k x
+
Hooke’s Law: F = - k x
+
Hooke’s Law: F = - k x
+
Hooke’s Law: F = - k x
Review: Energy in a Mass-Spring 2 2 2
1 1 1
2 2 2K mv U kx kAE
Energy of Mass-Spring
The total mechanical energy is constant. Energy is continuously being transferred between potential energy stored in the spring and the kinetic energy of the block.
Importance of Simple Harmonic Oscillators
Simple harmonic oscillators are good models of a wide variety of physical phenomena
Molecular example
If the atoms in the molecule do not move too far, the forces between them can be modeled as if there were springs between the atoms
The potential energy acts similar to that of the SHM oscillator
1: (# / sec), [ ]Frequency f cycles f Hz
Review Wave Terms:
Displacement
of Mass
: / , [ ] secPeriod T time cycle T
: [ ]Amplitude A m
2 : 2 , [ ] /Angular Frequency f rad s
Review: Circular Motion
( ) cos ( )x t A t ( ) sinv t A t 2( ) cosa t A t
222
, , t t c
R vv R a R a R
R
Springs and Pendulums Obey Hooke’s Law and exhibit Oscillatory Motion. Find the equations of motion:
Position vs Time: Sinusoidal
Mass-Spring Systems that obey Hooke’s Law exhibit Simple Harmonic Motion
Position vs Time: Sinusoidal
F ma2( )m x
2k m
k
m
kx
angular
frequency
kx
2m
Tk
Simple Harmonic Motion k
m
2T
Does the period
depend on the
displacement, x?
Both the angular
frequency and
period depend only
on how stiff the
spring is and how
much inertia there is.
Position Equation for SHM
( ) cos ( )x t A t
A is the amplitude of the motion
is called the angular frequency
Units are rad/s
is the phase constant or the initial phase angle
A, , are all constants
Motion Equations for Simple Harmonic Motion
22
2
( ) cos ( )
sin( t )
cos( t )
x t A t
dxv A
dt
d xa A
dt
2a x
Notice:
Simple Pendulum For small angles, simple pendulums exhibit SHM because
for small angles
Two ways to find .
Rectilinear Coordinates:
2a x
( ) cos ( )x t A t
0( ) cos ( )t t
/ /s L x L
2
Angular Coordinates:
They are equivalent since ,a r x r
Simple Pendulum: Rectilinear
The arclength s = L is the displacement from equilibrium, x.
2L
g
sinF mg mg
/ /s L x L
xF mg ma
L
ga x
L
Accelerating & Restoring Force in the
tangential direction, taking cw as positive
initial displacementdirection:
2a x
2x
g
L
( ) cos ( )x t A t
Simple Pendulum: Angular
is the displacement from equilibrium, x.
2L
g
r F I
g
L
Accelerating & Restoring Torque in
the angular direction:
2
g
L
sinLmg Lmg
2I mL2mL mgL
0( ) cos ( )t t
2
Physical Pendulum: Rods & Disks If a hanging object oscillates
about a fixed axis that does not pass through the center of mass and the object cannot be approximated as a particle, the system is called a physical pendulum It cannot be treated as a
simple pendulum The gravitational force provides
a torque about an axis through O
The magnitude of the torque is mgd sin I is the moment of inertia about
the axis through O
Physical Pendulm Sample Problem
1. A uniform thin rod (length L = 1.0 m, mass = 2.0 kg) is suspended from a pivot at one end. Assuming small oscillations, derive an expression for the angular frequency in terms of the given variables (m, L, g), and then solve for a numerical value in rad/s. Show all your work. Sketch a diagram showing angle, lengths, lever arms, etc, and explain whatever is needed for a fantastic solution.
Physics 41 Chapter 15 Lecture Problems
2. A uniform disk (R = 1.0 m, m = 2.0 kg) is suspended from a pivot a distance 0.25 m above its center of mass. Ignore air resistance and any other frictional forces. Starting from Newton’s Second Law and assuming small oscillations, derive a reduced expression for the angular frequency in terms of the given variables: (R, m, g), and then solve for a numerical value in rad/s. Show all your work. Sketch a diagram showing angle, lengths, lever arms, etc, and explain whatever is needed for a fantastic solution.
Energy in a Mass-Spring 2 2 2
1 1 1
2 2 2K mv U kx kAE
2 2 2 2 2kv A x A x
m
Damped Oscillations
In many real systems, nonconservative forces are present
This is no longer an ideal system (the type we have dealt with so far)
Friction is a common nonconservative force
In this case, the mechanical energy of the system diminishes in time, the motion is said to be damped
Damped Oscillation
One example of damped motion occurs when an object is attached to a spring and submerged in a viscous liquid
The retarding force can be expressed as R = - b v where b is a constant b is called the damping
coefficient
Damping Oscillation
The position can be described by
The angular frequency will be
2 cos( )b
tmx Ae t
2
2
k b
m m
2
2
02
b
m
Damping Oscillation
When the retarding force is small, the oscillatory character of the motion is preserved, but the amplitude decreases exponentially with time
The motion ultimately ceases
A graph for a damped
oscillation
The amplitude decreases with time
The blue dashed lines represent the envelope of the motion
Types of Damping
Graphs of position versus time for (a) an underdamped
oscillator
(b) a critically damped oscillator
(c) an overdamped oscillator
For critically damped and overdamped there is no angular frequency
2 cos( )b
tmx Ae t
2
2
02
b
m
02
b
m
02
b
m
02
b
m
0
k
m
Forced Oscillations
It is possible to compensate for the loss of energy in a damped system by applying an external force
The amplitude of the motion remains constant if the energy input per cycle exactly equals the decrease in mechanical energy in each cycle that results from resistive forces
After a sufficiently long period of time, Edriving = Elost to
internal Then a steady-state condition is reached
The oscillations will proceed with constant amplitude
Forced Oscillations
The amplitude of a driven oscillation is
0 is the natural frequency of the
undamped oscillator
This has damping
0
22
2 2
0
FmA
b
m
Resonance Resonance (maximum
peak) occurs when driving frequency equals the natural frequency
The amplitude increases with decreased damping
The curve broadens as the damping increases
The shape of the resonance curve depends on b
0
22 2
0
FmA
When the driving vibration matches
thenatural frequency of an object, it
produces a Sympathetic Vibration -
it Resonates!
Natural Frequency & Resonance
http://www.youtube.com/watch?v=17tqXgvCN0E