Physics 1501 – Lecture 27 Physics 1501: Lecture 27 Today s Agenda Homework #9 (due Friday Nov. 4) Midterm 2: Nov. 16
Katzenstein Lecture: Nobel Laureate Gerhard t’Hooft Friday at 4:00 in P-36 …
Topics SHM Damped oscillations Resonance 1-D traveling waves Physics 1501: Lecture 27, Pg 1
Simple Harmonic Oscillator
Spring-mass system
k
k d2x = !! 2 x where ! = 2 m dt
F = -kx m
a
x
z-axis
x(t) = Acos(ωt + φ)
Pendula
R
d 2! = !" 2! dt 2
θ d
Mg
θ = θ0 cos(ωt + φ) General physical pendulum ! = » Simple pendulum Torsion pendulum k != I
xCM
MgR I
wire θ
τ
I
Physics 1501: Lecture 27, Pg 2
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Physics 1501 – Lecture 27 What about Friction?
Friction causes the oscillations to get smaller over time This is known as DAMPING. As a model, we assume that the force due to friction is proportional to the velocity.
F f = !b
dx dt
! F = ma
" LHS = F = FS + F f = !kx ! b ! kx ! b
dx dt
RHS = ma = m
d 2x dt 2
dx d 2x =m 2 dt dt Physics 1501: Lecture 27, Pg 3
What about Friction? ! kx ! b
dx d 2x =m 2 dt dt
We can guess at a new solution. x = A exp(#
bt ) cos("t + ! ) 2m
With, 2
k & b # & b # 2 (= '$ ! = (o ' $ ! m % 2m " % 2m "
2
Physics 1501: Lecture 27, Pg 4
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Physics 1501 – Lecture 27 What about Friction? x = A exp(#
bt ) cos("t + ! ) 2m
What does this function look like? (You saw it in lab, it really works) 1.2 1 0.8 0.6 0.4
A
0.2 0 -0.2 -0.4 -0.6 -0.8 -1
!t
Physics 1501: Lecture 27, Pg 5
What about Friction? x = A exp(#
bt ) cos("t + ! ) 2m
There is a cuter way to write this function if you remember that exp(ix) = cos x + i sin x . x = A exp(i ("t + ! #)) 2
k & b # b (= '$ ! +i m % 2m " 2m
" = " o2 # ! 2 + i! Physics 1501: Lecture 27, Pg 6
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Physics 1501 – Lecture 27 Damped Simple Harmonic Motion Active Figure
" = " o2 # ! 2 ± i!
Frequency is now a complex number! What gives? Real part is the new (reduced) angular frequency Imaginary part is exponential decay constant
"o > !
"o = !
underdamped
critically damped
"o < !
overdamped Physics 1501: Lecture 27, Pg 7
Driven SHM with Resistance
To replace the energy lost to friction, we can drive the motion with a periodic force. (Examples soon). F = F0 cos(ωt)
Adding this to our equation from last time gives, F0 cos("t ) ! ks ! b
ds d 2s =m 2 dt dt
Physics 1501: Lecture 27, Pg 8
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Physics 1501 – Lecture 27 Driven SHM with Resistance
So we have the equation, F0 cos("t ) ! ks ! b
ds d 2s =m 2 dt dt
As before we use the same general form of solution, s = A cos("t+ ! )
Now we plug this into the above equation, do the derivatives, and we find that the solution works as long as, A=
F0 / m (! 2 " ! 02 ) 2 + (
b! 2 ) m Physics 1501: Lecture 27, Pg 9
Driven SHM with Resistance
So this is what we need to think about, I.e. the amplitude of the oscillating motion, A=
F0 / m (! 2 " ! 02 ) 2 + (
Note, that A gets bigger if Fo does, and gets smaller if b or m gets bigger. No surprise there.
b! 2 ) m
Something more surprising happens if you drive the pendulum at exactly the frequency it wants to go,
!= k m
Then at least one of the terms in the denominator vanishes and the amplitude gets real big. This is known as resonance. Physics 1501: Lecture 27, Pg 10
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Physics 1501 – Lecture 27 Driven SHM with Resistance
Now, consider what b does, F0 / m A= b! (! 2 " ! 02 ) 2 + ( ) 2 m
b small
b middling b large ω = ω0
ω
Physics 1501: Lecture 27, Pg 11
Dramatic example of resonance
In 1940, turbulent winds set up a torsional vibration in the Tacoma Narrow Bridge
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Physics 1501 – Lecture 27 Dramatic example of resonance
when it reached the natural frequency
Physics 1501: Lecture 27, Pg 13
Dramatic example of resonance
it collapsed !
Other example: London Millenium Bridge Physics 1501: Lecture 27, Pg 14
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Physics 1501 – Lecture 27 Lecture 27, Act 1 Resonant Motion
Consider the following set of pendula all attached to the same string
A D
B
If I start bob D swinging which of the others will have the largest swing amplitude ? (A)
(B)
C
(C) Physics 1501: Lecture 27, Pg 15
Chap. 13: Waves What is a wave ?
A definition of a wave: A wave is a traveling disturbance that transports energy but not matter.
Examples: Sound waves (air moves back & forth) Stadium waves (people move up & down) Water waves (water moves up & down) Light waves (what moves ??) Animation Physics 1501: Lecture 27, Pg 16
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Physics 1501 – Lecture 27 Types of Waves
Transverse: The medium oscillates perpendicular to the direction the wave is moving. Water (more or less) String waves
Longitudinal: The medium oscillates in the same direction as the wave is moving Sound Slinky
Physics 1501: Lecture 27, Pg 17
Wave Properties
Wavelength: The distance λ between identical points on the wave. Amplitude: The maximum displacement A of a point on the wave.
Wavelength λ
Amplitude A A Animation Physics 1501: Lecture 27, Pg 18
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Physics 1501 – Lecture 27 Wave Properties...
Period: The time T for a point on the wave to undergo one complete oscillation.
Speed: The wave moves one wavelength λ in one period T so its speed is v = λ / T.
v=
Animation
! T
Physics 1501: Lecture 27, Pg 19
v=λ/T
Wave Properties...
We will show that the speed of a wave is a constant that depends only on the medium, not on amplitude, wavelength or period λ and T are related !
λ=vT
Recall
or λ = 2π v / ω
(since T = 2π / ω )
or λ = v / f
(since T = 1/ f )
f = cycles/sec or revolutions/sec ω = rad/sec = 2πf Physics 1501: Lecture 27, Pg 20
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Physics 1501 – Lecture 27 Lecture 27, Act 2 Wave Motion The speed of sound in air is a bit over 300 m/s, and the speed of light in air is about 300,000,000 m/s. Suppose we make a sound wave and a light wave that both have a wavelength of 3 meters. What is the ratio of the frequency of the light wave to that of the sound wave ?
(a) About 1,000,000 (b) About .000,001 (c) About 1000 Physics 1501: Lecture 27, Pg 21
Wave Forms
So far we have examined continuous waves that go on forever in each direction !
We can also have pulses caused by a brief disturbance of the medium:
v
v
v
And pulse trains which are somewhere in between.
Physics 1501: Lecture 27, Pg 22
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Physics 1501 – Lecture 27 Mathematical Description y
Suppose we have some function y = f(x): x
0
y
f(x-a) is just the same shape moved a distance a to the right: 0
y
Let a=vt Then f(x-vt) will describe the same shape moving to the right with speed v.
x
x=a v
0
x
x=vt
Physics 1501: Lecture 27, Pg 23
Math... y
Consider a wave that is harmonic in x and has a wavelength of λ.
λ
A x
If the amplitude is maximum at y (x ) = A cos & 2 ( x # $ ! x=0 this has the functional form: % ' " y
Now, if this is moving to the right with speed v it will be described by:
v x
& 2) (x ' vt )#! y (x , t ) = A cos $ ( % "
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Physics 1501 – Lecture 27 Math...
So we see that a simple harmonic & 2) (x ' vt )#! y (x , t ) = A cos $ wave moving with speed v in the x ( % " direction is described by the equation:
By using v =
! !" = from before, and by defining T 2#
we can write this as:
k!
2" #
y (x , t ) = A cos (kx " !t )
(what about moving in the -x direction ?) Physics 1501: Lecture 27, Pg 25
Movie (twave)
Math Summary y
The
formula y (x , t ) = A cos (kx " !t ) describes a harmonic wave of amplitude A moving in the +x direction.
λ
A x
Each
point on the wave oscillates in the y direction with simple harmonic motion of angular frequency ω. 2" k
The wavelength of the wave is ! =
The speed of the wave is v =
The quantity k is often called wave number .
! k
Physics 1501: Lecture 27, Pg 26
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