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

Physics 1501: Lecture 27, Pg 12

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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 $ ( % "

Physics 1501: Lecture 27, Pg 24

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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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