Physics 1501: Lecture 27, Pg 1 Physics 1501: Lecture 27 Today’s Agenda l Homework #9 (due Friday...

Physics 1501: Lecture 27, Pg 1

Physics 1501: Lecture 27Physics 1501: Lecture 27TodayToday’’s Agendas Agenda

Homework #9 (due Friday Nov. 4) Midterm 2: Nov. 16

Katzenstein Lecture: Nobel Laureate Gerhard t’HooftFriday at 4:00 in P-36 …

TopicsSHMDamped oscillationsResonance1-D traveling waves


Spring-mass system

Pendula

General physical pendulum

» Simple pendulumTorsion pendulum

Simple Harmonic OscillatorSimple Harmonic Oscillator

dMg

z-axis

R

xCM

= 0 cos(t + )

k

x

m

FF = -kx aa

I

wire

x(t) = Acos(t + )

where


What about Friction?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.


What about Friction?What about Friction?

We can guess at a new solution.

With,



What does this function look like?(You saw it in lab, it really works)



There is a cuter way to write this function if you remember that exp(ix) = cos x + i sin x .


Damped Simple Harmonic MotionDamped Simple Harmonic Motion

Frequency is now a complex number! What gives?Real part is the new (reduced) angular frequencyImaginary part is exponential decay constant

underdamped critically damped overdamped

ActiveFigure


Driven SHM with ResistanceDriven SHM with Resistance

To replace the energy lost to friction, we can drive the motion with a periodic force. (Examples soon).

Adding this to our equation from last time gives,

F = F0 cos(t)


Driven SHM with ResistanceDriven SHM with Resistance

So we have the equation,

As before we use the same general form of solution,

Now we plug this into the above equation, do the derivatives, and we find that the solution works as long as,


Driven SHM with ResistanceDriven SHM with Resistance So this is what we need to think about, I.e. the amplitude

of the oscillating motion,

Note, that A gets bigger if Fo does, and gets smaller if b or m gets bigger. No surprise there.

Then at least one of the terms in the denominator vanishes and the amplitude gets real big. This is known as resonance.

Something more surprising happens if you drive the pendulum at exactly the frequency it wants to go,


Driven SHM with ResistanceDriven SHM with Resistance Now, consider what b does,

b small

b middling

b large


Dramatic example of resonanceDramatic example of resonance In 1940, turbulent winds set up a torsional vibration in the

Tacoma Narrow Bridge


Dramatic example of resonanceDramatic example of resonance

when it reached the natural frequency


Dramatic example of resonanceDramatic example of resonance

it collapsed !

Other example: London Millenium Bridge


Lecture 27, Lecture 27, Act 1Act 1Resonant MotionResonant Motion

Consider the following set of pendula all attached to the same string

D

A

B

CIf I start bob D swinging which of the others will have the largest swing amplitude ?

(A) (B) (C)


Chap. 13: WavesChap. 13: Waves

What is a wave ?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


Types of WavesTypes 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 movingSoundSlinky


Wave PropertiesWave Properties

Wavelength

Wavelength: The distance between identical points on the wave.

Amplitude A

Amplitude: The maximum displacement A of a point on the wave.

A

Animation


Wave Properties...Wave Properties...

Period: The time T for a point on the wave to undergo one complete oscillation.

Speed: The wave moves one wavelength in one period T so its speed is v = / T.

Animation


Wave Properties...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

and T are related !

v = / T

= v T or = 2 v / (sinceT = 2 /

or v / f (since T = 1/ f )

Recall f = cycles/sec or revolutions/sec rad/sec = 2f


Lecture 27, Lecture 27, Act 2Act 2Wave MotionWave 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


Wave FormsWave Forms

So far we have examined “continuous wavescontinuous waves” that go on forever in each direction !

v

v We can also have “pulses” caused by a brief disturbanceof the medium:

v And “pulse trains” which are

somewhere in between.


Mathematical DescriptionMathematical Description

Suppose we have some function y = f(x):

x

y

f(x-a) is just the same shape moveda distance a to the right:

x

y

x=a0

0

Let a=vt Then

f(x-vt) will describe the same shape moving to the right with speed v. x

y

x=vt 0

v


Math...Math...

Consider a wave that is harmonic in x and has a wavelength of .

If the amplitude is maximum atx=0 this has the functional form:

y

x

A

Now, if this is moving tothe right with speed v it will be described by:

y

x

v


Math...Math...

By using from before, and by defining

So we see that a simple harmonic wave moving with speed v in the x direction is described by the equation:

we can write this as:

(what about moving in the -x direction ?)


Math SummaryMath Summary

The formula describes a harmonic wave ofamplitude A moving in the +x direction.

y

x

A

Each point on the wave oscillates in the y direction withsimple harmonic motion of angular frequency .

The wavelength of the wave is

The speed of the wave is

The quantity k is often called “wave number”.

Movie (twave)

Physics 1501: Lecture 27, Pg 1 Physics 1501: Lecture 27 Today’s Agenda l Homework #9 (due Friday...

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