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

Chapter 15 Simple Harmonic Motion

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