Review of Classical PhysicsReview of Classical Physics

By the late part of the 19th century, By the late part of the 19th century, physics consisted of two great pillars: physics consisted of two great pillars: a) mechanics including thermodynamics a) mechanics including thermodynamics and b) electromagnetism.and b) electromagnetism.

However, a series of problems that However, a series of problems that

should have been solvable continued to should have been solvable continued to perplex physicists. perplex physicists.

Modern physics is the study of the two Modern physics is the study of the two great revolutions (relativity and quantum great revolutions (relativity and quantum mechanics) that solved these problems and our mechanics) that solved these problems and our continuing effort to find the ultimate rules of the continuing effort to find the ultimate rules of the game.game.

Physics ProblemPhysics Problem

Fast Fast ObjectObject

Large Large ObjectObject

Quantum Quantum MechanicsMechanics

NoNo NoNo

YesYes

LargeLargeObjectObject

Relativity & QMRelativity & QM

NoNo

YesYesRelativityRelativity

Classical PhysicsClassical Physics

Quantum MechanicsQuantum Mechanics

RelativityRelativity

OrOr

OrOr

YesYes

Classical PhysicsClassical Physics• General Ideas of MechanicsGeneral Ideas of Mechanics

1.1. A A reference framereference frame is a coordinate axis and is a coordinate axis and origin used by an observer to describe the motion of origin used by an observer to describe the motion of an object.an object.2.2. An An inertial reference frameinertial reference frame is one in which the is one in which the observer is observer is not accelerating not accelerating (i.e. one in which (i.e. one in which Newton's Laws are valid without adding fake forces to Newton's Laws are valid without adding fake forces to your free-body diagram)your free-body diagram)

Question:Question: Accelerating with respect to what?Accelerating with respect to what? dtpd

Fext

Galilean RelativityGalilean Relativity All inertial reference frames are equivalent! Another way of All inertial reference frames are equivalent! Another way of stating this principle is that stating this principle is that only relative motion can be only relative motion can be detecteddetected..

Transformation Equations Transformation Equations If you know what an observer in a particular reference frame If you know what an observer in a particular reference frame observes then you can observes then you can predict the observations made by predict the observations made by an observer in any other reference frame. The an observer in any other reference frame. The equations that equations that enable you to make these calculations are called enable you to make these calculations are called Transformation EquationsTransformation Equations..

InvarianceInvariance Since the labeling of your coordinate axis and its origin Since the labeling of your coordinate axis and its origin location is arbitrary, the location is arbitrary, the equations of physics should equations of physics should have the same form regardless when you rotate or have the same form regardless when you rotate or translate your axis set. Equations that have this property translate your axis set. Equations that have this property are said to be invariant to the transformation. are said to be invariant to the transformation.

It wasIt was shown that Maxwell's Equations are not shown that Maxwell's Equations are not invariant under a Galilean Transformation so E&M invariant under a Galilean Transformation so E&M and Mechanics are not consistent.and Mechanics are not consistent.

General Ideas of E&MGeneral Ideas of E&M 1.1. Light is a Light is a transverse wavetransverse wave..2.2. The The speed of lightspeed of light depends depends onlonly on y on the mediumthe medium through through which it travels and which it travels and not upon the observernot upon the observer..

3.3. The light propagates from the sun to the earth through The light propagates from the sun to the earth through the the luminiferous etherluminiferous ether..4.4. Properties of the Ether FluidProperties of the Ether Fluidi)i) non-viscous - Earth doesn't slow down while traveling non-viscous - Earth doesn't slow down while traveling through the ether.through the ether.ii)ii) incompressible - speed of light is very fastincompressible - speed of light is very fastiii)iii) masslessmassless

1c

Galilean TransformationsGalilean Transformations

A.A. TimeTime

All observers measure the same All observers measure the same timetime. This was assumed without proof . This was assumed without proof and used to derive our equations and used to derive our equations

t = t't = t'

B.B. PositionPositionLet us consider a ball being measured by two different Let us consider a ball being measured by two different observers as shown below:observers as shown below:

BallBall

y'y'

x'x'

yy

xxTomTom

SuSueeR

r'r

B.B. PositionPositionBy vector subtraction, we see that the location of the ball By vector subtraction, we see that the location of the ball according to Sue is given by:according to Sue is given by:

BallBall

y'

x'

y

xTomTom

SuSueeR

r'r

wherewhere is the position of the ball as seen is the position of the ball as seen by Sueby Sue

is the position of the ball as seen by Tomis the position of the ball as seen by Tom is the position of Sue as seen by Tomis the position of Sue as seen by Tom

Rr'r

'r

r

R

C.C. VelocityVelocityWe now apply the time derivative operator to both sides of our We now apply the time derivative operator to both sides of our position.position.

R

dtdr

dtd'r

dtd

Applying the definition of velocity, we can rewrite Applying the definition of velocity, we can rewrite the right-hand side of the equation asthe right-hand side of the equation as

uv'rdtd

where where is the velocity of the ball as seen by is the velocity of the ball as seen by TomTom

is the velocity of Sue as seen by Tom is the velocity of Sue as seen by Tom

v

u

C.C. VelocityVelocitySince the time measured by both Sue and Tom is the same, Since the time measured by both Sue and Tom is the same, we can replace t with t' and apply the definition of velocity to we can replace t with t' and apply the definition of velocity to the left-hand side of the equation. the left-hand side of the equation.

uv'rdt'd

where is the velocity of the ball as seen by where is the velocity of the ball as seen by Sue.Sue.

uv'v

'v

You should note that the definition of velocity requires You should note that the definition of velocity requires that that bothboth the the timetime and and positionposition be measured be measured by the by the same observersame observer! !

D.D. AccelerationAcceleration We now follow the same procedure as in part We now follow the same procedure as in part CC to obtain the to obtain the relationship between the ball's acceleration as measured by relationship between the ball's acceleration as measured by Sue and as measured by Tom.Sue and as measured by Tom.

udtdv

dtd'v

dtd

Aa'vdtd

Aa'v'dt

d

Aa'a

Where Where is the acceleration of the ball as seen by Tomis the acceleration of the ball as seen by Tom is the acceleration of the ball as seen by Sueis the acceleration of the ball as seen by Sue

is the acceleration of Sue as seen by Tom is the acceleration of Sue as seen by Tom

a

'a

A

D.D. AccelerationAcceleration

Aa'a

Where Where is the acceleration of the ball as seen by Tomis the acceleration of the ball as seen by Tom is the acceleration of the ball as seen by Sueis the acceleration of the ball as seen by Sue

is the acceleration of Sue as seen by Tom is the acceleration of Sue as seen by Tom

a

'a

A

Note:Note: If Sue and Tom are not accelerating with respect to If Sue and Tom are not accelerating with respect to each other (ie each other (ie ), they will agree on the ), they will agree on the acceleration of the ball and Newton's Laws! The last acceleration of the ball and Newton's Laws! The last term on the right-hand side is the reason we add fake term on the right-hand side is the reason we add fake forces when using non-inertial reference frames.forces when using non-inertial reference frames.

0A

Problem: Assuming that , who is accelerating Problem: Assuming that , who is accelerating (Tom, Sue, or both)?(Tom, Sue, or both)?

E. Special Case of Two Observers in 1-D Uniform E. Special Case of Two Observers in 1-D Uniform Motion Motion In developing special relativity, we find it convenient to simplify In developing special relativity, we find it convenient to simplify the math by considering the motion of an object as see by two the math by considering the motion of an object as see by two observers who are in 1-D uniform motion with respect to each observers who are in 1-D uniform motion with respect to each other. Thus, we will assume the following:other. Thus, we will assume the following:

1) Tom and Sue are both located at the origin at1) Tom and Sue are both located at the origin at t = 0t = 0

(We can arbitrarily start measuring time whenever we (We can arbitrarily start measuring time whenever we want want so this doesn't limit our results)so this doesn't limit our results)

2) Sue is traveling at constant speed2) Sue is traveling at constant speed uu in thein the +x+x direction direction as seen by Tom (Thus, we don't consider acceleration. as seen by Tom (Thus, we don't consider acceleration.

This This is what will be special about special relativity!) is what will be special about special relativity!)

0A

The position vector of Sue as seen by Tom is given at any instantThe position vector of Sue as seen by Tom is given at any instant t t byby

Inserting our results above into our previous results for the Inserting our results above into our previous results for the position equation of the Galilean Transformation, we haveposition equation of the Galilean Transformation, we have

We will see how these equations must be modified for high speed We will see how these equations must be modified for high speed problems when we study the Lorentz Transformation.problems when we study the Lorentz Transformation.

k0j0ituR

zz'yy'

tuxx'tt'

IV. Important Physics Problems of Late 19th IV. Important Physics Problems of Late 19th CenturyCenturyModern Physics was developed as the solution to Modern Physics was developed as the solution to some extremely important problems in the late 19th some extremely important problems in the late 19th century that stumped physicists. We will study these century that stumped physicists. We will study these important problems and how they have caused us to important problems and how they have caused us to change our notions of time, space, and matter. Some change our notions of time, space, and matter. Some of these important problems include of these important problems include a) the ether problem, a) the ether problem, b) stability of the atom, b) stability of the atom, c) blackbody radiation, c) blackbody radiation, d) photoelectric effect, and d) photoelectric effect, and e) atomic spectra.e) atomic spectra.

V. Binomial ApproximationV. Binomial Approximation

The Binomial Expansion is a powerful method for The Binomial Expansion is a powerful method for approximating small effects in physics and engineering approximating small effects in physics and engineering problems. It is extremely useful in both special problems. It is extremely useful in both special relativity and electromagnetism problems even when you relativity and electromagnetism problems even when you have a calculator.have a calculator.

The expansion of the nThe expansion of the n thth power of (1+x) is given by power of (1+x) is given by

The Binomial approximation states that when The Binomial approximation states that when x << 1x << 1

...x2

1-nnxn1nx1 2

xn1x1 n

VI.VI. Work and Energy ConceptsWork and Energy ConceptsA.A. WorkWork

The work done by a force, , upon a body in displacing the The work done by a force, , upon a body in displacing the body an amount is defined by the equationbody an amount is defined by the equation

B.B. EnergyEnergyEnergy is the ability of a body to perform work. (i.e. Stored Energy is the ability of a body to perform work. (i.e. Stored Work!!)Work!!)

C.C. Kinetic EnergyKinetic EnergyKinetic energy is Kinetic energy is defineddefined as the as the energyenergy that an object has that an object has due due toto its its motionmotion!!

F

sd

final

initialsdFW

D.D. Work-Energy TheoremWork-Energy Theorem The work done by the net external force upon an object (or equivalently the net The work done by the net external force upon an object (or equivalently the net work done by all forces upon the object) is equal to the change in the objects work done by all forces upon the object) is equal to the change in the objects kinetic energy!!kinetic energy!!

The work-energy theorem is the heart of all energy concepts as it relates the connection between Newton II, work and energy!!

We used this theorem to derive the conservation of mechanical energy and to We used this theorem to derive the conservation of mechanical energy and to develop the classical formula for computing the kinetic energy of a body.develop the classical formula for computing the kinetic energy of a body.

KsdFWfinal

initialnet

E.E. Classical Formula For Finding The Kinetic Classical Formula For Finding The Kinetic Energy of an ObjectEnergy of an Object

The kinetic energy of an object traveling at speeds much less The kinetic energy of an object traveling at speeds much less than the speed of light (ie classical physics) can be obtained than the speed of light (ie classical physics) can be obtained using the formulausing the formula

Proof:Proof:Inserting Newton II into the work energy theorem, we have Inserting Newton II into the work energy theorem, we have thatthat

2vM21K

KsddtpdW

final

initialnet

dtsdpdΔK

final

initial

We now apply the definition of velocity and linear momentum to We now apply the definition of velocity and linear momentum to our equationour equation

In classical mechanics, the mass of a particle is constant (an In classical mechanics, the mass of a particle is constant (an assumption we will have to re-examine in special relativity) soassumption we will have to re-examine in special relativity) so

We can simplify our equation by using the following CalculusWe can simplify our equation by using the following Calculus

vvmdΔKfinal

initial

vvdmΔKfinal

initial

vvd2vdv vvdvvdvd 2

Thus, our equation is simplified toThus, our equation is simplified to

By comparing the individual terms, we obtain our classical formula for kinetic energy.By comparing the individual terms, we obtain our classical formula for kinetic energy.

2vdm2

1ΔK

final

initial

2vdm2

1ΔK

final

initial

2

initial

2

finalinitialfinalmv

2

1vm

2

1KK

F.F. Temperature and Average EnergyTemperature and Average EnergyIn our study of thermodynamics, the relationship is In our study of thermodynamics, the relationship is developed between average energy of a monatomic gas developed between average energy of a monatomic gas molecule with three degrees of motion and the temperature molecule with three degrees of motion and the temperature of the gas. The constant of proportionality is called the of the gas. The constant of proportionality is called the Boltzman constant and is related to the ideal gas constantBoltzman constant and is related to the ideal gas constant RR

The calculation of the average energy of the particle The calculation of the average energy of the particle involves two separate steps: involves two separate steps:

1)1) determining the number of degrees of freedom for the determining the number of degrees of freedom for the particle and particle and

2)2) determining the average energy for each degree of freedom. determining the average energy for each degree of freedom.

Tk23Eaverage

We can rewrite our previous result for a gas particle with three We can rewrite our previous result for a gas particle with three degrees of freedom asdegrees of freedom as

Thus, we see that each degree of freedom (translation in the Thus, we see that each degree of freedom (translation in the xx, , yy, and, and z z directions) have on average of energy. This directions) have on average of energy. This fact can be generalized as the equipartition theorem.fact can be generalized as the equipartition theorem.

Tk

213Eaverage

Tk21

Equipartion TheoremEquipartion Theorem

The average energy of any degree of freedom involving The average energy of any degree of freedom involving square of a generalized co-ordinate is .square of a generalized co-ordinate is .

The equipartition theorem is very useful in classical calculations The equipartition theorem is very useful in classical calculations of systems containing many particles like gases. We will use this of systems containing many particles like gases. We will use this concept when considering the classical computation of the concept when considering the classical computation of the thermal radiation of black-bodies. thermal radiation of black-bodies. The ability to correctly calculate the number of degrees of The ability to correctly calculate the number of degrees of freedom for more complicated systems is considered in freedom for more complicated systems is considered in Mechanics. The application of this material in determining Mechanics. The application of this material in determining macroscopic properties of systems based upon their atomic macroscopic properties of systems based upon their atomic nature is considered in Advanced Thermodynamics, Statistical nature is considered in Advanced Thermodynamics, Statistical Mechanics, and in Solid State Physics.Mechanics, and in Solid State Physics.

Tk21

Classical Physics Review - Objectives 1.  1.    Know the meanings of basic mechanics terms including   Know the meanings of basic mechanics terms including "event," "observer," and "reference frame" "event," "observer," and "reference frame" 2. 2.    Know the Galillean Transformation equations and the    Know the Galillean Transformation equations and the principle assumption behind their development principle assumption behind their development 3. 3.    Review Newton's Laws including inertial frames of    Review Newton's Laws including inertial frames of reference. reference. 4. 4.    Review the basic concepts of particles concerning linear    Review the basic concepts of particles concerning linear

momentum, energy, location, and trajectory momentum, energy, location, and trajectory 5. 5.    Review Maxwell's Equations and know the basic properties    Review Maxwell's Equations and know the basic properties

of light (speed, transverse wave, etc). of light (speed, transverse wave, etc). 6.6.    Review the basic concepts of energy and linear     Review the basic concepts of energy and linear momentum and be able to apply these concepts momentum and be able to apply these concepts 7. 7.    Review the basic properties of waves including energy,    Review the basic properties of waves including energy, and interference. and interference. 8.8.    Understand the conflict between mechanics and     Understand the conflict between mechanics and electromagnetic theory concerning light. electromagnetic theory concerning light. 9.9.    Review the kinetic theory of gases and its application to     Review the kinetic theory of gases and its application to determining specific heat determining specific heat 10.10.  Know the Binomial Expansion and be able to apply it to   Know the Binomial Expansion and be able to apply it to problems problems