(a) Chemicalenergy

(b) Heatenergy

(d) Manyothers:electrical,potential,pressure,etc...

• Energyistheabilitytodowork:work = (force)× (distancemoved)

06.Dynamics--Energy1.Energy

(c) Kineticenergy

PrincipleofConservationofEnergyThetotalenergyofaclosedsystemisconserved(doesn'tchangeovertime).

• ForaconstantforceF,asmallchangeinworkdW isrelatedtoasmallchangeindistancedx by

dW = Fdx

• Manyformsofenergy,allinterconvertible:

Topics:1. Energy2. NuclearPhysics3. Mass/EnergyEquivalence

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KineticEnergyK• Energyduetothemotionofanobject.• Workdonebyaforceinacceleratingtheobjectfromresttosomevelocity.

NewtonianKineticEnergyK𝐾 = !

"𝑚𝑣"

and p =mv.- Recall: 𝐹 =𝑑𝑝𝑑𝑡

- So:AsmallchangeinkineticenergydK isrelatedtoasmall

changeindistancedx by 𝑑𝐾 =𝑑𝑝𝑑𝑡𝑑𝑥 =

𝑑𝑥𝑑𝑡𝑑𝑝 = 𝑣𝑑(𝑚𝑣)

- Thisyields: 𝐾 = !"𝑚𝑣

"

- Nowintegrate: -𝑑𝐾 = -𝑣𝑑(𝑚𝑣)

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RelativisticKineticEnergyKSRKSR =mc2(γ − 1)

KineticEnergyK• Energyduetothemotionofanobject.• Workdonebyaforceinacceleratingtheobjectfromresttosomevelocity.

- So:AsmallchangeinrelativistickineticenergydKSR isrelatedtoasmall

changeindistancedx by 𝑑𝐾#$ =𝑑𝑝#$𝑑𝑡

𝑑𝑥 =𝑑𝑥𝑑𝑡𝑑𝑝#$ = 𝑣𝑑(𝑚𝛾𝑣)

- Nowintegrate: -𝑑𝐾#$ = -𝑣𝑑(𝑚𝛾𝑣)

and pSR =mγv,where- Recall: 𝛾 =1

1 − %!&!

𝐹 =𝑑𝑝#$𝑑𝑡

- Thisyields:KSR =mc2(γ − 1).

3

• Recall:Forv = c,wecanapproximateγ by1 + %!

"&!

• So:Whenv = c,𝐾#$ ≈ 𝑚𝑐" 1 + %!

&!− 1 =

"!𝑚𝑣

"

• Thus:RelativistickineticenergyreducestoNewtoniankineticenergywhenv = c (the"Newtonianlimit").

KineticEnergyK• Energyduetothemotionofanobject.• Workdonebyaforceinacceleratingtheobjectfromresttosomevelocity.

RelativisticKineticEnergyKSRKSR =mc2(γ − 1)

NewtonianKineticEnergyK𝐾 = !

"𝑚𝑣"

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RelativisticEnergy• Relativistickineticenergy:KSR =mc2(γ − 1),or:

KSR =mc2γ −mc2

(energyofmotion)= (partdependingonv)− (partindependentofv)

• Rewriteas:

mc2γ = KSR +mc2

(totalenergy)= (energyofmotion)+ ("rest"energy)

E = KSR + E0

• TotalEnergy:E =mc2γ- Inunitsinwhichc = 1,thissays"Relativisticmass(mg)= totalenergy".

• RestEnergy:E0 =mc2.- Inunitsinwhichc = 1,thissays"Restmass(m)= restenergy".

• Ingeneral:E =Mc2,whereM iseitherrelativisticmassorrestmass!5

RelativisticEnergy

E =Mc2

• AnincreaseΔE inenergyentailsanincreaseinmassby ΔM= ΔE/c2.- Verysmallincreaseinmass,sincecissolarge!

• Whataboutreverseprocess:AnincreaseΔM inmassentailsanincreaseinenergybyΔE = ΔMc2.- Verylargeincreaseinenergy,sincecissolarge!

(Chargebattery)⇒ (increaseinchemenergy)⇒ (very smallincreaseinmass)

(Heatwateronstove)⇒ (increaseinheatenergy)⇒ (verysmallincreaseinmass)

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2.NuclearPhysics

• Theoreticalresultin1905:E =Mc2.- Experimentalconfirmation--1930'snuclearchemists.

matter

atoms nucleuselectrons

+++

+

positiveprotons

neutrons

nucleus

• The#ofprotonsandneutronsfixesthechemicalcharacteroftheelement.

Ex:Carbonhas6protonsandcomesintwoversions:Onewith6neutrons(regularCarbon): !"#𝐶

Onewith7neutrons(Carbon-13): !"$𝐶

• Nuclearreactionsconvertelementsintoeachother(Alchemists'dream):

Ex: %"&𝑁 + #&𝐻𝑒 '

"%𝑂 + ""𝐻

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Bindingenergy(BE) = energyneededtobindnucleustogether.

• Specialrelativityentailsthatbindingenergyhasmass!

(massofnucleus)= (massofprotons)+ (massofneutrons)+ (massofBE)

"Large"elementsUranium(235U),Gold(197Au),etc...

"Medium"elementsIron(56Fe),etc...

"Small"elements:Hydrogen(1H),Helium(2He),etc...

• Experimentalresult:BE ofmedium-sizedelementsisgreaterthanBE oflargeorsmallelements.- Medium-sizedelementsaremoretightlybound.

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Twoconsequences(1) Whenalargeelementbreaksapart,thepieces aremoretightlybound.

- Thusthere'sanincreaseinrestmass!

(2) Ifsmallelementsarefused,theresultantwhole ismoretightlybound.- Thusthere'sanincreaseinrestmass!

(massof235U)= (massofParts)+(massofBEdifference)

smallerBE greaterBE

235U Parts

(massof1H)+ (massof1H)= (massofWhole)+ (massofBEdifference)

smallerBE greaterBE

1H + Whole1H

Big energy release!

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Process(1):Fission

92235U

• Naturalradioactivedecay:Energyreleasedasheat.

fragments

heat

n

- Triggerinitialreactionwithfreeneutron.• Togetlargeramountsofenergyreleased,setupa"chainreaction":

nn n

- Additionalproducts:2-3moreneutrons!- Usethesetotriggermorereactions.

U

U

U

U

U

U

U

n

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Two methodstopreventneutronsfromescaping

(a) FissionBomb:Put"criticalmass"inonespot.

(b) FissionReactor:Slowdownneutronswithmoderator(Cadmium)sotheytriggermorereactions.

Process(2):Fusion

hydrogen deuterium helium

+ 11H 1

2H 23He + enormous energy release

• Verydifficulttoinitiate:Requiresenormousenergyinput!

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(c) Stars- Intensegravitationalpressureinitiatesandsustainsfusionreactions.

(i) InertialConfinement- Use(giant)lasertoblastadropletofwaterfromallsides.

Threemethodsforproducingfusionreactions

(b)FusionReactor- Noreliablecost-effectiveprocess:Experimentalreactorsusemoreenergythantheyproduce.

(a)FusionBomb- Usefissionbombastrigger.

(ii) MagneticBottle- Usemagnetstosqueezeaplasma.

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3.Mass/EnergyEquivalence

What does it mean to say mass and energy are "equivalent"?

• Recall:

mγ = relativisticmass(varieswithvelocity)m = restmass(constant)E =mγc2 = totalenergy(varieswithvelocity)E0 =mc2 = restenergy(constant)

Option(1):Theyarethesamething.

Option(2):Theyaredifferentthingsbutcanbeconvertedintoeachother.

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Option(1a).Same-PropertyInterpretation

Claim:InNewtonianphysics,massandenergyaredifferent properties.InSpecialRelativity,massandenergyaredifferentnamesforthesame property.

• Objection1.Weusedifferentunitsformass(kg)andenergy(kg ⋅m2/s2).- Doesn'tthismeanthey'redifferentproperties?- No!Canchooseunitsinwhichc= 1.- Intheseunits,E=mγ,E0=m,andmassandenergyaremeasuredinthesameunits(kg,say).

• Objection2.Whenc = 1,distanceandtimearealsomeasuredinthesameunits.Shouldwethusallowthatdistanceandtimearealsoequivalentandaredifferentnamesforthesameproperty?

PossibleCounter-Claim:- Distanceandtimeintervalsaredifferent inertial-frame-dependentaspects ofaninvariantproperty,spatio-temporallength.

- Similarly,massandenergyaredifferentinertial-frame-dependentaspects ofaninvariantproperty,"energy-momentum".

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Option(1b).One-StuffInterpretation

InNewtonianphysics:- Matterhasbothmassandenergy,whilefieldshaveenergyonly.- Hencematterisdistinctfromfields:Therearetwotypesoffundamentalstuff.

Distinctions:mass= property matter= substanceenergy= property field= substance

InSpecialRelativity:

WeakClaim:Mass/energyequivalenceentailswecannottreatmatterandfieldsdifferently.(Onlyonetypeofstuff.)

StrongClaim:Energy/fieldsarefundamental.(Fields aretheonlytypeofstuff.)

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• Objection1.WeakClaim assumesmatterisessentially characterizedbymassandenergy,whereasfieldsareessentially characterizedbyenergy(hencewecannottreatthemdifferentlyifmassandenergyareequivalent).- Butwhymakethisassumption?

WeakClaim:Mass/energyequivalenceentailswecannottreatmatterandfieldsdifferently.(Onlyonetypeofstuff.)

StrongClaim:Energy/fieldsarefundamental.(Fields aretheonlytypeofstuff.)

• Objection2.Ifmassandenergyareequivalentbybeingdifferentaspectsofthesameproperty(energy-momentum),thentheStrongClaim owesusanexplanationofwhytheenergyaspectofthisquantityismorefundamentalthanthemassaspect.

PossibleCounter-Claim:- Thereisatypeofsubstance,distinctfrommatterorfields,thathasthepropertyofenergy-momentum.

- Thissubstanceappearsindifferentaspects(matterorfield)inanalogywiththewaythepropertyofenergy-momentumappearsinmassandenergyaspects. 16

Option(2).ConversionInterpretation

• Note:Thisrejectsa"spacetime"interpretationofmassandenergyasdifferentaspectsofaninvariantpropertyenergy-momentum (inanalogywithdistanceandtimeasdifferentaspectsofaninvariantproperty,spatiotemporallength).

• But:Perhapsthisanalogydoesn'tworkinthecaseofdynamics.Perhapsthedistinctionbetweenkinematicsanddynamicsisn'tasclearaswe'vebeenassuming...

Claim:Massandenergyaredifferentproperties,asinNewtonianphysics,butSpecialRelativityindicatesthattheycanbeconverted intoeachother.

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