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1、GREstudysheetDec.10,2005StevenByrnes(10/2007:Putinafewminorcorrectionsandaddenda.)1.ClassicalMechanics-20%(suchaskinematics,Newton’slaws,workandenergy,oscillatorymotion,rotationalmotionaboutafixedaxis,dynamicsofsystemsofparticles,centralforcesandcelestialmechanics,three-dimensionalparti
2、cledynamics,LagrangianandHamiltonianformalism,noninertialreferenceframes,elementarytopicsinfluiddynamics)•vintermsofxinuniformacceleration.22v=v0+2a(x−x0)Derivation:((v+v0)/2)∆t=vavg∆t=(x−x0)and(v−v0)/∆t=a.Multiplythesetwoequationstogether.•Lagrangian.LetT=kineticenergy,U=potentialenerg
3、y.L=L(q,q,t˙)=T−U•Euler-Lagrangeequationsofmotion.d∂L∂L=dt∂q˙∂q•Action.PickthepathwiththecorrectendpointsthatminimizesZt1S=L(q(t),q˙(t),t)dtt0•Conjugatemomentum.∂Lp=∂q˙•Hamiltonian.H=pq˙−L=T+U•Hamiltonianequationsofmotion.∂Hq˙=∂p∂Hp˙=−∂q1GREstudysheetDec.10,2005StevenByrnes•Bernoulli
4、’sEquationforfluidflow.(Assumeincompressible,nonviscous,laminarflow;thisequationholdsalongflowlines,oriftheflowisirrotational,itholdseverywhere.)Takegaspositive.12p+ρgy+ρv=constant2Derivation:Conservationofenergy.Thesecondtwotermsareenergyperunitvolume,thefirsttermrelatestothefactthatwhenpre
5、ssuredecreasesalongaflowline,there’sanetforwardforceonthefluid,whichincreasestheenergy.•Fictitiousforcesfromrotatingreferenceframe.F~coriolis=−2m~ω×(~vinrotatingframe)F~centrifugal=−m~ω×(~ω×~r)HowtorememberCoriolisforce:Itmakesithard2movstraight(remeberωcorre-spondstothelettero).•Torque.
6、~τ=~r×F~Howtoremembertheorderof~randF~:“tariff”or“traffic”or“turf”or“teriffic”or“thrifty”or“trefoil”.•Angularmomentum.L~=~r×~p•Momentofinertia.risdistancefromaxis.ZZX222I=miri=rdm=rρdVi•Frequencyofapendulumofarbitraryshape.Iismomentofinertiaaboutthesupport,risdistancefromsupporttocentero
7、fmass.rmgrω=IInparticular,forapointmassattheendofamasslesspendulumoflength`,I=m`2,r=`,sorgω=`Derivation:Iθ¨=τ=~r×F~=−(r)(mg)(sinθ)≈−rmgθ.2GREstudysheetDec.10,2005StevenByrnes2.E&M-18%(suchaselectrostatics,currentsandDCcircuits,magneticfieldsinfreespace,Lorentzforce,induction,Maxwell’s
