l1 – center of mass, reduced mass, angular...
TRANSCRIPT
L1–Centerofmass,reducedmass,angularmomentum
Readcoursenotessec6ons2‐6Taylor3.3‐3.5
Newton’sLawfor3bodies(orn)
m1d 2r1dt 2
=F1 + 0 +
f12 +
f13
m2d 2r2dt 2
=F2 +
f21 + 0 +
f23
m3d 2r3dt 2
=F3 +f31 +
f32 + 0
Whatare mi ?
ri ?Fi ?
fij ?
Centerofmass&itsmo6on
• Posi6onofcenterofmassisthe(mass)weightedaverageofindividualmassposi6ons
RCM ≡
m1
Mr1 +
m1
Mr2 +
m1
Mr3 + ... =
mn
Mrn
n∑
mnd 2rndt 2n
∑ =Fn
n∑
Md 2R
dt 2=
Fn
n∑
Md 2R
dt 2= 0⇒ ?
2‐bodyproblemis“solvable”CoM
r1
r ≡ r2 −r1
r2
m1
m2
RCM
RCM ≡m2
Mr2 +
m1
Mr1
2‐bodyproblemis“solvable”CoM
r1
r2
m1
m2
RCM
RCM ≡m2
Mr2 +
m1
Mr1
CoMmovesasifallthemassislocatedthere.Inabsenceofextforces,VCM=constantWecanchooseconstvel=0RCM=constandwecanchoose=0.
2‐bodyproblemis“solvable”rela6vecoordinates
r1
r ≡ r2 −r1
r2
m1
m2
RCM
RCM ≡m2
Mr2 +
m1
Mr1 = 0
can choose if Fext=0
r2 =m1
m1 + m2
r ; m2r2 = µr
r1 = −m2
m1 + m2
r ; m1r1 = −µr
REDUCED MASS
µ ≡m1m2
m1 + m2
Checklimits,direc6onsforsense;Studentstoderivemathema6callyathome.
2‐bodyproblemis“solvable”rela6vecoordinates
r1
r ≡ r2 −r1
r2
m1
m2
RCM
m2r2 = µr ; m1
r1 = −µrµ ≡m1m2
m1 + m2
Rela6vemo6onobeysasingle‐par6cleequa6on.Fic66ouspar6cle,massµmovinginacentralforcefield.VectorrismeasuresRELATIVEdisplacement;wehaveto“undo”togetactualmo6onofeachofpar6cles1and2.O^enm1>>m2.
m1d 2r1dt 2 =
f12
m2d 2r2dt 2 =
f21
µd 2rdt 2 =
f21 = f r( ) r̂ for central force
• Definecentralforce
• f12 reads “the force on 1 caused by 2”• Dependsonmagnitudeofsepara6onandnotorienta6on
• Pointstowardsoriginina1‐par6clesystem• Derivablefrompoten6al(conserva6ve)
• Workdoneispathindependent
Centralforce
f21 = −
f12 = f (r)r̂; gravity: f (r) = −
Gm1m2
r2
f12 = −∇U r( ); gravity: U(r) = −
Gm1m2
r
Angularmomentum• Define(mustspecifyanorigin!)
L ≡ r × p; τ = r ×
F
LTOT = ri ×
pii∑ = r1 × m1
dr1dt
+ r2 × m2dr2dt
= r1 × −µdrdt
⎛⎝⎜
⎞⎠⎟+ r2 × µ
drdt
= r1 −r2( ) × µd
rdt
LTOT = r × µv; d
LTOTdt
= 0
LTOT = ri ×
pii∑
Showthelaststatementistrueforcentralforce.AngularmomentumisconservedifnoexternalTORQUE.
Conserva6onofAM&Kepler’s2ndlaw
• Equalareasequal6mes(cons.angularmom.)• Later(oncewe’veexploredpolarcoordinates).
Summary
• DefineRCM;totalmassM• Definer,rela6vecoord;µ=m1m2/M
• Noextforces=>CoMconstantmomentum
• Noexternaltorque=>AMconserved
• AMconserved‐>whatisvalue?(seelater)
• AMconserved‐>Orbitinaplane(hwk)