Lecture 23: Central Force Motion

Transcription

1 Lectue 3: Cental Foce Motion Many of the foces we encounte in natue act between two paticles along the line connecting the Gavity, electicity, and the stong nuclea foce ae exaples These types of foces ae called cental foces We ll stat by consideing the siple case of a syste with only two paticles: = - Note that thee see to be six degees of feedo

2 We assue that the only foce acting is in the diection We also assue that the foce is consevative In that case, the potential enegy can only depend on the agnitude of : ( ) ( ) U = U = U And theefoe the Lagangian has the fo: L= ( ) + U This Lagangian doesn t depend on how we choose the oigin, so let s ake the easiest choice: put it at the cente of ass of the syste. With that oigin, we have: + = 0

3 We also can find and in tes of : and siilaly: + = + = = + = = + We can now substitute these expessions into the Lagangian: L= ( ) U + + +

4 L= ( ) ( ) U ( + + ) ( ) = U µ U( ) + The educed ass of the syste The fact that we can wite the Lagangian in this fo eans that we can teat the syste as though it consisted of a single paticle of ass µ If we want to find the otion of the individual paticles, we can easily find and once we know

5 Reduced Mass: The Extee Cases To get a feel fo what the educed ass epesents, let s conside two extee cases Case : + µ = = Makes sense: it s a good appoxiation to teat the heavie paticle as stationay Case : = µ = = = + Note that the educed ass is always less than the ass of eithe paticle in the syste Explains why it s called educed!

6 Conseved Quantities We can lean quite a bit about the otion of the two-body syste siply by consideing the consevation laws we ve aleady studied even if we don t know the fo of the foce acting between the paticles Fist, note that with the oigin at the cente of ass, thee is no toque acting anywhee on the syste so the angula oentu, L, ust be constant But ecall that L is defined as p So fo L to be constant, and p ust always lie in a given plane Theefoe we can educe the thee-diension poble to two diensions we ve eliinated a degee of feedo

7 This eans we need just two genealized coodinates to descibe the syste A convenient choice is the set (, θ) With these coodinates, the Lagangian becoes: L= µ + θ U ( ) ( ) Note that the vaiable θ doesn t appea anywhee in the Lagangian This eans that the genealized oentu associated with θ ust be constant: L p θ = = 0 θ L pθ = = µ θ = const = l θ

8 Once again, this allows us to lean soething inteesting about cental foces Conside a paticle oving past soe oigin: Path da da = ( θ) = dθ da ( = θ + d θ ) dt So the aeal velocity is just: da dt = θ da is the aea swept out by the path in a tie dt This te is 0, since dθ is infinitely sall

9 But this is closely elated to the conseved quantity l: da dt l µ = θ = In othe wods the aeal velocity is constant You ay ecall this as Keple s Second Law of planetay otion He noted this as an expeiental fact decades befoe Newton developed his laws of otion We now see that Keple s Second Law holds fo any cental foce, not just the / fo of gavity

10 Enegy Enegy is also conseved fo the types of foces we e consideing hee: E = T + U = µ + θ + U ( ) ( ) ( µ θ) ( ) = µ + + µ l = µ + + U( ) µ U Note that the consevation of angula oentu allows us to wite the enegy in a fo that depends on only one vaiable