CHAPTER 21 CENTRAL FORCES AND EQUIVALENT POTENTIAL

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1 1 1.1 Intoduction CHAPTER 1 CENTRA FORCES AND EQUIVAENT POTENTIA Wen a paticle is in obit aound a point unde te influence of a cental attactive foce (i.e. a foce F () wic is diected towads a cental point, wit no tansvese component) it expeiences, wen efeed to an inetial efeence fame, a centipetal acceleation. If, oweve, te system is descibed wit espect to a co-otating efeence fame, tee is no centipetal acceleation; ate, it appeas as toug an additional foce, te centifugal foce, is pusing it away fom te cente of attaction. In te co-otating fame, tis foce depends only on te distance of te paticle fom te cente of attaction, and it is teefoe a consevative foce and, like any consevative foce, it can be descibed by te negative of te deivative of a potential enegy function. Wen descibing te motion wit espect to te co-otating fame, we must add tis potential to any additional eal potentials (suc as oiginate fom te gavitational fields of ote bodies), to fom an equivalent potential wic constains te motion of te paticle. An excellent example of tis metod is te analysis of te esticted tee-body poblem given in some detail in Capte 16 of my notes on Celestial Mecanics (ttp://oca.pys.uvic.ca/~tatum/celmecs.tml). But I deal fist, by way of example, wit some simple poblems involving cental foces, in wic we sall be able, by simple aguments, to deduce some basic caacteistics of te motion. 1. Motion Unde a Cental Foce I conside te two-dimensional motion of a paticle of mass m unde te influence of a consevative cental foce F(), wic can be eite attactive o epulsive, but depends only on te adial coodinate. Recalling te fomula & θ & fo acceleation in pola coodinates (te second tem being te centipetal acceleation), we see tat te equation of motion is m & mθ& = F( ) Tis descibes, in pola coodinates, two-dimensional motion in a plane. But since tee ae no tansvese foces, te angula momentum m θ & is constant and equal to, say. Tus we can wite equation 1..1 as m & = F( ) m Tis as educed it to a one-dimensional equation; tat is, we ae descibing, elative to a co-otating fame, ow te distance of te paticle fom te cente of attaction (o epulsion) vaies wit time. In tis co-otating fame it is as if te paticle wee subject

2 not only to te foce F(), but also to an additional foce. m foce on te paticle (efeed to te co-otating fame) is In ote wods te total F '( ) = F( ) m Now F(), being a consevative foce, can be witten as minus te deivative of a dv potential enegy function, F =. ikewise, is minus te deivative of d m. Tus, in te co-otating fame, te motion of te paticle can be descibed as m constained by te potential enegy function V', wee V ' = V m Tis is te equivalent potential enegy. obiting paticle, tis becomes If we divide bot sides by te mass m of te Φ ' = Φ Hee is te angula momentum pe unit mass of te obiting paticle, Φ is te potential in te inetial fame, and Φ' is te equivalent potential in te cootating fame. 1. Invese Squae Attactive Foce Tis is dealt wit in detail in Capte 9 of my notes on Celestial Mecanics (ttp://oca.pys.uvic.ca/~tatum/celmecs.tml). Hee we investigate some geneal popeties of te motion. GMm GMm If F = ten V =, and ence GMm V ' = m I sketc tis in figue XXI.1. Te total enegy (potential + kinetic) is constant (independent of ) and is geate tan (o equal to) te potential enegy. If te total enegy is less tan zeo, you can see fom te gap tat as a lowe (peielion) and uppe (apelion) limit; tis coesponds to an elliptic obit. But if te total enegy is

3 positive, as a lowe limit, but no uppe limit; tis coesponds to a ypebolic obit. If te total enegy is equal to te minimum of V', only one value of is possible, and te obit is a cicle. FIGURE XVI.1 XXI.1 /(m ) 0 V' -GMm/ 1.4 Hooke s aw We imagine a paticle wiling aound on te end of a sping, oscillating in and out as it does so. Te foce constant of te sping is k, te foce on te paticle is k and te potential (elastic) enegy is V = 1 k. Te effective potential enegy is teefoe V ' = 1. k m I sketc tis in figue XVI.. Te total enegy (potential + kinetic) is constant (independent of ) and is geate tan (o equal to) te potential enegy. Te distance of te paticle fom te cente of attaction is bounded above and below. Te motion is a issajous ellipse, wit te cente of attaction at te cente (not te focus) of te ellipse. Te lowe bound is te semi mino axis and te uppe bound is te semi majo axis.

4 4 FIGURE XXI. FIGURE XVI. V' /(m ) k / k / /(m ) An invese squae foce (e.g. a gavitational foce, o a Coulomb s law electostatic foce) and a Hooke s law foce (kx) ae obvious examples of eal foces in natue. In wat follows we sall investigate te beaviou of a paticle unde te influence of ote foce laws, suc as invese fout powe and invese cube foces. It is difficult to imagine wete suc foces actually exist in natue (te field of an electic dipole falls off as te cube of te distance - but te field is not adial, and te foce is not a cental foce), and to tat extent muc of wat follows is an execise in matematics moe tan in pysics. But invese squae and Hooke s law foces ae cetainly not te only foces to opeate in natue. Wat is te foce law, fo example, fo te esidual stong inteactions between nucleons in an atomic nucleus, o te foce law between te quaks witin a nucleon? It will be wotwile investigating te simple ypotetical foces to be discussed ee in ode to undestand te pinciples and metods tat may be applicable to a moe difficult poblem.

5 5 1.5 Invese fout powe attactive foce a a If F = ten V =, and ence 4 ' a V = m I sketc tis in figue XXI.. Te total enegy (potential + kinetic) is constant (independent of ) and is geate tan (o equal to) te potential enegy. If te total enegy is negative, te distance as an uppe limit, but te only lowe limit is te oigin, o te cente of attaction, and paticle will eventually end tee. If te total enegy is geate tan te maximum of V', te motion is completely unbounded. If te total enegy is positive but less tan V' max, te motion depends on te initial value of. Fo small te motion is bounded above, and te paticle will eventually end at te oigin. Fo lage, tee is a minimum distance to wic te paticle can appoac te oigin, and te paticle will eventually wande off to infinity. Fo total enegy in tis ange, tee is a ange of tat is not possible. FIGURE XVI. XXI. V' /(m ) 0 -a/

6 6 1.6 A geneal cental foce et us suppose tat we ave a paticle tat is moving unde te influence of a cental foce F (). Te equations of motion ae Radial: m (& θ & ) = F( ) 1.6. Tansvese: & θ + & θ& = Tese can also be witten & θ & = a( ) θ & = Hee a is te adial foce pe unit mass (i.e. te adial acceleation) and is te (constant) angula momentum pe unit mass. [If you ae unsue of wy equations 1.6. and ae te same, diffeentiate equation wit espect to time.] Tese ae two simultaneous equations in, θ, t. In pinciple, if we could eliminate t between tem, we would obtain a elation between and θ, wic would tell us te sape of te pat pusued by te paticle. In Capte 9 of my Celestial Mecanics notes we do tis fo te gavitational case, and we find tat te pat is an ellipse of te fom l =. O peaps we could eliminate and ence find out ow te angle 1 + e cos θ θ canges wit time. O again we migt be able to eliminate θ and ence get a elation telling us ow vaies wit te time. Yet again we migt be told te sape of te pat (θ), and asked to find te foce law F (). O again, ate tan te foce, we migt be given te fom of te potential enegy V (), wic is elated to te foce by F = dv / d. Te potential Φ is te potential enegy pe unit mass, and d Φ / d is te adial foce pe unit mass - i.e. it is te adial acceleation a() of te obiting paticle. Te angula momentum of te paticle, wic is constant, is = m θ&, and te angula momentum pe unit mass is = θ&, wic is twice te ate at wic te adius vecto sweeps out aea. We migt also emembe tat, if we ae given te potential enegy V o te potential Φ in an inetial fame, we migt also want to wok in a co-otating fame, making use of te equivalent potential enegy V ' = V + o te equivalent potential m Φ ' = Φ +.

7 7 One last ting to bea in mind befoe stating any poblems of tis class. It tuns out tat, vey often, a cange of vaiable u = 1/ tuns out to be useful. Consevation of angula momentum ten takes te fom θ & / u =. Also d du d du θ& du du & = = = = du dt du dt u d du d du d du d u d u and & & = = = =. u. = u. dt dt dt Equations and now become 1 d u u + θ& = a( ) dt u and θ & = u We can now easily eliminate te time wic was one of ou aims: u d u + u = a( ) [As eve, ceck te dimensions.] Tis equation, wic does not contain te time, wen integated will give us te (, θ) equation to te pat. Wit tese emaks in mind, let us ty a few poblems. Fo example: 1.7 Invese cube attactive foce A paticle moves in a field suc tat te attactive foce on it vaies invesely as te cube of te distance fom a cente of attaction. Wat is te sape of te pat? How does te angle θ vay wit time? et s suppose tat te adial acceleation is a( ) = k / = k u. (I want te coefficient of 1/ to be negative, so tat te foce is attactive, wic is wy I ave witten te coefficient as k. Besides, te dimensions of k ae ten T 1, wic ae te same as tose of, te angula momentum pe unit mass, wic elps to make te algeba simple.) Te diffeential equation to te pat (equation ) is ten

8 8 d u u + u = k u o d u + u = k u d u k Tat is, = u Te fom of te motion evidently depends on wete k > (a stongly attactive foce, o a small angula momentum), o if k < (a weak foce, o a lage angula momentum.) If we stat te paticle olling wit just te igt amount of angula momentum ( k = ), tee will evidently be zeo adial acceleation, and te paticle will move in a cicle. Befoe integating equation 1.7.1, let us look at te equivalent potential. Fo 1 a( ) = k /, te potential in te inetial fame is Φ = k / povided we take te potential at infinity to be zeo. Te equivalent potential is ten (see equation 1..5) ' k Φ = We see tat, if k =, te potential is zeo and independent of distance. If < k, te equivalent potential is negative, inceasing to zeo as, and te paticle acceleates towads te cente of attaction. If > k, te potential is positive, deceasing to zeo as, and te paticle acceleates away fom te cente of attaction. Tis sounds like a contadiction, but wat is appening is tat > k means tat te paticle as initially been given a lage angula momentum, and, in te cootating fame, te centifugal foce is lage tan te attactive foce. If < k, te equation of motion (equation 1.7.1) is d u = c u, k wee c =

9 9 Te geneal solution is c θ + cθ u = Ae Be du If te initial conditions ae tat at t = 0, = 0, u = u0, = 0 (tis last condition means tat te paticle was launced in a diection at igt angles to te adius vecto, tis solution becomes u = u cos c θ Tat is, = sec c θ I ave dawn tis below fo c = 0. 1; tat is, fo k And fo c = 0. 5 ; tat is, fo k 1., a smalle angula momentum. FIGURE XXI.4 c = 0.1

10 10 FIGURE XXI.5 c = 0.5 We also need to conside te case > k, in wic case te geneal solutuion is of te fom u = Acos cθ + B sin cθ. Alas, I aven t ad te enegy to do tis yet. Peaps some view can beat me to it, and let me know at jtatum at uvic.ca