CHAPTER 3 SYSTEMS OF PARTICLES

Transcription

1 HAPTER 3 SYSTEMS O PARTILES 3. Intoducton By systes of patcles I ean such thngs as a swa of bees, a sta cluste, a cloud of gas, an ato, a bck. A bck s ndeed coposed of a syste of patcles atos whch ae constaned so that thee s ey lttle oton (apat fo sall apltude batons of the patcles elate to each othe. In a syste of patcles thee ay be ey lttle o no nteacton between the patcles (as n a loose assocaton of stas sepaated fo each othe by lage dstances o thee ay be (as n the bck stong foces between the patcles. Most (pehaps all of the esults to be deed n ths chapte fo a syste of patcles apply equally to an appaently sold body such as a bck. Een f scentsts ae wong and a bck s not coposed of atos but s a genune contnuous sold, we can n ou agnaton suppose the bck to be ade up of an nfnte nube of nfntesal ass and olue eleents, and the sae esults wll apply. What sot of popetes shall we be dscussng? Pehaps the splest one s ths: The total lnea oentu of a syste of patcles s equal to the total ass tes the elocty of the cente of ass. Ths s tue, and t ay be obous but t stll eques poof. It ay be equally obous to soe that the total knetc enegy of a syste of patcles s equal to M, whee M s the total ass and s the elocty of the cente of ass but ths one, howee obous, s not tue! Befoe we get ound to popetes of systes of patcles, I want to clafy what I ean by the oent of a ecto such as a foce o oentu. You ae aleady fala, fo haptes and, wth the oents of ass, whch s a scala quantty. 3. Moent of a oce st, let s look at a fala two-densonal stuaton. In fgue III. I daw a foce and a pont O. The oent of the foce wth espect to O can be defned as oce tes pependcula dstance fo O to the lne of acton of. θ sn θ IGURE III. O

2 Altenately, (fgue III. the oent can be defned equally well by Tansese coponent of foce tes dstance fo O to the pont of applcaton of the foce. cos θ θ sn θ IGURE III. O Ethe way, the agntude of the oent of the foce, also known as the toque, s sn θ. We can egad t as a ecto, τ, pependcula to the plane of the pape: τ. 3.. Now let e ask a queston. Is t coect to say the oent of a foce wth espect to (o about a pont o wth espect to (o about an axs? In the aboe two-densonal exaple, t does not atte, but now let e oe on to thee densons, and I shall ty to clafy. In fgue III.3, I daw a set of ectangula axes, and a foce, whose poston ecto wth espect to the ogn s. IGURE III.3

3 3 The oent, o toque, of wth espect to the ogn s the ecto τ. 3.. The x-, y- and z-coponents of τ ae the oents of wth espect to the x-, y- and z- axes. You can easly fnd the coponents of τ by expandng the coss poduct 3..: ( y z yˆ ( z x ˆ( x y, τ xˆ z 3..3 z y x whee xˆ, y, ˆ zˆ ae the unt ectos along the x, y, z axes. In fgue III.4, we ae lookng down the x-axs, and I hae dawn the coponents y and z, and you can see that, ndeed, τ x y z. z y z z y x z y y z x? y IGURE III.4 The densons of oent of a foce, o toque, ae ML T, and the SI unts ae N. (It s best to leae the unts as N athe than to expess toque n oules. 3.3 Moent of Moentu In a sla way, f a patcle at poston has lnea oentu p, ts oent of oentu wth espect to the ogn s the ecto l defned by l p, 3.3.

4 4 and ts coponents ae the oents of oentu wth espect to the axes. Moent of oentu plays a ole n otatonal oton analogous to the ole played by lnea oentu n lnea oton, and s also called angula oentu. The densons of angula oentu ae ML T. Seeal choces fo expessng angula oentu n SI unts ae possble; the usual choce s J s (oule seconds. 3.4 Notaton In ths secton I a gong to suppose that we n patcles scatteed though theedensonal space. We shall be deng soe geneal popetes and theoes and, to the extent that a sold body can be consdeed to be ade up of a syste of patcles, these popetes and theoes wll apply equally to a sold body. In the fgue III.5, I hae dawn ust two of the patcles, (the est of the ae left to you agnaton and the cente of ass of the syste. z O y x IGURE III.5 A gen patcle ay hae an extenal foce actng upon t. (It ay, of couse, hae seeal extenal foces actng on t, but I ean by the ecto su of all the extenal foces actng on the th patcle. It ay also nteact wth the othe patcles n the syste, and consequently t ay hae ntenal foces actng upon t, whee goes fo to n except fo. I defne the ecto su as the total extenal foce actng upon the syste.

5 5 I a gong to establsh the followng notaton fo the puposes of ths chapte. Mass of the th patcle Total ass of the syste M Poston ecto of the th patcle efeed to a fxed pont O: x xˆ y yˆ z zˆ Velocty of the th patcle efeed to a fxed pont O: & o (Speed Lnea oentu of the th patcle efeed to a fxed pont O: p Lnea oentu of the syste: P p Extenal foce on the th patcle: Total extenal foce on the syste: Angula oentu (oent of oentu of the th patcle efeed to a fxed pont O: l p Angula oentu of the syste: L l p Toque on the th patcle efeed to a fxed pont O: τ Total extenal toque on the syste wth espect to the ogn: τ τ. Knetc enegy of the syste: (We ae dealng wth a syste of patcles so we ae dealng only wth tanslatonal knetc enegy no otaton o baton: T Poston ecto of the cente of ass efeed to a fxed pont O: xxˆ yyˆ zzˆ The cente of ass s defned by M Velocty of the cente of ass efeed to a fxed pont O: & o (Speed

6 6 o poston ectos, unped sngle-subscpt sybols wll efe to O. Ped snglesubscpt sybols wll efe to. Ths wll be clea, I hope, fo fgue III.5. Poston ecto of the th patcle efeed to the cente of ass : ' Poston ecto of patcle wth espect to patcle : (Intenal foce exeted on patcle by patcle : (Intenal foce exeted on patcle by patcle : If the foce between two patcles s epulse (e.g. between electcally-chaged patcles of the sae sgn, then and ae n the sae decton. But f the foce s an attacte foce, and ae n opposte dectons. Accodng to Newton s Thd Law of Moton (Lex III, Total angula oentu of syste efeed to the cente of ass : L Total extenal toque on syste efeed to the cente of ass : τ o the elocty of the cente of ass I ay use ethe & o. O s an abtay ogn of coodnates. s the cente of ass. Note that and theefoe that s to say 3.4. & & & ; Note also that Note futhe that ( M M That s, the total lnea oentu wth espect to the cente of ass s zeo.

7 7 Hang establshed ou notaton, we now oe on to soe theoes concenng systes of patcles. It ay be oe useful fo you to conue up a physcal pctue n you nd what the followng theoes ean than to eoze the detals of the deatons. 3.5 Lnea Moentu Theoe: The total oentu of a syste of patcles equals the total ass tes the elocty of the cente of ass Thus: P ( M oce and Rate of hange of Moentu Theoe: The ate of change of the total oentu of a syste of patcles s equal to the su of the extenal foces on the syste. Thus, consde a sngle patcle. By Newton s second law of oton, the ate of change of oentu of the patcle s equal to the su of the foces actng upon t: Now su oe all the patcles: p&. ( 3.6. P & ( ( But, by Newton s thd law of oton, 0, so the theoe s poed. oollay: If the su of the extenal foces on a syste s zeo, the lnea oentu s constant. (Law of onseaton of Lnea Moentu. 3.7 Angula Moentu Notaton: L angula oentu of syste wth espect to cente of ass. L angula oentu of syste elate to soe othe ogn O. poston ecto of wth espect to O.

8 8 P lnea oentu of syste wth espect to O. (The lnea oentu wth espect to s, of couse, zeo. Theoe: L L P Thus: L p ( ( ' ( ' ( ' ( ' ' p' M ( 0 L. 0 L L P. Exaple. A hoop of adus a ollng along the gound (fgue III.6: ω /a P M O IGURE III.6 The angula oentu wth espect to s L I ω, whee I s the otatonal neta about. The angula oentu about O s theefoe L I ω Ma I ω Ma ω ( I Ma Iω, whee I I Ma s the otatonal neta about O. 3.8 Toque Notaton: τ ecto su of all the toques about. τ ecto su of all the toques about the ogn O. ecto su of all the extenal foces.

9 9 Theoe:. τ τ 3.8. Thus: ( ' τ. '. τ τ 3.9 opason At ths stage I copae soe soewhat sla foulas. P ' τ' ' ' L τ L τ τ P L L & & & 3.0 Knetc enegy We end ouseles that we ae dscussng patcles, and that all knetc enegy s tanslatonal knetc enegy. Notaton: Τ knetc enegy wth espect to the cente of ass. T knetc enegy wth espect to the ogn O. Theoe:. M T T 3.0. Thus: ( (. ' ' ' ' T. M T T oollay: If 0, T T. (Thnk about what ths eans. oollay: o a non-otatng gd body, T 0, and theefoe. M T (Thnk about what ths eans.

10 0 3. Toque and Rate of hange of Angula Moentu Theoe: The ate of change of the total angula oentu of a syste of patcles s equal to the su of the extenal toques on the syste. Thus: L p 3.. L & & p p&. 3.. But the fst te s zeo, because & and p ae paallel. Also p& L&. But 0 by Newton s thd law of oton, and so s also zeo. Also, τ, and so we ae at L & τ, 3..4 whch was to be deonstated. oollay: If the su of the extenal toques on a syste s zeo, the angula oentu s constant. (Law of onseaton of Angula Moentu.

11 3. Toque, Angula Moentu and a Mong Pont ' ' O IGURE III.7 In fgue III.7 I daw the patcle, whch s ust one of n patcles, n of whch I haen t dawn and ae scatteed aound n 3-space. I daw an abtay ogn O, the cente of ass of the syste, and anothe pont, whch ay (o ay not be ong wth espect to O. The queston I a gong to ask s: Does the equaton L & τ apply to the pont? It obously does f s statonay, ust as t apples to O. But what f s ong? If t does not apply, ust what s the appopate elaton? The theoe that we shall poe and ntepet s We stat: L ( ( L & τ M ' & &. 3.. [ ]. 3.. L& ( ( & & ( & & ( The second te s zeo, because ontnue: &.

12 L & ( & & & Now &, so that the fst te s ust τ. ontnue: L& τ τ τ M && M ( &&. & M M && & L & τ M' & &..E.D Thus n geneal, L & τ, but L& τ unde any of the followng thee ccustances:. ' 0 - that s, concdes wth.. & & 0 - that s, s not acceleatng.. & & and ' ae paallel, whch would happen, fo exaple, f O wee a cente of attacton o epulson and wee acceleatng towads o away fo O. 3.3 The Val Theoe st, let e say that I a not sue how ths theoe got ts nae, othe than that y Latn dctonay tells e that s, s eans foce, and ts plual fo, es, u s geneally tanslated as stength. The te was appaently ntoduced by Rudolph lausus of theodynacs fae. We do not use the wod stength n any patcula techncal sense n classcal echancs, although we do talk about the tensle stength of a we, whch s the foce that t can suon up befoe t snaps. We use the wod enegy to ean the ablty to do wok; pehaps we could use the wod stength to ean the ablty to exet a foce. But enough of these dle speculatons. Befoe poceedng, I defne the quantty I 3.3. as the second oent of ass of a syste of patcles wth espect to the ogn. As dscussed n hapte, ass s (apat fo soe ncetes n geneal elatty

13 3 synonyous wth neta, and the second oent of ass s used so often that t s nealy always called sply the oent of neta, as though thee wee only one oent, the second, woth consdeng. Note caefully, howee, that you ae pobably uch oe used to thnkng about the oent of neta wth espect to an axs athe than wth espect to a pont. Ths dstncton s dscussed n hapte, secton 9. Note also that, snce the sybol I tends to be healy used n any dscusson of oents of neta, fo oent of neta wth espect to a pont I a usng the sybol I. I can also wte equaton 3.3. as Dffeentate twce wth espect to te: I ( I & ( &, and I & ( & & o I & 4T &, whee T s the knetc enegy of the syste of patcles. The sus ae undestood to be oe all patcles -.e. fo to n. & & s the foce on the th patcle. I a now gong to suppose that thee ae no extenal foces on any of the patcles n the syste, but the patcles nteact wth each othe wth conseate foces, beng the foce exeted on patcle by patcle. I a also gong to ntoduce the notaton, whch s a ecto dected fo patcle to patcle. The elaton between these thee ectos n shown n fgue III.8. IGURE III.8 Ogn

14 4 I hae not dawn the foce, but t wll be n the opposte decton to f t s a epulse foce and n the sae decton as f t s an attacte foce. The total foce on patcle s, and ths s equal to & &. Theefoe, equaton becoes Now t s clea that I & 4T < Howee, n case, lke e, you fnd double subscpts and suatons confusng and you hae eally no dea what equaton eans, and t s by no eans at all clea, I wte t out n full n the case whee thee ae fe patcles. Thus: ( ( ( 3 ( 4 ( Now apply Newton s thd law of oton: ( ( ( 3 ( 4 ( Now bea n nd that, and we see that ths becoes

15 and we hae aed at equaton Equaton then becoes I & T < Ths s the ost geneal fo of the al equaton. It tells us whethe the cluste s gong to dspese (I & poste o collapse (I & negate though ths wll edently depend on the natue of the foce law. Now suppose that the patcles attact each othe wth a foce that s nesely popotonal to the nth powe of the dstance apat. o gatatng patcles, of couse, n. The foce between two patcles can then be wtten n aous fos, such as k k ˆ ˆ, n n and the utual potental enegy between two patcles s nus the ntegal of d, o k U n ( n I now suppose that the foces between the patcles ae gatatonal foces, such that G Now etun to the te, whch occus n equaton 3.3.8: k k (. n n U n 3.3. Thus equaton becoes

16 6 I & 4T ( n U, whee T and U ae the knetc and potental eneges of the syste. Note that fo gatatonal nteacton (o any attacte foces, the quantty U s negate. Equaton s the al theoe fo a syste of patcles wth an attacte foce between the. The syste wll dspese o collapse accodng the sgn of I &. o a syste of gatatonally-nteactng patcles, n, and so the al theoe takes the fo I & 4T U Of couse, as the nddual patcles oe aound n the syste, I, T and U ae all changng fo oent to oent, but always n such a anne that equaton s satsfed. In a stable, bound syste, by whch I ean that, oe a long peod of te, thee s no long-te change n the oent of neta of the syste, and the syste s nethe eesbly dspesng o contactng, that s to say n a syste n whch the aeage alue of I & oe a long peod of te s zeo (I ll defne long soon, the al theoe fo a stable, bound syste of n patcles takes the fo T ( n U 0, and fo a stable syste of gatatonally-nteactng patcles, T U 0, Hee the angula backets ae undestood to ean the aeage alues of the knetc and potental eneges oe a long peod of te. By a long peod we ean, fo exaple, long copaed wth the te that a patcle takes to coss fo one sde of the syste to the othe, o long copaed wth the te that a patcle takes to oe n an obt aound the cente of ass of the syste. (In the absence of extenal foces, of couse, the cente of ass does not oe, o t oes wth a constant elocty. o exaple, f a bound cluste of stas occupes a sphecal olue of unfo densty, 3GM the potental enegy s (see equaton 5.9. of elestal Mechancs, so the al 5a theoe (equaton wll enable you to wok out the ean knetc enegy and hence speed of the stas. A globula cluste has oughly sphecal syety, but t s not of unfo densty, beng centally condensed. If you assue soe functonal fo fo the densty dstbuton, ths wll ge a slghtly dffeent foula fo the potental enegy, and you can then stll use the al theoe to calculate the ean knetc enegy.

17 7 A tal exaple s to consde a planet of ass ong n a ccula obt of adus a aound a Sun of ass M, such that <<M and the Sun does not oe. The potental enegy of the syste s U GM/a. The speed of the planet s gen by equatng a GM to, fo whch T GM/(a, so we easly see n ths case that T U 0. a