Section : Newton s Law of Gavitation In 1686 Isaac Newton published his Univesal Law of Gavitation. This explained avity as a foce of attaction between all atte in the Univese, causin e.. apples to fall fo tees and the oon to obit the Eath. (See also 1X Dynaical stonoy) Conside two asses and, sepaated by distance 1 1 ( we inoe fo the oent the physical extent of the two asses i.e. we say that they ae point asses ) 1 Gavitational foce on due to is F G 1 1 ˆ 1 (.1)
Notes 1. The avitational foce is a vecto i.e. it has both anitude and diection.. ˆ 1 is a unit vecto fo to. In othe wods, F 1 acts 1 alon the staiht line joinin the two asses. 3. The avitational constant is a fundaental constant of G natue, believed to be the sae eveywhee in the Univese. G 6.673 10 11 N k - (.) 4. The avitational foce on due to 1 is of equal anitude, but in the opposite diection, i.e. F 1 F 1 (.3) 5. Gavity is descibed as an Invese-Squae Law. i.e. the avitational foce between two bodies is invesely popotional to the squae of thei sepaation.
6. The avitational foce pe unit ass is known as the avitational field, o avitational acceleation. It is usually denoted by side We shouldn t be too supised that is an acceleation: Newton s nd law states that Foce ass x acceleation. Howeve, Newton s nd law concens inetial ass while Newton s law of avitation concens avitational ass. That these two quantities ae easued to be identical to each othe is a vey pofound fact, fo which Newton had no explanation, but which uch late led Einstein to his theoy of elativity. See 1 dynaics & elativity, and special elativity lanets and stas ae not point ass objects. To deteine the net foce on 1 due to we ust add toethe the foces fo all pats of. Suppose that is spheical, and its density (aount of ass 1 pe unit volue) depends only on distance fo the cente of. e.. oe dense Less dense We say that the density is spheically syetic
In this special case, the net avitational foce on 1 due to is exactly the sae as if all of the atte in wee concentated in a point at the cente of. 1 Section 3: Suface Gavity and Escape Speed Conside, theefoe, a spheical planet of adius and total ass which has a spheically syetic density distibution. The avitational field the planet s suface is diected adially towads the planet s cente, and at G anitude of (3.1)
The anitude of denoted by is known as the suface avity, often just (i.e. it is not a vecto). Fo exaple Eath 5.974 10 6.378 Eath 10 4 6 k This eans (assuin the Eath is spheical) - Eath 9.80s (3.) easues the ate of acceleation of fallin objects (nelectin ai esistance). Fo any othe body to wite (e.. anothe planet o oon) it is useful Eath Eath Eath (3.3) e.. fo as : as 0. 533 as 0. 107 Eath Eath So - as 0.377 Eath 3.69s (3.4)
Execise: Use the table of planetay data fo the textbook and Section 1 to copute fo all the planets. We can also expess in tes of aveae density ρ ass ρ volue 4 3 π 3 i.e. 4 π 3 3 ρ (3.5) So fo (3.1) G 4 π G ρ (3.6) 3 Fo eq. (3.6) 1. If two planets have the sae aveae density, the lae planet will have the hihe suface avity.. If two planets have the sae adius, the dense planet will have the hihe suface avity.
Escape Speed Conside a pojectile launched vetically upwads at speed υ fo the suface of a planet, with suface avity. s the pojectile clibs, the planet s avity slows it down its kinetic eney conveted to potential eney. In the aniation we see the pojectile slow to a stop, then acceleate back to the suface. If the initial speed is hih enouh, the pojectile will neve etun to the suface. We say that the pojectile escapes the planet s avity. The iniu speed equied to achieve this escape is known as the escape speed, and υ escape G ass of planet adius of planet (3.7) See 1X Dynaical stonoy
Fo the Eath υ escape 11. ks -1 Fo Jupite υ escape 59.6 ks -1 Note that the escape speed does not depend on the ass of the pojectile. Section 4: Tidal Foces In Section we pointed out that planets and oons (and indeed stas) ae not point ass objects. Consequently, they will be subjected to tidal foces since diffeent pats of thei inteio and suface expeience a diffeent avitational pull fo neihbouin bodies. We see this diffeential effect with e.. the Eath s tides, due to the oon (and the Sun). We now conside biefly the aths of tidal foces, befoe late exploin soe applications to planets and oons in the Sola Syste.
Suppose that planet and oon ae sepaated by distance (cente to cente) S Conside a sall ass at position and S C C Q Tidal foce between and due to the planet is equal to the diffeence in the avitational foce on and due to Let the distance fo to equal, and assue C C C << Gavitational foce on due to : Gavitational foce on due to : F G C C F G ( ) (We needn t woy about foces bein vectos hee, since, C and Q lie alon a staiht line) So F F C G G ( ) G G ( 1 / ) G 1 1 ( 1 / ) (4.1)
We can wite eq. (4.1) as F F C G [( 1 ) 1] If we now use that << then ( ) 1+ 1 (4.) ( side: Eq. (4.) follows fo the Binoial expansion fo 1+ nx x << 1 which is appoxiately if ) ( 1+ x) n So F G FC (4.3) 3 The ipotant point hee is that the anitude of the tidal foce is an invese-cube law: i.e. it falls off oe apidly with distance than does the foce of avity. So, if the planet is fa fo the oon, the tidal foce expeienced by the oon (and vice vesa) will be sall. S Convesely, howeve, if the oon lies vey close to the planet, then the tidal foces on its inteio ay be consideable. In a late section we will exploe the consequences of this fo the stability of oons in the Sola Syste.