Recap: Newton s Gravitational Law

Transcription

1 Recap: Newton s Gavitational Law The gavitational foce between two objects is popotional to thei masses and invesely popotional to the squae of the distance between thei centes. F = G m 1 m (Newtons) F is an attactive foce vecto acting along line joining the two centes of masses. F 1 F G = Univesal Gavitational Constant G = 6.67 x N.m /kg (vey small) m 1 Note: G was not measued until > 100 yeas afte Newton! - by Heny Cavendish (18 th cen.) m (F 1 = -F )

2 m e e F How is Weight Related to Gavitation? F m e = adius of Eath = 6370 km m = mass of an object m e = mass of Eath = 5.98 x 10 4 kg Gavitational foce of attaction: if m = 150 kg, F = 147 N (o ~ 330 lbs wt) But this foce ceates the object s weight: By Newton s nd law (F=ma) we can also calculate weight: W = m g = 9.81 x 150 = 147 N By equating these expessions fo gavitational foce: m g = G m e m o at suface: g = G m e e F = G m 1 m (N) e Result: g is independent of mass of object!!

3 Thus acceleation due to gavity g is: 1. Constant fo a given planet and depends on planets mass and adius.. Independent of the mass of the acceleating object! (Galileo s discovey). Howeve, the gavitational foce F is dependent on object mass. In geneal, the gavitational acceleation (g) of a planet of mass (M) and adius (R) is: g = G M R Planet Mecuy Venus Mas This equation also shows that Jupite g will decease with altitude: e.g. At 100 km height g = 9.53 m/s Satun Uanus At moon s obit g =.7 x 10-3 m/s Neptune Pluto Eath Moon g m/s no solid suface

4 Newton s 3 F F d law: Each body feels same foce m 1 m Gm 1 m F= acting on it (but in opposite diections) Thus each body expeiences an acceleation! Example: Boy 40 kg jumps off a box: Foce on boy: F = m g = 40 x 9.81 = 39 N Foce on Eath: F = m e a = 39 N 39 3 o a = = x 10 4 x 10-3 m/s ie. almost zeo! Example: 3 billion people jumping off boxes all at same time (mass 100 kg each) 3 x 10 9 x 100 x a = x 10 4 = 5 x m/s Conclusion: The Eath is so massive, we have essentially no effect on its motion!

5 Planetay Motions & Obits (Chapte 5) Heavenly bodies: sun, planets, stas How planets move? Geeks: Stas emain in the same elative position to one anothe as they move acoss the sky. Seveal bight stas exhibit motion elative to othe stas. Bight wandees called planets. Planets oam in egula but cuious manne. Hypothesis: Geocentic Eath-centeed univese! Sun moves aound the Eath - like on a long ope with Eath at its cente. Stas lying on a giant sphee with Eath at cente. Moon too exhibits phases as it obits Eath.

6 Plato: Concentic sphees sun, moon and 5 known planets each move on a sphee centeed on the Eath. Big poblem Planets do not always behave as if moving continuously on a sphee s suface. Retogade motion (happens ove seveal months) e.g. Mas, Jupite, Satun Planet appeas to go backwads! Fixed stas Solution (Ptolemy nd centuy AD) Planet Eath Epicycle Eath Planet Epicycles cicula obits not on sphees. Planets moved in cicles that olled aound lage obits - still centeed on Eath.

7 Heliocentic Model: Copenicus (16 th centuy) Sun centeed view that was late poven by Galileo using telescope obsevations of Jupite and its satellite moons. Bad news: demoted Eath to status of just anothe planet! Revolutionay concept equied the Eath to spin (to explain Sun s motion). If Eath spinning why ae we not thown off? (at 1000 mph). Good news: no moe need fo complex epicycles to explain etogade motion! Mas obit Eath obit Same diection Result: Eath moves faste in obit and Mas appeas to move backwads at cetain times. Fixed stas field

8 Copenicus heliocentic model assumed cicula obits but caeful obsevations by Tycho Bahe (the last geat naked eye astonome) showed not tue Keple (17 th centuy, Bahe s student) developed thee laws based on empeical analysis of Bahe s extensive data 1. Obits of planets aound the sun ae ellipses with Sun at one focus. Note: A cicle is a special case of an ellipse with foci coincident. Sun two foci In eality, the planets obits ae vey close to cicula but nevetheless ae slightly elliptical. planet

9 nd law: Descibes how a planet moves faste when neae the Sun. fast Sun planet slow. A adius vecto fom Sun to planet sweeps out equal aeas in equal times. 3 d law: Lots of numbes late (tial and eo) Keple discoveed that to a vey high appoximation, the peiod of obit is elated to aveage adius of obit: 3. Τ = 3 constant Same value fo constant fo all planets (except the Moon)

10 This means that the oute planets (i.e. futhe fom the Sun than Eath) all have much lage obital peiods than Eath (and vice vesa). i.e. Τ 3 So if we know Τ (by obsevations) we can find fo each planet! Conclusion: These caeful obsevations and new fomula set the scene fo Newton s theoy of gavitation F = G m 1 m

11 Using Keple s 3 d law, Newton calculated: Ô 3 = 4 D G m = constant (fo a given ' m') whee: m = mass of Sun fo the planetay motions, but m = mass of Eath fo the Moon s motion. Hence Keple s diffeent esult fo the constant Τ 3 = a constant numbe fo moon compaed with othe planets!

12 Atificial Satellites Τ Each of the planets will have its own value of fo its satellites (as constant has a 1/m dependence). 3 But in each case as Τ 3 the lowe the altitude of the satellite the shote its obital peiod. Example: 1 st Eath satellite Sputnik (1957) Launched into a vey low altitude obit of ~ 70 km (any lowe and atmospheic dag would pevent obital motion). = =6640 km => Τ = 90 min (1.5 hs to obit Eath) 70 km Sputnik e beep beep

13 Example: Geosynchonous Obit Obital peiod = 4 hous Pemits satellites to emain stationay ove a given equatoial longitude. Fo Τ = 4 hs => = 4,000 km (to cente Eath) i.e. altitude 7 R e (compaed with 60 R e fo Moon.) equato Geostationay obit is vey impotant fo Eath obseving and communications satellites a vey busy obit! In geneal thee ae many, many possible obits, e.g.: Cicula and elliptical Low Eath obits (LEO) Geostationay obit Pola obit e.g. GPS system uses many obiting satellites. cicula pola elliptical

14 We can equate centipetal foce to gavitational attaction foce to detemine obital speed (v o ). Fo cicula motion: Centipetal foce = gavitational foce (F C = F G ) m v Obital Velocity o = v o = G M m G M M = planet s mass m = satellite s mass M» m Results: Any satellite egadless of its mass (povided M» m) will move in a cicula obit o adius and velocity v o. The lage the obital altitude, the lowe the equied tangential velocity!

15 Example: Low Eath obit: altitude 600 km) v o ( " 11) ( " ! 10! 5.98! 10 ) G M = = ! 10 = (o km/h) Planetay Obital Velocity: v o 7.6 km/s M km/s Lage the obit, the lowe the speed M = mass of the Sun Mean Distance (Astonomical Units)

16 Qu: How to achieve obit? Launch vehicle ises initially vetical (minimum ai dag). Gadually olls ove and on sepaation of payload is moving tangentially at speed v o poduces cicula obit. If speed less than v o, caft will descend to Eath in an (decaying) elliptical obits. If speed geate than v o it will ascend into a lage elliptical obit. If speed geate than v o it will escape eaths gavity on paabolic obit! Eath cicula (v o ) paabolic elliptical

17 Not a simple ellipse due to gavitational foce of Eath and Sun. Sun Luna Obit 1.5 x 10 8 km ( 400 R moon ) F 1 Eath Moon Gavitational attaction between Eath and Moon povides centipetal acceleation fo obit. Sun s gavitation distots luna obital ellipse. (Obit oscillates about tue elliptical path.)

18 Moon The only satellite Newton could study and played a key ole in his discoveies Phases known since pehistoic times Moonlight is eflected sunlight. Sun light New moon Eath Half moon Full moon Phases simply due to geomety of sun, moon Half moon and Eath. Phases epeat evey 7.3 days (luna obit). Full moon: Moon on opposite side of Eath fom Sun. Fully illuminated disk. Rises at sunset and sets at sunise (due to Eath s 4 h otation). Luna eclipse only duing full moon.

19 At othe times duing moon s obit of Eath, we see only a pat of illuminated disk. New moon: Moon on same side of Eath as the Sun. Essentially invisible (cescent moons seen eithe side of New moon). Rises at sunise and sets at sunset (i.e. up all day). Sola eclipse condition. When moon is in-between full and new phase, it can often be seen duing daylight too. Example: half moon ises at noon and sets at midnight (and vice vesa). Unde good obseving conditions (at sunset o sunise) you can see dak pats of moon illuminated by Eathshine!