Chapter 12. Gravitation

Transcription

1 Chapte 1 Gavtaton 1 Gavty the oce that holds the Moon n ts obt s the same that makes an apple all not only does Eath attact an apple and the Moon but evey body n the Unvese attact evey othe body Isaac Newton

2 Newton s law o Gavtaton Isaac Newton 1665 Evey patcle attacts any othe patcle wth a gavtatonal oce whose magntude s gven by m1m F G m 1 and m ae the masses o the patcles, s the dstance between them, G s the gavtatonal constant G N m /kg Two oces o attacton ae equal n magntude but opposte n decton (Newton s thd law) Example Fnd the oce o gavty between a student (70 kg o 154 lb) and a textbook ( kg o 4.4 lb) when they ae sepaated by a dstance o 0. m (o 0.98 t) F ( Nm / kg (70.0kg)(.0kg) ) (0.m) the oce to keep 1 dme (Fmg) s about N F d (9.81m / s ) (0.001kg) N.e. F F d Gavtatonal oce deceases wth dstance as 1/ An ncease n dstance by a acto o 10 esults n a decease n the oce 4by a acto o 100.

3 Questons and poblems 1. Why do we cae, the oce o gavty between object aound us s so small?. How to calculate the gavtatonal oce between eal objects,.e. that ae not pont-lke ones?. What to do thee ae moe than two patcles? 4. Moton and gavty 5. What would happen as ->0? 5 Gavtaton and the pncple o supeposton The pncple o supeposton: the net eect s the sum o the ndvdual eects Fo n nteactng patcles the net oce on patcle 1 can be wtten as + F1, net F1 F1 F14 KF1 n + + beng pactcal see chapte Vectos (addng vectos usng components) 6

4 Gavtatonal attacton between eal objects 1. I the szes o the objects ae small compaed to the dstance between them we may consde the objects lke patcles (good appoxmaton o Moon-Eath) R Eath 6,70 km R Moon 1,061 km 8,000 km 7 Gavtatonal attacton between eal objects. How about the apple-eath? Newton s shell theoem: A unom sphecal shell o matte attacts a patcle that s outsde the shell all the shell s mass wee concentated at ts cente Mm F G 8 4

5 Gavtaton nea Eath s suace Let us assume that Eath s a unom sphee o mass M. The magntude o the gavtatonal oce on a patcle o mass m, at a dstance om Eath s cente Newton s second law Mm F G F mg then the ee-all acceleaton g 9 Gavtaton nea Eath s suace The ee-all acceleaton s not a constant! g Besdes - the actual ee-all acceleaton g s deent om the equaton above 1. Eath s not unom (densty vaes). Eath s not a sphee. Eath s otatng 10 5

6 Gavtaton nsde Eath o Moe om Newton s shell theoem Jouney to the Cente o the Eath A unom sphecal shell exets no net gavtatonal oce on a patcle located nsde t g g 4π M ρ V ρ G 4π ρ 4π g Gρ o 0 g0, o the net gavtatonal oce nsde Eath 0 11 The eect o the Eath s otaton Unom ccula moton: the peod o the otaton 4 hous N F g then N Fg v ge g Mm G mg Vaaton o g e Canal zone Pttsbugh, PA Geenland e F g N Lnea speed o a pont on the Eath s suace due to the otaton: Noolk: 87 mph Equato: 100 mph Speed aound the Sun: 66,000 mph 1 6

7 Gavtatonal potental enegy Potental enegy s enegy that can be assocated wth the conguaton o a system o objects that can exets oces on one anothe. I the conguaton o the system changes, the potental enegy o the systems can also change. Gavtatonal potental enegy U U() appoaches zeo as appoaches nnty o any nte value o, the value o U() s negatve connecton between the oce and the potental du d F d d 1 Compae two dentons o the Eath Gavtatonal potental enegy U and U mgh ΔU << + U m Δ ( ) mgh The omula we have used n the past, U mgh, s vald only when <<. 14 7

8 Gavtatonal potental enegy o many patcles Gavtatonal potental enegy o a system wth moe than two patcles U Gm1m 1 Gm1m + 1 Gmm + 15 Pojectles, Satelltes and Planets How to explan the moton? 16 8

9 Pat 1: Vetcal moton Consevaton o enegy Vetcal moton beoe (potental enegy om zeo at h0) + mgh K + U K + U + mgh Vetcal moton now (potental enegy om zeo at ) 17 Pat 1: Vetcal moton v Consevaton o enegy o vetcal moton K + U K + U 18 9

10 Vetcal moton: Escape speed Vetcal pojectle moton: up, up and up Thee s a cetan mnmum ntal speed that wll cause a pojectle to move upwad oeve, theoetcally comng to est only at nnty (g s not a constant!). Fom consevaton o enegy R v R R 0 escape speed o Eath: M5.98*10 4 kg, R 6.7*10 6 m, v11. km/s 19 Pat : Obtal moton It s a matte o ntal condtons! h v x 0 then vetcal moton at some v x v c ccula moton ( + h) vc m + h v c + h Fo a shuttle: v7.9 km/s 0 10

11 Satelltes: Obts and Enegy As a satellte obts Eath on ts ellptcal path, both ts speed, whch xes ts knetc enegy K, and ts dstance om the cente o Eath, whch xes ts gavtatonal potental enegy U, luctuate wth xed peods. Howeve, the mechancal enegy E o the satellte emans constant. Fo ccula obts (second Newton s law) v m K K U 1 Satelltes: Obts and Enegy How to change the obt? E K + U Less knetc enegy less total enegy E (moe negatve) less adus 11

12 Example A satellte s obtng the Eath as shown below. At what pat o the obt, any, ae the ollowng quanttes lagest? # Knetc enegy # Potental enegy # Total enegy # Obtal velocty # Gavtatonal oce # Angula momentum Checkpont An astonaut wokng outsde o the space staon obtng the Eath eleases a wench. Neglectng a esstance, the wench wll: A) stke Eath unde the satellte at the nstant o elease B) stke Eath unde the satellte at the nstant o mpact C) stke Eath ahead o the satellte at the nstant o mpact D) stke Eath behnd the satellte at the nstant o mpact E) neve stkes Eath 4 1

13 Planets and satelltes: Keple s Laws Thee laws om Johannes Kepple ( ) 1. The law o obts: All planets move n ellptcal obts, wth the Sun at one ocus A ccle s just a specal case o an ellpse. 5 Planets and satelltes: Keple s Laws (cont.). The law o aeas: A lne that connects a planet to the Sun sweeps out equal aeas n the plane o the planet s obt n equal tmes. Keple s second law s totally equvalent to the law o consevaton o angula momentum 6 1

14 Planets and satelltes: Keple s Laws (cont.). The law o peods: The squae o the peod o any planet s popotonal to the cube o the sem majo axs o ts obt. The peod does not depend on the mass o the obtng object. Applyng Newton s second law T m( ω ) π ω T π 4 o T π 7 / Fo a planet o adus R and densty ρ 4π mass o a planet M ρ R ee all acceleaton g R escape speed v e 4π ρ GR 8πGρ R R Obtal speed just above the gound v o R v e 8 14

15 Black Holes Escape om a sta v 8πGρ R escape R example: escape speed om the suace o the sun s about. mllon km/h o 1.5 mllon mph Fo lght c R s R s c R s s the Schwazschld adus (nothng, not even lght can escape om that body/sta) 9 Geosynchonous satelltes example Many satelltes ae movng n a ccle n the eath s equatoal plane. They ae at such heght that they always eman above the same pont. Fnd the alttude o such satelltes above the eath s suace T π T 4π h R E E E / 1/ *600 s m 7 m 0 15

16 Mllennum Smulaton: The bggest and most detaled supecompute smulaton o the evoluton o the Unvese om a ew hunded thousand yeas ate the Bg Bang to the pesent day. The Mllennum Smulaton used 10 bllon patcles to tack the evoluton o 0 mllon galaxes ove the hstoy o the unvese. A -dmensonal vsualzaton o the Mllennum Smulaton. The move shows a jouney though the smulated unvese. Dung the two mnutes o the move, we tavel a dstance o whch lght would need moe than.4 bllon yeas. 1 Physcs o Gavtaton seach o gavtatonal waves 16