Gravitation. Definition of Weight Revisited. Newton s Law of Universal Gravitation. Newton s Law of Universal Gravitation. Gravitational Field

Transcription

1 Defnton of Weght evsted Gavtaton The weght of an object on o above the eath s the gavtatonal foce that the eath exets on the object. The weght always ponts towad the cente of mass of the eath. On o above anothe astonomcal body, the weght s the gavtatonal foce exeted on the object by that body. The decton of the weght (o gavtatonal foce) ponts towads the cente of mass of that body. SI Unt of Weght: Newton (N) Newton s Law of Unvesal Gavtaton m W G () whee W s the weght of an object wth mass m due to the eath s gavtatonal foce, G s the unvesal gavtatonal constant.7x0 - m /kg, s the mass of the eath, s the dstance between the object and the cente of mass of the eath. Wte W n the usual fom, W mg We get g G.38x0 m 5.97x0 kg Newton s Law of Unvesal Gavtaton. The gavtatonal acceleaton, g depends on the dstance,, between the object and the eath s cente of mass.. quaton () can be genealzed fo the gavtatonal foce between two objects wth masses m and, fo whch n eqn. () s eplaced by and the dstance epesents the dstance between the centes of mass of the two objects. 3. By Newton s 3 d law, the object acts on the eath wth a foce havng the same magntude but pontng n the opposte decton. G.7x0 - m kg - 3 Gavtatonal Feld The gavtatonal feld, g, at a pont s the gavtaton foce an object expeences when placed at that pont dvded by the object s mass. Fo gavtatonal feld comng fom the eath, m g G m g G whee g s n unts of m/s and s the dstance the pont s fom the cente of mass of the eath. Ths esult shows that the gavtatonal feld s the same as the 5 gavtatonal acceleaton. Gavtatonal Feld Snce the gavtatonal feld s essental the gavtatonal foce expeenced by a unt mass at the pont of nteest, t should have a decton. The decton of the gavtatonal feld s ponted towads the body that poduces the feld. In othe wods, gavtatonal foce always attacts the object towads the body poducng the feld. Note that gavtatonal foce s a knd of nteacton foces. So both the object and the body nvolved expeence the same magntude of attactve foce fom each othe.

2 Gavtatonal Feld F m g m m F m g m < m Gavtatonal feld poduced by m at a dstance of : g Gm / Gavtatonal feld poduced by m at a dstance of : g Gm / (< g snce m < m ) 7 Gavtatonal foce on the eath s suface Fnd the gavtatonal feld g on the eath s suface. g G ( ) ( ) kg.7 0 N m kg (.38 0 m) ( ) ( ) 5.98kg +.7 N m kg m s (.38 m) 8 Gavtatonal foce actng on the eath by a ca Fnd the gavtatonal acceleaton actng on the eath by a ca wth mass 500 kg unnng on ts suface. The gavtatonal foce, F, actng on the eath by the ca s (500kg)(9.8m/s ) 700N ponted fom the eath s cente towads the ca. The acceleaton on the eath due to ths foce s F/ mg +mg 700N/(5.98x0 kg).x0 - m/s, whch would be too small to be detected. 9 ath and oon Usng the fact that the gavtatonal feld at the suface of the ath s about sx tmes lage than that at the suface of the oon, and the fact that the ath s adus s about fou tmes the oon s adus, detemne how the mass of the ath compaes to the mass of the oon. Gm g g /m g /m (: eath, : moon) m /m ( / ) (g /g ) (/) (/) /9 So the mass of the moon s /9 tmes of that of the eath. 0 Thee masses on a staght lne (a) Thee masses, of mass,, and 3 ae equally 3 spaced along a lne, as shown. The only foces each mass expeences ae the foces of gavty fom the othe two masses. () Whch mass expeences the lagest magntude net foce? qual fo all thee 5. Net foce magntude on s equal to that on 3 but bgge than that on Thee masses on a staght lne 3 () What s the magntude of the net foce expeenced by mass? Soluton We can just add the foces fom the othe two objects. The net foce on the object s dected ght wth a magntude of:

3 A tangle of masses Thee pont objects, though 3 wth dentcal mass, ae placed at the cones of an equlateal tangle. 3 In what decton s the net gavtatonal feld at pont A, halfway between objects and 3? 3 Net gavtatonal feld at pont A The net gavtatonal feld at pont A comes fom thee souces, objects,, and 3. The dagam below shows the coespondng thee gavtatonal felds, g, g and g 3. Obvously, g and g 3 cancel. The net feld at A s due only to g, whch ponts up. g g g 3 3 g Gm/ A / A whee A s the dstance between pont A and object, m s the mass of the objects. Net gavtatonal foce at pont A Fnd the net gavtatonal foce expeenced by an object of mass at pont A n tems of the mass of the thee souce objects m and the length of each sde of the equlateal tangle, L. Soluton Fom the above dscusson, the net gavtatonal feld s g, whch s Gm/ A. But A Lsn0 o ( 3/)L. So, g Gm/(3L ). The net gavtatonal foce, F, expeenced by an object wth mass s tmes the feld at that pont. So, F ()g Gm/(3L ). 5 Velocty of Obtng Satelltes Thee s only one speed, v, that a satellte can have f the satellte s to eman n a ccula obt wth adus,. What s the elaton between v and? What s v at 0? Fo the satellte to eman n a ccula obt wth adus, the acceleaton of the satellte must equal a c v / fo othewse the unbalanced foce, F net ma c wll cause the satellte to move away fom the (ccula) obt. v m m mg G G v F net mg Thee masses on a staght lne 3 Velocty of Obtng Satelltes Substtute 0 to fnd v n the obt: () Whch of the followng changes would cause the magntude of the foce expeenced by the object to ncease by a facto of? Select all that apply. [ ] double the mass of all thee objects [ ] change the mass of the object to 8, wthout changng the mass of the othe objects [ ] double [ ] ove the system to a paallel unvese whee the value of the unvesal gavtatonal constant s fou tmes lage than ts value n ou unvese 7 v G -.7x0 m kg x5.97x0 0x.38x0 m.50 km/s Ths s faste than the speed of any plane eve flown on eath (whch s ~ km/s)! kg -.7m kg 5.97kg ( + ) /.38m 0 8 3

4 The Obtal adus fo Synchonous satelltes In many applcatons of satelltes such as dgtal satellte system televson, t s desable that the moton of the satellte follows a ccula obt and be synchonzed wth the eath s self otaton (so that the satellte s always at the same locaton above the eath s suface). Ths eques that the peod, τ, of the satellte be exactly one day,.e., 8. x 0 s. What s the heght, H, of the satellte above the eath s suface? The Obtal adus fo Synchonous satelltes G Fom last example, v π But, fo unfom ccula moton, v τ π τ G.38x0 m 5.97x0 kg τ G G.7x0 - m kg - 3/ () π. x 0 7 m So, H 3.58 x 0 7 m 9 0 Geneal Obtal otons of satelltes. They often tace out an ellpse. Theefoe, the gavtatonal foce s not always pependcula to the satellte s velocty.. K + U s conseved thoughout the obt 3. Angula momentum s conseved thoughout the obt.. Lnea momentum s not conseved snce thee s gavtatonal foce. 5. The obt peod does not depend on the mass of the satellte.. They obey the Keple s nd Law. That s, equal aeas of the obt ae swept out n equal tme Gavtatonal potental enegy The gavtatonal nteacton o potental enegy of two objects wth masses m and and sepaaton s: Gm Ug The negatve sgn tells us that the nteacton s attactve. Note that wth ths equaton the potental enegy s defned to be zeo when nfnty. What mattes s the change n gavtatonal potental enegy. Fo small changes n heght at the ath s suface,.e., fom to +h, the equaton above gves the same change n potental enegy as mgh, whee g G/ as found befoe. ntevals. Fou objects n a squae Fou objects of equal mass, m, ae placed at the cones of a squae that measues L on each sde. How much gavtatonal potental enegy s assocated wth ths confguaton of masses? 3 Fou objects n a squae Label the fou objects by,, 3, as shown. Imagne that you stat wth no object on the squae. () You fst bng object n place. That causes no enegy change snce thee s no othe masses n the space. () Then you bng object n place. The potental enegy (P) nvolved s mg, whee g Gm/L s the gavtatonal feld poduced by object at whee object s placed. (3) Then you bng object 3 n. The P nvolved s m[g 3 + g 3 ]. () Fnally, you bng object n. The P equed s m[g + g + g 3 ]. Among these, U, U 3, U 3, and U all equal -Gm /L And U 3 and U equal -Gm /( L) The gavtatonal P s the sum of all the sx U j s, whch s -(+ )Gm /L. Convnce youself that the answe s ndependent of the ode by whch we bng the chages n. 3

5 Gavtatonal potental enegy of a dstbuton of masses In geneal, the gavtatonal potental enegy, U, of a dstbuton of masses compsng masses m (whee,, 3,, N) s: U -G[m m / + m m 3 / m / N + m m 3 / 3 + m m / + + m / N / N-,N ], scape speed How fast would you have to thow an object so t neve came back down? Ignoe a esstance. Fnd the escape speed - the mnmum speed equed to escape fom a planet's gavtatonal pull. Appoach to use: Foces ae had to wok wth hee because the sze of the foce changes as the object gets fathe away. negy s ease to wok wth n ths case. whee j s the dstance between m and m j (, j,, 3,, N.) 5 Consevaton of enegy: U + K + Wnc Uf + Kf scape speed Smulaton Whch tems can we coss out mmedately?. Assume no esstve foces, so W nc 0.. Assume that the object baely makes t to nfnty, so both U f and K f ae zeo. Ths leaves: U + K 0 -Gm / + ½ mv escape 0 v escape G / x( / ) (G / ) scape speed Smulaton Whch tems can we coss out mmedately?. Assume no esstve foces, so W nc 0.. Assume that the object baely makes t to nfnty, so both U f and K f ae zeo. Ths leaves: U + K 0 -Gm / + ½ mv escape 0 v escape G / x( / ) (G / ) 9.8 m/s v escape (9.8 m/s )(.38x0 m) v escape (5x0 )m/s. km/s 9.8 m/s v escape (9.8 m/s )(.38x0 m) v escape (5x0 )m/s. km/s 7 8 5