Physics 201, Lectue 22 Review Today s Topics n Univesal Gavitation (Chapte 13.1-13.3) n Newton s Law of Univesal Gavitation n Popeties of Gavitational Foce n Planet Obits; Keple s Laws by Newton s Law Of Univesal Gavitation q The Law: Any pai of objects in the Univese attact each othe with a foce that is popotional to the poducts of thei masses and invesely popotional to the squae of thei distance F 12 = G m m 1 2 ˆ 2 12 on F 21 = G m 1m 2 ˆ 2 21 = F 12 Univesal Gavitational Constant: G = 6.673x10-11 Nm 2 /kg 2 m 2 F 12 =-F 21 F 12 F 21 12 12 = 12 ˆ 12 m 1 Univesal Gavity and Gavity on Eath q Gavity on eath (fee fall gavity) is just one example of the Univesal Gavity M m F g = GMm/ 2 = m( GM/ 2 ) On Eath: g= GM E /R E 2 e.g. On eath suface: g=g (5.98x10 24 kg)/ (6.37x10 6 m) 2 = 9.83 m/s 2 On top of Eveest: g=g (5.98x10 24 kg)/ (6.37x10 6 + 8848m) 2 = 9.81 m/s 2 On moon: g moon =G (7.35x10 22 kg)/ (1.74x10 6 ) 2 = 1.62 m/s 2 ~ 1/6 g Eath g 1
Motion Unde Gavitational Foce q Foce on m is always pointing to M Ø Equation of Motion: m d 2 (t) dt 2 = GMm 2 Don t woy about the exact fom of this diffeential eq. This is the same as a c =GMm/ 2 ˆ M v m F g Majo axis Ellipse a: semimajo axis b: semimino axis c: semi-focallength a 2 =c 2 +b 2 eccenticity e=c/a Ø Final solution also depending on v t=0, t=0 depending on initial condition, the object (m) can be eithe: Obiting in elliptical path (cicula been an example) Escaping in hypebolic path Focuses Mino axis Obiting Escaping ellipse equation x 2 /a 2 +y 2 /b 2 =1 Conceptual undestanding only. Math details fo Ellipse not equied fo this couse Ellipse And Cicle q An Ellipse Becomes a Cicle When a=b (= ) Planet Obits Keple s Laws b a q In Geneal, Obits fo Planets ae elliptical, but many ae close to be cicula. Johannes Keple (1571-1630) (1643-1727) q Half centuy befoe Newton, Keple summaized all available empiical obsevation data on planet obits into thee laws: 1. All planets move in elliptical obits with the sun at one focal point 2. The adius vecto fom sun to a planet sweeps out equal aeas in equal time intevals. 3. Squae of the obital peiod of any planet is popotional to the cube of semimajo axis a. (a= in cicula case) T 2 =(4π 2 /GM S )a 3 =K s a 3 q Keple s laws led to and can be deived diectly fom Newton s Law of Gavity and Newton s 2 nd Law of Motion. q Solute to Keple and Newton! 2
Keple s Fist Law v All planets move in elliptical obits with the sun at one focus Obits of Eath s Planets v All planets move in elliptical obits with the sun at one focal point an ellipse planet s obit Ø Fun quiz: What is on the othe focus? (answe: nothing!) Ø Fun quiz: What is on the othe focal point? (answe: nothing!) Planet Obit Paametes Fig. 13.5, p. 380 3
Keple s Second Law v The adius vecto fom sun to a planet sweeps out equal aeas in equal time intevals. Execise on Keple s Second Law q Pe Keple s 2 nd law Aea 1 = Aea 2 How to compae v 1 to v 2? Aea 1 Aea 2 Aea 1=Aea 2 2 1 v Aea 1 Aea 2 v 1 2 v Keple s 2 nd Law is a diect consequence of angula momentum consevation. (see me fo moe details) Answe: v 1 : v 2 ~ 2 : 1 (see boad) Keple s 2 nd Law And Angula Momentum Consevation q Keple s 2 nd Law can be deived fom angula momentum consevation: Ø F G // τ = 0 dl/dt =0 i.e L=constant (conseved.) Ø See figue: The swept aea da = dsinφ = x d ecall L= x mv = m x d/dt è L = m da/dt è L= constant da/dt =constant. i.e. equal time equal aea swept. (note: the deivation is not equied fo the couse) φ Keple s Thid Law v Squae of the obital peiod (T) of any planet is popotional to the cube of semimajo axis a. (a= in cicula case) T 2 =(4π 2 /GM S )a 3 =K s a 3 (Keple s constant K s = 4π 2 /GM S ) q Keple s 3 d Law is a diect esult of invese squae law (F g ~ 1/ 2 ) ² A bief deivation fo those inteested (not equied fo the couse): F Centipetal = F g = GM sm p, a 2 Centipetal = v 2 = ω 2 = ( 2π T )2 Newton's 2nd Law : F Centipetal = M p a Centipetal GM sm p 2 = M p ( 2π T )2 T 2 = 4π 2 GM s 3 4
Moon and Eath Execise: Satellite s Obiting Speed q Moon obits aound the Eath: ME = 3.85x10 8 m v = (GM E / ME ) ½ T = 2π ME /v = 2π ME 3/2 / (GM E ) ½ = 2.36x10 6 s = 27.3 days Chinese Luna Calenda: 1 month=28 days ME q Newton s 2 nd Law with gavitational foce: mv 2 / = GM E m/ 2 à v = (GM E /) ½ = (GM E /(R E +h)) ½ (=7.9 km/s if h<<r E ) à peiod T = 2π/v = 2π 3/2 / (GM E ) ½ Ø T = 5.4x10 3 s ~ 90 min. fo ~ R E = 6.37 x10 6 m Ø Quiz: at which height h T= 1day? Answe: = 42164 km, h=-r E = 35786 km q Moon causes ocean tides: Eath also pulled by moon, but not as much F G Moon Geostationay obit fo communication satellite wate left behind Eath wate pulled fowad by moon Will be explained in details in next lectue. 5