Chapter 5: Circular Motion : Earlier in the semester. Universal Law of Gravitation: Today. Newton s Universal Law of Gravitation

Transcription

1 Chapte 5: Cicula otion : Ealie in the semeste Univesal Law of Gavitation: Today 1 Newton s Univesal Law of Gavitation 1

2 Newton s Law of Univesal Gavitation Fo a pai of point masses Diection: towads each othe m 1 F 1 d law pai of foces Displacement fom m 1 to m agnitude: 1 F 1 m G m m F1 1 1 Diection: Foces point to each othe Attactive foce Gavitational Constant G is small: G=6.67x10-11 Nm /kg Gavitational foce: Too small to notice between most human-scale objects and smalle Lage if at least one of two objects is massive, like -human, -sun Gavitational foce mg you leaned so fa: A special case of univesal gavitation law Gavitational foce between and objects nea the suface of the Fgavity mg applies only between and objects nea the suface of G m m F1 elation: 1 1 g G applies to any objects Eath Eath

3 This image cannot cuently be displayed. This image cannot cuently be displayed. Gavitational foce by multiple point masses: The net foce on a point mass when thee ae many othes neaby is the vecto sum of the foces taken one pai at a time m 1 F F F F F on 1 i, 1, 1, 1 4, 1 i1 All gavitational effects ae between pais of masses. m 1 F 1 1 F F m m 4 iclicke Quiz m 4 m1 = m = m = m4 = m5 =1kg Distances fom m1 to m, m, m4, m5 ae all 1 m. m m 1 m 5 m What is magnitude of the net gavitational foce on m1 by m, m, m4, and m5? A)5 G x kg^/m^, B)4 G x kg^/m^ C) G x kg^/m^ D) G x kg^/m^ E)zeo

4 This image cannot cuently be displayed. iclicke Quiz m1 = m = m =1kg Distances fom m1 to m, and m ae all 1 m. m 1 m m What is magnitude of the net gavitational foce on m1 by m and m? Angle conneting m-m1- m is 90 degee. A) G x kg^/m^, B) G x kg^/m^ C)Sqt() G x kg^/m^ D) G x kg^/m^ E) zeo adius of the Eath s cicula obit is 1.5 x metes. Using G=6.67x10-11 Nm /kg calculate the mass of the Sun. 4

5 Gavitational foce between a point-like object and continuous mass distibution with spheical symmety Accoding to Shell Theoem : x m x m F m G Cente of sphee Gavitation nea the suface of the Eath: m h When m is on o nea the suface: What do g and weight = mg mean? Eath s mass acts as like a point mass at the cente (by the Shell Theoem) adius of Eath = Object with mass m is at altitude h above the suface, so = + h g mg m h G ( h ) h o, in othe wods h G whee =6,70 km oe geneally, fee fall acceleation depends on altitude g h G ( h ) _ = 6 x 10^4 kg 5

6 Fee fall acceleation 9. What is the magnitude of the fee-fall acceleation at a point that is a distance e above the suface of the Eath, whee e is the adius of the Eath? a) 4.8 m/s b) 1.1 m/s c). m/s d).5 m/s e) 6.5 m/s a g G m ( h) e e at any altitude g 9.8 m/s Keple s Laws and Planetay otion Keple s laws of planetay motion Obit Law Aea Law Peiod Law Satellite and planetay obits 6

7 Keple s laws of planetay motion - summay 1. Obit Law: Planets all follow elliptical obits with the Sun at one focus Bound states, E mech < 0 Cicle is special case Semi-majo axis = a a S. Aea Law: Angula momentum consevation The line fom the sun to a planet sweeps out equal aeas in equal time intevals Aeal velocity is constant da dt A t aea swept time t A S 1 A t. Peiod Law: The squae of the obital peiod of any planet is popotional to the cube of the semi- majo axis of the elliptical obit 4 T G. Aea Law: Equivalent to angula momentum consevation L is constant since gavity cannot exet a toque aound the cental mass da aea swept out duing time dt 1 1 base height v dt 1 1 mv dt L dt m m So, p mv ds v dt slow aphelion v m x v Tiangle aea da fast peihelion " aeal velocity" da dt ate at which adius sweeps out aea L m constant at all points of the obit Compae speeds at aphelion (fa) and peihelion (close): Lpeihelion mp vp, mp vp v a p 1, so v aphelion v L m v m v v aphelion a a, a a p a peihelion 7

8 Apogee and peigee An Eath satellite is placed in an obit whose apogee distance is 9 times its peigee distance. Find the speed at peigee ove the speed at apogee.: A) 10 B) 9 C) 1/9 D) 1/10 E) slow apogee x apo pei fast peigee L a mv a a L p mv p p. Peiod Law: Simplified poof fo cicula obits Equate gavitational and centipetal foces m mv Fg G F cent ecall: v whee T peiod T mass m 4 cancels: G T F g v m Keple Thid Law: 4 T G. is mass of the cental body. Holds also fo eccentic obits with = semi-majo axis T Also: 4 G, constant fo all satellites aound the same object with mass. Fo the Sola System: sun = x 10 0 kg K T S yeas mete G 11 0 a sun / 8

9 Same value fo each planet in the sola system! Semi-majo Axis Peiod T /a =K sun Planet a (10 10 m) T (y) (10-4 y /m ) ecuy Venus Eath as Jupite Satun Uanus Neptune Pluto G sun The value of K depends on the object being obited Fo example, fo the oon aound the Eath, K Sun is eplaced with K Eath Example: Find Jupite s peiod in Eath-yeas Given: Jupite s mean distance fom the Sun is 5.19 A. U. Define: 1 AU = 1 Astonomical Unit = Aveage adius of the Eath s obit = 9,000,000 miles = 1.5 x metes 9

10 Obital adius fo Halley s Comet 10.. Since about 40 BC, Halley s comet has eappeaed in the sky all ove the Eath evey 76 yeas on the aveage. The latest appeaance was in The obit is quite eccentic, but it still obeys Keple s d Law. What is the appoximate, semi-majo axis of the obit in A. U.? A) 66 A. U. B).00 A. U. C).056 A. U. D) 4. A. U. E) 17.9 A. U. T a 4 G sun fo any object obiting the Sun 10