Gravitation. Chapter 12. PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman. Lectures by James Pazun

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1 Chapte 12 Gavitation PowePoint Lectues fo Univesity Physics, Twelfth Edition Hugh D. Young and Roge A. Feedman Lectues by James Pazun Modified by P. Lam 5_31_2012

2 Goals fo Chapte 12 To study Newton s Law of Gavitation Use Newton s Law of Gavitation and Newton s Second Law of Motion (F=ma) to calculate obital speed of satellites and planets Use enegy method to detemine escape velocity Use Newton s Law of Gavitation and Newton s Second Law of Motion to explain Keple s Laws of planetay motion

3 Newton s Law of Gavitation The gavitational foce is always attactive and its magnitude depends on both the masses of the bodies involved and thei sepaations.! F g = Gm 1 m 2 2 G = x10 "11 N m 2 /kg 2

4 Supeposition pinciple The total foce on m1= foce on m1 by m2 + foce on m1 by m3 Find F! 1, F! 2 and the net foce F! m3 Expess the foces in ˆ i, ˆ j. m1 m2

5 Compae little g with big G Conside the gavitational foce on a mass (m) by the Eath: F = GmM E 2 = ma! a = GM E 2 Nea Eath's suface! ~R E! a = GM # E " 9.8 m & 2 R E $ % s 2 ' ( = g (fee ) fall acelleation) The magnitude of "g" deceases as the object gets futhe away fom the Eath.

6 Satellite motion Obital motion = pojectile motion with vey lage initial velocity How lage an initial velocity is equied? Answe on next slide

7 Calculate satellite obits - assume cicula obit m M E This example conside satellite aound the Eath (but the same method applies to motion of satellite aound any lage mass even fo a planet aound the Sun) (1) Daw fee diagam on the satellite (2) Set up Newton's 2nd Law F = ma equation (a = v 2 /) " F = m v 2 (3) Apply Newton's Law of Gavitaion : F = GmM E 2 (4) Combine " GmM E = m v 2 2 (5) Solve fo v in tems of " v = GM E Fo an obit nea the suface of the Eath ( # R E = 6.4x10 6 m), v # 8x10 3 m /s ~ 18,000mi /h

8 Calculate planetay obits - assume cicula obit m p M S Gm p M S v 2 = m 2 p v 2 = GM S v = 2" ; T = obital peiod T $ # 2" ' & ) % T ( # 3 2 = GM S T = GM S = same numbe fo all 2 2 4" planets in the sola system. (This called Keple's 3d Law)

Gavitational potential enegy The expession fo the gavitaitonal potential enegy fo a mass at height h fom the suface of the Eath : U = mgh is only valid fo an object nea the suface of the Eath.

9 Gavitational potential enegy The expession fo the gavitaitonal potential enegy fo a mass at height h fom the suface of the Eath : U = mgh is only valid fo an object nea the suface of the Eath. The moe geneal expession is: U = " GmM E = potential enegy fo the mass - Eath system sepaated by a distance of. Nea the suface of the Eath = R E + h # U(R E + h) "U(R E ) = " GmM & # E % ( " " GmM & E % ( ) m GM E 2 $ R E + h ' $ ' R E R E h = mgh

10 Calculate escape velocity What is the minimum initial velocity of a ocket fo it to escape fom the gavitational effect of the Eath? K i + U i = K f + U f 1 2 mv 2 i M EV 2 i " GmM E R E = M EV 2 f " GmM E # V f $ V i % 1 2 mv 2 i " GmM E = 0 R E % v i = 2GM E R E & 25,300mi /h Note : If a sta is vey dense such that it has a lage mass but a small adius, then v escape > speed of light!! Such a sta is a "black hole". Howeve, coect teatment of black hole equies Einstein's theoy of Gavitation (Geneal Relativity). Newton's theoy of gaviation is only appoximate.

11 Newton s Law of gavitation How did Newton come up with the Law of Gavitation?! F = Gm 1m 2 2 Newton elied on the astonomical data of Keple.

Keple s laws fo planetay motion 1st Law: Each planet moves in an elliptical obit with the sun at one of the foci 2nd Law: A line connecting the sun to a given planet sweeps out equal aeas

12 Keple s laws fo planetay motion 1st Law: Each planet moves in an elliptical obit with the sun at one of the foci 2nd Law: A line connecting the sun to a given planet sweeps out equal aeas in equal times. 3d Law: The peiods of the planets ae popotional to the 3 / 2 powes of the majo axis lengths in thei obits. (Planets futhe away fom the Sun takes longe to go aound the Sun)

13 Keple s 1st Law - elliptical obits Elliptical obits ae consequence of the 1/ 2 dependence of the gavitational foce. The poof equies advanced calculus. Cicula obit is a special case of elliptical obit.

14 Keple s 2d Law 2nd Law: A line connecting the sun to a given planet sweeps out equal aeas in equal times => a planet moves faste when it is close to the sun.

15 Keple s 2d Law - Consevation of angula momentum Keple's 2nd Law : da dt = constant Relate da dt to angula momentum (! L ) of the planet about the Sun! da dt = 1! " v! = 1! " m! L v = 2 2m 2m da dt = constant # L! = constant # small means lage v.! L = constant when toque = 0. Newtons concluded that the gavitaional foce must be diected along the "adius" fom the Sun to the planet. Since F! and! ae along the same diection,! " F! = 0.

16 Keple s 3d Law and Newton s Law of Gavitation Deived in an ealie slide, epeat hee: Newton's 2nd Law: F=ma =m p v 2 Newton's Law of gavitation: F= Gm pm S v 2 = m 2 p! v 2 = GM S Assume cicula obit! v = 2!, T=obital peiod T "! 2! % $ ' # T & 2 = GM S (2! )3/2! T = ( 3/2 (Keple's 3d Law) GM S Newton did this backwad, he stated with Keple's 3d Law and deduce that gavitaional foce = GMm/ 2. Note: We can use Keple's 3d law to calculate the mass of the Sun based on the adius and the peiod of the Eath's obit aound the Sun.

17 Sun and planet evolve about the cente of mass. We have assumed the Sun was stationay while the planet obits aound the Sun. In fact, a bette way to view thei motion is that they both evolve about thei cente of mass.

Chapter 13: Gravitation

Chapter 13: Gravitation v m m F G Chapte 13: Gavitation The foce that makes an apple fall is the same foce that holds moon in obit. Newton s law of gavitation: Evey paticle attacts any othe paticle with a gavitation foce given