Chapter 13 Gravitation

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1 Chapte 13 Gavitation Newton, who extended the concept of inetia to all bodies, ealized that the moon is acceleating and is theefoe subject to a centipetal foce. He guessed that the foce that keeps the moon in its obit has the same oigin as the foce that causes the apple to fall. He ecalled some thity yeas late: I deduced that the foces that keep the planets in thei obs must be ecipocally as the squaes of thei distances fom the centes about which they evolve, and theeby compaed the foce equied to keep the Moon in he ob with the foce of the gavity at the suface of the Eath, and found them to answe petty nealy. 1 Newton s Discoveies Newton used the peiod of the moon s obit (7.3 days) to calculate its centipetal acceleation: a m 1/360 m/s. How? Next, he assumed that the gavitational foce between two bodies vaies as the invese squae of the distance between them, that is, F 1/ --- an idea that had been aound since Thus, the ation of the foce on the moon to that on the apple at the suface of the eath should be R E / M. He knew that M 60R E. How do we detemine R E?

2 13.1 Newton s Law of Gavitation Newton s law of univesal gavitation: Gm m F 1 ˆ 1 1 It eadily apply to point masses, but how to apply this equation to bodies with abitay mass distibutions. 3 Pinciple of Supeposition Expeiment shows that when seveal paticles inteact, the foce between a given pai is independent of the othe paticles pesent. Pinciple of supeposition: F + 1 F1 + F1 3 + L F1N The net foce on m1 is the vecto sum of the paiwise inteactions. 4

3 13. Gavitational and Inetia Mass Mass has appeaed in two entiely diffeent contexts: Netwon s second law of motion and Newton s law of gavitation. F m a I F What is the diffeence between inetia mass m I and gavitational mass m G? m G GM G On the fee-fall case, foce is expessed as Fm G g. Substituting to foce equation, we get a(m G /m I )g. Is the mass atio vaying with diffeent mateials? What is the ole of the gavitational constant G? 5 Pinciple of Equivalence No expeiment can distinguish the effects of a gavitational foce fom that of an inetia foce in an acceleated fame. The latest expeiments show that m I and m G ae equivalent to within 1 pat in Example: the peiod of a simple pendulum, T π m L I / m G g 6

4 13.3 The Gavitational Field Stength How does two paticles inteact diectly with each othe though fee space? What is the action at a distance? The same poblem of inteaction without actual contact occus with electic chages and magnets. In 1830s Faaday developed the concept of a field, esolving the poblem of action at a distance. The distibution of values ove a egion of space is call a field. Pessue and tempeatue fom scala fields, wheeas velocity and foce give ise to vecto fields. 7 The Gavitational Field Stength Suppose a stationay paticle of mass M. What is the foce exeting on a test paticle of mass m whee m can be placed at diffeent position? GM F ( ˆ ) m gm It is convenient to conside the foce pe unit mass F/m. The quantity g, measued in N/kg, is called the gavitational field stength at position with espect to M. 8

5 Acceleation due to gavity Although the unit N/kg educe to m/s, the gavitational field stength g is a concept diffeent fom the acceleation due to gavity g a. The gavitational foce is diected to the cente and seves two functions: It causes the body to fall with acceleation g and it poduces the centipetal acceleation a c. mg m( g ) a + ac 9 Gavitational Foce Vaiation: Rotation and Non-unifom Mass Distibution At the equato acv / E 3.4 cm/s. Thus, the gavitational field stength at poles and equato is diffeent by 3.4 cm/s. Measuement, howeve, show that the diffeence is 5. cm/s. Why disagee? Non-unifom mass distibution. If the Eath otates vey fast, all the gavitational field stength g is contibuted to the centipetal foce ac and gavitational acceleation g a is zeo (weightless condition). How many seconds a day in such condition? ω E 9.8 m/s T ( / 9.8) π 5000 sec 10

6 13.4 Keple s Laws of Planetay Motion Keple discoveed thee laws of planetay motion that futhe stengthened the idea that the eath obits the sun athe than vise vesa. He was found though a laboious analysis of data left by his teache, Tycho Bahe. Law 1. The planets move aound the sun in elliptical obits with the sun at one focus. 11 Second Law: Consevation of Angula Momentum Law. The line joining the sun to a planet sweeps out equal aeas in equal times. 1

7 Example 1.8**: Accoding to Keple s second law of planetay motion, the line joining the sun to a planet sweeps out equal aea in equal time intevals. Show that this is a consequence of the consevation of angula momentum. Solution: A 1 1 h vsinθ t Angula momentum l psinθ m(vsinθ ) A t l m const. 13 Thid Law: Gavitational Foce Povides Centipetal Motion Law 3. The squae of peiod of a planet is popotional to the cube of its mean distance fom the sun. v T GM, 4π ( ) GM v 3 π T How to deive the thid law fo elliptical obit? Exta bonus! 14

8 Enegy in an Elliptical Obit Angula momentum consevation v P P A v A Mechanical enegy consevation E GmM / 1 1 mva GmM / A mv P P We substitute We get P + A a GMm E a # Ty to deive this esult by youself Continuous Distibution of Mass Each infinitesimal mass element contibutes to the gavitational field stength is: dg Gdm 16

9 Example 13.3: Find the field stength at the cente of a thin semicicula ing of adius R and mass M, as shown in Fig The linea mass density is λ kg/m. Solution: Gdmsinθ dg y dg sinθ R The total field stength is g y π G sinθ dm 0 R Gλ R 17 Example 13.5: How does the field stength vay inside a unifom solid sphee of density ρ kg/m 3 and adius R? Solution: Gauss law inside M ( ) g( ) GM ( ) 4πρ 3 / 3 18

10 Execises and Poblems Ch.13: Ex.11, 19, 7 Pob. 4, 6, 7, 9, 17 19

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it. Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing