Newtons Law of Universal GravitationIsaac Newton1666, England (45 years after Keler!" #$estionsIf t%e lanets are or&iting t%e s$n, w%at force is 'eeing t%e( in or&it)*%at force 'ees t%e (oon in its or&it)+o$ld t%e force of gravity &e $niversal)Newtons Law of Universal Gravitation,ny two o&-ects attract eac% ot%er wit% a gravitational force, roortional to t%e rod$ct of t%eir (asses and inversely roortional to t%e s.$are of t%e distance &etween t%e("/%e force acts in t%e direction of t%e line connecting t%e centers of t%e (asses"Newtons Law of Universal Gravitation%tt011science&logs"co(1startswit%a&ang1$load123341351(eet6o$r6second6(oon143378NewtonsLaw9fUniversalGravitation"svg"ngEffect of Mass on Gravitational Forcem m m 2mddF2F 2m 2md4F 3mm3FdEffect of Mass on Gravitational ForceEffect of Distance on Gravitational Forcem mdF1/4F 2d mmEffect of Distance on Gravitational Force d4Fmm2.7F 2m 3m1 dInverse Sqare !a"FG = G m1 m2r 2The law of gravitation is called an inverse square law because the magnitude of the force is inversely proportional to the square of the separation.If the masses are moved twice as far apart, the force of gravity between is cut by a factor of four. What if each mass and the separation were all quadrupled?answer:no change in the force #niversal Gravitational $onstant

G re%resents t&e universal gravitational constantG ' (.(7 ) 1*+11 ,-m2/./20enr1 $avendis&2se7eri(ent deter(ined t%e roortionality constant G in 154:"%tt011www"newscientist"co(1data1i(ages1arc%ive116;4116;43131"-gSinceGis only 6.67 10-11 Nm2 / kg2, the measurements ha to !e very "recise.3ro4lem 1/wo s%eres of (ass ;5'g are 63( aart",! *%at force does one e7ert on t%e ot%er)ieldsNear (arth)s sur%ace the gravitational %iel is a""ro*imately uni%orm.&ar %rom the sur%ace it looks more like a sea urchin.+arthThe field lines2 are raial, rather than parallel,and point toward center of +arth.2 get farther apart farther from the surface, meaning thefield is weaer there.2 get closer together closer to the surface, meaning the field is stronger there.On a large scale, gravity is an attractive radial feldEarths surfaceOn a small scale gravity can be regarded as a uniform feldField lines give the direction of the feld. The feld is a vector.Gravitational Field Strength gMmr!rGMmF =!rGMg =From Newtons nd !awF " mgg " acceleration due to gravity #m$s

%g " gravitational &eld strength #N$'g%mFg =Gravitational 3otential Ener/1Gravitational 3otential Ener/1Gravitational %otential ener/1 of an o4=ect of mass m a distance r from t&e Eart&2s center>?er1 close to t&e Eart&2s srface: t&e /ravitational %otential increases linearl1 "it& altitde>Gravitational %otential ener/1: =st li.e all ot&er forms of ener/1: is a scalar. It t&erefore &as no com%onents@ =st a si/n.Gravitational 3otential Ener/15otal mec&anical ener/1 of an o4=ect of mass m a distance r from t&e center of t&e Eart&>7s an o4=ect a%%roac&es t&e Eart&: it moves faster and faster.Ener/1 $onservationEner/1 $onservation7not&er "a1 of visaliAin/ t&e /ravitational %otential "ell>Ener/1 $onservationEsca%e s%eed5&e initial %"ard s%eed a %ro=ectile mst &ave in order to esca%e from t&e Eart&2s /ravit1Ener/1 $onservationIf an o4=ect is sfficientl1 massive and sfficientl1 small: t&e esca%e s%eed "ill eqal or e)ceed t&e s%eed of li/&t.!i/&t itself "ill not 4e a4le to esca%e t&e srface.5&is is a 4lac. &ole.Ener/1 $onservationB&at are Clac. &olesDKeplers Laws of Planetary MotionEarl1 7stronomersIn the !nd century 34 the 3le/andrian astronomer 5tolemy put forth atheory that +arth is stationary and at the center of the universe and that the sun, moon, and planets revolve around it.Though incorrect, it was accepted for centuries.In the early 1#000s the 5olish astronomer %icolaus 6opernicus boldly re,ected 5tolemy0s geocentric model for a heliocentric one.7is theory put the sun stated that the planets revolve around the sun in circular orbits and that +arth rotates daily on its a/is. In the late 1#000s the 4anish astronomer Tycho 8rahe made better measurements of the planets and stars than anyone before him.The telescope had yet to be invented.7e believed in a 5tolemaic96operican hybrid model in which the planets revolve around the sun, which in turn revolves around the +arth.Earl1 7stronomersIn the late 1#000s and early 16000s the Italian scientist :alileo was one of the very few people to advocate the 6opernican view, for which the 6hurch eventually had him placed under house arrest.3fter hearing about the invention of a spyglass in 7olland, :alileo made a telescope and discovered four moons of ;upiter, craters on the moon,and the phases of ie :alileo, =epler believed in the heliocentricsystem of 6opernicus, but using 8rahe0s planetary data he deduced that the planets move in ellipses rather than circles.This is the first of three planetary laws that =epler formulated based on 8rahe0s data.+oth ,alileo an -e"ler contri!ute greatly to $ork o% the (nglish scientist Sir .saac Ne$ton a generation later.Earl1 7stronomersKeplers Laws of Planetary Motion1.5lanets move around the sun in elliptical paths with the sun at one focus of the ellipse.!.?hile orbiting, a planet sweep out equal areas in equal times.3.The square of a planet0s period 'revolution time* is proportional to the cube of its mean distance from the sun@ T 2R ;Summary of Keplers 3 Laws:These laws apply to any satellite orbiting a much larger body.Ee%ler2s First !a"Planets move around the sun in elliptical paths with the sun at one focus of the ellipse.3n ellipse has two foci, A1 and A!.Aor any point5on the ellipse, A1 5 B A! 5is a constant.The orbits of the planets are nearly circular 'A1and A! are close together*, but not perfect circles.3 circle is a an ellipse with both foci at the same point99the center.6omets have very eccentric 'highly elliptical* orbits. A1A!Cun5lanet5Ee%ler2s Second !a" While orbiting a planet sweep out e!ual areas in e!ual times.Cun 6438The blue shaded sector has the same area as the red shaded sector.Thus, a planet moves from6to4in the same amount of time as it moves from3to8.This means a planet must move faster when it0s closer to the sun.Aor planets this affect is small, but for comets it0s quite noticeable, since a comet0s orbit is has much greater eccentricity.Ee%ler2s 5&ird !a""he s!uare of a planets period is proportional to the cube of its mean distance from the sun:T2 R 33ssuming that a planet0s orbit is circular 'which is not e/actly correct but is a good appro/imation in most cases*, then the mean distance from the sun is a constant99the radius.Fis the force of gravity on the planet.Fis also the centripetal force.If the orbit is circular, the planet0s speed is constant, andv = 2 R / T. Therefore, ?$n=lanet FRMmG M mR 2

m v 2R =G MR 2

m @2 R / TA 2R =+ancelms and si(lify04 2 R T2 =Bearrange0G MT2 4 2= R3?ince G, M, andare constants, T2 R ;"G MT2 4 2= R 3/r!ital "erio e"ens on the mass o% the central !oy 0$hich %or a "lanet is its star1 !ut not on the mass o% the or!iting !oy&arther a$ay a "lanet is %rom its star, the longer it takes to com"lete an or!it.2here are t$o reasons %or this31. 2he %arther a$ay the satellite is, the %arther it must travel to com"lete an or!it42. 2he %arther out its or!it is, the slo$er it moves, as sho$n3G M mR 2

m v 2R =v = G M R Ee%ler2s 5&ird !a"The EndSam%le 3ro4lemsWhat i" the force of gravitational attraction bet$een t$o 10(kg !a""e" "e&arate b) a i"tance of 100 ! bet$een their center" of !a""*The force of gravitational attraction bet$een t$o !a""e"+ !1 an !2+ "e&arate b) a i"tance + i" 4, ne$ton".What i" the force of gravitational attraction be$t$een the"e t$o !a""e" if the i"tance bet$een the! i" ecrea"e b) 1-.. A "atellite $eight" 200 ne$ton" on the "urface of /arth. What i" it" $eight at a i"tance of one /arth raiu" above the "urface of /arth* A "&ace "huttle $eigh" 2.01 % 107 ne$ton" (4., !illion &oun"0) on /arth1" "urface.What i" it" $eight in ne$ton" $hen it i" orbiting at a i"tance of 6.4. % 10, ! above /arth1" "urface*What i" the force of gravitational attraction bet$een 2aturn an /arth* Saturn StatisticsMass=5.87 x 1026 kgRadius=6.00 x 107 mMean distance frm Earth !surface t surface"= 1.20x 1012 m