Background - United States Naval Academy

Ch. 13 - GravitationDan Finkenstadt, finkenst@usna.eduNovember 1, 2015BackgroundBy the time he was 23, Issac Newton hadinvented calculus, optics and mechanics, and thatwas when he began to formulate his law ofuniversal gravitation. His theory had to agreewith the empirical Kepler’s Laws1. Planetary orbits are stable ellipses (almostcircles) with Sun at one focus.2. Radial vector from Sun to planet will sweepout equal areas in equal time (this is conservation of L).3. For circular orbits,r3 T 2 .I Newton’s law of universal gravitation hadalso to explain gravity near the surface of13.1 – Newton’s Law of GravitationG m1m2Fg r2 IIIIr̂It is an (always) attractive force, henceminus signG 6.67 10 11 Nm2 /kg2It acts between two bodies of mass m1 andm2r is the distance between the bodies, r̂ is aunit vector along the line connecting thebodies

13.2 – Principle of SuperpositionIINewton’s Law of Gravitation obeys animportant linear relation in physics:Principle of SuperpositionActive Learning ExerciseProblem: GravitationThree 5.0 kg masses arelocated at the points in thex-y plane as shown in thefigure. What is the1. force vector, and2. force magnitudeacting on the mass at theorigin?If several bodies are attracting a givenobject, the net force on the object is the sumof the forces: net F 1 F 2 F 3 .FIThis strict linear separation of the forces(each force not depending on the strength ofthe others) facilitates calculation by breakingthe problem into smaller pieces, and thenyou add the total result.Useful Data: Earth & SunDo Newton’s Laws reproduce little g ?from F maIIIMEarth 5.98 1024 kg ,MSun 1.99 1030 kgREarth 6380 kmrEarth–Sun 1 AU 150 million km 6.67 10 11 Nm2/kg2 (5.98 1024 kg) m mg62(6.38 10 m)9.799 m/s2 gYES!!!!

Can we determine g on other planets?Active Learning ExerciseRecall: F ma.Fg ma mGM ma R2GM a gR2Problem: Gravitational Acceleration(not necessarily 9.8!)Mmoon 7.3477 1022 kgRmoon 1740 kmOn the moon, g 1.6 m/s2On all planets, an object’s gravitationalacceleration will be independent of itsmass! Depends on planet’s ratio of M toR2 .How much does Earth weigh?What is Gravitational Field Inside Earth?Principia, Book III, The System of the World,Proposition 10, Theorem 10 (Newton, 1687):If the earth were not denser than the seas, it wouldemerge from those seas and, according to the degree of itslightness, a part of the earth would stand out from thewater, while all those seas flowed to the opposite side. Bythe same argument the spots on the sun are lighter thanthe solar shining matter on top of which they float. Andin whatever way the planets were formed, at the timewhen the mass was fluid, all heavier matter made for thecentre, away from the water. Accordingly, since theordinary matter of our earth at its surface is about twiceas heavy as water, and a little lower down, in mines, isfound to be about three or four or even five times heavierthan water, it is likely that the total amount of matter inthe earth is about five to six times greater than itwould be if the whole earth consisted of water.Newton figured out that:I A uniform spherical shell of matter attractsa particle outside of the shell as if all of theshell’s mass were at its center.I That same shell of matter does not attractparticles inside of the shell at all!

What is Gravitational Field Inside Earth?How Much Work to Place Earth in Orbit?What about the field inside our planet?Imagine the Sun must pull a big rock fromInfinity to a point that is a distance R away:Z 4πr3G ρEmG(ME )m3Fg kE rr2r2(Hooke’s Law!)How Much Work to Place Earth in Orbit?The work is positive, and the Earthaccelerates(increase K.E.)Let’s get a number:GMS ME R6.67 10 11 Nm2 /kg2 (1.99 1030 kg)(5.98 1024 kg)150 000 000 000 m33 5.29 10 JRR How Much Work to Place Earth in Orbit?IIZGMS ME g · d r Wg r̂ · d rF2r Z R11 GMS MEdr GMM SE2r r 11GMS ME GMS ME R RIIRThe Earth clearly had great potential whensituated at Infinity (5.29 1033 J, in fact), atleast when compared to where it currentlyorbits.IIOne could DEFINE the potential at Infinityto be zeroIIThe potential of mass m orbiting body M isGM mUg r

How Much Velocity to Orbit Earth?How Much Velocity to Orbit Earth?IAs an example let’s try r 7000 kmIIv vcrit (Closed orbit)v vcrit(ProjectileMotion)Fg F c2GM mvcrit mr2r rGMvcrit rHow Much Velocity to ESCAPE Earth?REMEMBER: this is only some 600 km inaltitude!rGMvcrit r r6.67 10 11 Nm2 /kg2 (5.98 1024 kg) 7 000 000 m 7500 m/sActive Learning ExerciseProblem: SatellitesE0 Ef00 U K0 U0 K ff 1 2mMmv G 02Rr2GMvesc RFun fact: a black hole occurs when vesc cSuppose that the head of zombieBertrand Russel orbits the Earthwith an orbital radius half that ofthe Earth-Moon distance. If theperiod of the Moon orbiting theEarth is 28 days, what is theperiod of orbit for the zombiehead?

Einstein and MercuryIIIIAfter a thorough study of the orbits of theplanets we can see:Newton’s theory of gravitational attractionbreaks down near very massive objects.Very near the sun, the mass of the suneffectively “bends” space time meaning weneed general relativity to solve for thegravitational attraction.The same phenomenon occurs (and has beenobserved) near neutron stars, black holes,and supermassive stars.Template Summary SlideNameDefinitionNewton’s Universal Gravitationforce between massive objectspotential form:Eqns. / Examples g G M m r̂Fr2G MmUg rKepler’s Laws:FirstPlanetary orbits are ellipses with Sun at one foci.SecondRadial vector sweeps out equal areas in equal time.ThirdFor circular orbits, r3 T 2 .

Principia, Book III, The System of the World, Proposition 10, Theorem 10 (Newton, 1687): If the earth were not denser than the seas, it would emerge from those seas and, according to the degree of its lightness, a part of the earth would stand out from the water, while all those seas owed to the opposite side. By