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InKeplerandNewtonwestudiedthebasicsofgravitation.Nowwewillexploresomeanalyticaltoolsthatsimplifythecalculationsinvolvedwithgravitatingsystems.Mostoftheseweredevelopedinthenineteenthcenturywhenphysicistssoughttosolveproblemsinvolvingmanyobjectsinteractingundercomplicatedconditions.Oneofthemostimportantconceptsthatcameoutoftheirmathematicalanalysiswasthenotionofenergyconservation.Theideaofenergyconservationnecessitatedtheconceptofpotentialenergy,which,unlikethekineticenergythatmanifestsitselfinmotion,wasconsideredtobethelatentabilityofthesystemtoproduceusefulwork.Asweshallsee,becausepotentialenergyisascalarandnotavector,itcangreatlysimplifythecalculationofthepotentialinherentinalargenumberofbodiesatanypoint--theproblemjustreducestosummingthecontributionsofalltheindividualbodies.Theforcecanthenbefoundbytakingthenegativeofthespatialderivativeintheusualway(seeNewton''''sSecondLaw)

WewillusetheconceptofthegravitationalpotentialenergytoproveNewton''''sShellTheorem,whichassertsthatasphericalmasscanbetreatedasifallitsmasswereconcentratedatitscenterforthepurposesofcalculatingthegravitationalforceonanobjectoutsideit,andthatamassive,thinshellexertsnogravitationalforceonamassinsideitself.Furthermore,wewillstatethePrincipleofEquivalence,whichstatesthatinertialmass,appearinginNewton''''sSecondLaw,isthesameasthegravitationalmassappearingtheUniversalLawofGravitation.

PrincipleofEquivalence-Assertsthatalltypesofmatterfallatthesamerate.Thatis,gforabrickisthesameasgforwater.ThismeansthattheinertialmassappearinginNewton''''sSecondLawisequivalenttothegravitationalmassappearingintheUniversalLawofGravitation.

Inertialmass-ThemassmithatappearsinNewton''''sSecondLawF=mia.

Gravitationalmass-ThemassthatappearsintheUniversalLawofGravitation.

ShellTheorem-Statesthatanysphericalmasscanbetreatedasthoughallitsmasswereconcentratedatitscenterforthepurposesofcalculatinggravitationalforce.Also,thatasphericalshellofmatterexertsnogravitationalforceonamassinsideit.