Force of gravity and its potential function

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1 F. W. Phs0 E:\Ecel files\ch gavitational foce and potential.doc page of 6 0/0/005 8:9 PM Last pinted 0/0/005 8:9:00 PM Foce of gavit and its potential function (.) Let us calculate the potential function U of the gavitational foce The gavitational foce of M on m is given b: u F mmg,, z + + z The coodinate sstem is centeed in the cente of mass of M We geneall omit the subscipts and when we ae dealing with a sola sstem, in which the motion of the planets is caused b the field of the sun. As the sun is bael acceleated b the planets this makes sense. We know that this function has a potential function because cul F 0 B (.) F 0 W F d gadu d du Δ U U A U B path fom path fom A A to B A to B This is a path-independent integal, theefoe we ae allowed to integate along an convenient path we choose. We choose as path the diect connection between location A ( ) and B( ) along the adius vecto B F d u + dθ u d u d θ d (.) Fd u ( ) d W -ΔU path path Theefoe, ΔU -W - (/ -/ ) We ae fee to chose ou oiginal point to establish ou potential function U() itself. We choose at infinit, such that / 0. The potential function fo the gavitational foce is theefoe:

2 F. W. Phs0 E:\Ecel files\ch gavitational foce and potential.doc page of 6 0/0/005 8:9 PM Last pinted 0/0/005 8:9:00 PM U () Uz (,,) ( + + z ) (.4) / Let us take as an eample that a satellite of mass 5,000 kg is moved fom location A which is an obit 500 km above the suface of the eath to an obit which is 5,000 km above the suface of the eath. An point on this cicula obit can be egaded as location B. G M m.994e E+6 m;.7e7 m. The wok done b gavit duing this movement is equal to ΔUU(A)-U(B) W Δ U U ( ) U ( ) GMm GMm (.5) GMm E8*( ).67 0 J Let us see what we get if we simpl use mg fo U. 7 7 W Δ U mg( ) 5000* (.6) W.04 0 J The eason wh we don t get the same esult is of couse the fact that the potential eneg function deceases with / as we go above the suface of the eath. Potential eneg function fo gavit: The gaph of this function looks as epected: ( SEE) U U() fo gavitation J As inceases, the potential eneg inceases fom moe negative values to less negative values, to each 0 in infinit, just as we set it up. This means that as an object moves to the ight its potential eneg inceases; du/d >0 positive. ΔU is positive, and W is negative. How is this consistent with the definition Seies of potential eneg Umg on the suface of the eath. Well, if we incease in U() on the suface of the eath the adius b a small amount d we should be able to find out b calculating the total change in U b using the total deivative: Let us calculate the total deivative of U:

3 F. W. Phs0 E:\Ecel files\ch gavitational foce and potential.doc page of 6 0/0/005 8:9 PM Last pinted 0/0/005 8:9:00 PM (.7) U () + + z U MG du d d m d m g d Δ U m g d m g m g 9.8 s Δ U mg( ) mg b setting 0 U Also, - F, which is the attactive negative foce. (the patial deivative is equal to the egula deivative, as thee is onl one vaiable) This makes sense, because the negative deivative (gadient) of U must give us the foce, which on the suface of the eath mg. If we set d - 0 we have the potential eneg of gavitation on the suface of the eath. To ecapitulate: the foce function of the gavitational foce is: (.8) u Fg ( i + j + zk) ; the component of the vecto field F is + + z F ( + + z ) Let us calculate the patial deivative of this function with espect to : It is, using poduct ule and chain ule, which hold fo patial deivatives just as fo egula deivatives: (.9) F / + + z / + + z 5 / Similal fo F, F and so on. Fo eample: (.0) F 5 z z Double check that F g cul F g 0

4 F. W. Phs0 E:\Ecel files\ch gavitational foce and potential.doc page 4 of 6 0/0/005 8:9 PM Last pinted 0/0/005 8:9:00 PM We know that the elationship fo eve potential eneg function U and the associated consevative foce F is U U U (.) F gadu,, ( U, U, zu) U z The potential eneg U of,, and z fo the gavitational foce is: (.) U ; with the efeence at ; U ( ) z One can check that accoding to (.) F - U The component of the foce of gavit F is given b: (.) F U() + + z ( z ) ( z ) which is indeed the coect component of F g If thee ae moe than one attacting masses to conside, the total potential eneg is simpl the sum of all the potential enegies of the masses m i in the gavitational potential fields: m m (.4) U U+ U M Fo a lage numbe of masses we find that thei total potential eneg is equal to the sum of thei potential enegies, which is the amount of wok equied to assemble them. mmg i j U Ui ; i<j (this means that we do not count i, j ij (.5) com binations twice). Fo five masses we get the following combinations: and, and, and 4, 5,, 4, 5, 4, 5, 45.

5 F. W. Phs0 E:\Ecel files\ch gavitational foce and potential.doc page 5 of 6 0/0/005 8:9 PM Last pinted 0/0/005 8:9:00 PM Concepts to be clea on: A) Minimal velocit fo a satellite in obit aound the eath: (.6) The satellite stas in a cicula obit if the gavitational attaction is equal to the foce of gavit: mv MG v g B) Escape velocit: (.7) v esc m MG 0 0 v g + kinetic eneg + potential eneg on the suface of the eath. C) Wok equied to put a satellite in obit with adius is equal to the change in kinetic eneg plus the change in potential eneg: mv W Δ K +Δ U Ein obit Eon suface of eath ( + + Ke) (.8) is the adius of the eath; Ke is the kinetic eneg of the satellite due to the otation of the eath. D) What is the foce on a mass m at half the adius of the eath (inside the eath): The onl foce acting is the foce due to the mass contained inside the shell with adius R: (.9) ' mm G F ; M ρv ρ ; ρ densit of the eath with adius, assumed unifom. R M M ' ρ R ; ρ mρ R G M R R F mgρ R mg R mg 0.5mg R Black hole situation: no photons can escape fom the suface, M is ulta-elativistic. (.0) mc 0+ 0 kinetic eneg + potential eneg on the suface of a black hole.

6 F. W. Phs0 E:\Ecel files\ch gavitational foce and potential.doc page 6 of 6 0/0/005 8:9 PM Last pinted 0/0/005 8:9:00 PM (.) E mc p c + m c pc; p mc m is not m!!!!! m 0 S MG c pg ; this adius S is also called the Schwazschild adius. c 0 The Schwazschild adius fo the sun is appoimatel 500 m. The nomal adius of the sun is 7E8 m. Fo the eath we get 4.4 mm. This means that fo the eath to become a black hole its adius has to shink to 4.4 mm. Note that these equations sa that distance is popotional to mass, eneg and momentum. This is etemel stange. Space, time, and matte cannot be sepaated an moe. Einstein s geneal theo of elativit coves this domain. (.) mg ) ΔΔ p ; c At these dimensions and Δ ae the same, also p Δp mg p pg c ; m ; p into ) c c c G c G G f m G c c

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

Gravitation. Chapter 12. PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman. Lectures by James Pazun 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 Goals fo Chapte 12 To study Newton s Law