ABHS PHYSICS (H) Gavitational Field of a Sphee Newton gavitational theoy tate that you ae attacted to evey paticle that ake up the eath. He wa able to how that the gavitational attaction to a unifo phee (alot the eath) wa a if the whole a of the phee wa at the cente of a of the phee. We will do a veion of that poof hee. To do thi poof, we will fit deive an expeion fo the gavitational attaction between an object and a unifo ing, then to a unifo pheical hell, and finally to a unifo phee. Unifo ing Fit we will deive an expeion fo the gavitational attaction between an object of a and a unifo ing of a M and adiu. The object i located a ditance fo the cente of the ing on the cental axi of the ing, a hown in figue 1. d α df θ dθ d = ρdθ ing ing = adiu of ing M = a of ing = a of econd object = ditance between cente of ing and econd object d = all chunk of the ing = ditance fo a,, to all chunk of ing, d α = angle fo a,, to all chunk of ing, d, Fit, iagine dividing up the ing into a lot of little tiny chunk (d.) To calculate the foce of attaction between the whole ing and the a, we need to take the vecto u of the foce between the a,, and each little chunk of ing, d. To do thi, we look at the yety of the aangeent, and notice that fo evey point on the ing, thee i a point exactly oppoite it. The non-axial coponent of each chunk of foce cancel out. All we need to add up ae the axial coponent of the attaction. So, the axial coponent of the foce between the a and the chunk of the ing i: df = G d coα Now we need to et up the integation. We will integate aound the ing, fo θ = to 2π. (See figue 2.) We need an expeion fo the chunk of a a it depend on θ. Defining the linea denity of the ing a we can ay that figue 1 figue 2 ρ = M ing C ing = M 2π d = ρdθ Side 1
ABHS PHYSICS (H) The integal thu follow: Gavitational Field of a Sphee F = df = G d coα 2π = G (ρdθ) 2π coα = G ρ coα dθ 2 = G ρ2π coα = G M coα We will ue the above eult in the integation fo the pheical hell, but let continue with oe ubtitution to ake it ue le vaiable. Fo the oiginal diaga we have the following identitie = + 2 coα = = + 2 Subtituting thee into ou eult leave the following expeion: M F = G ( + 2 ) 3 / 2 Side 2
ABHS PHYSICS (H) Gavitational Field of a Sphee Unifo Spheical Shell To calculate the gavitational attaction between an object of a and a pheical hell of a M and adiu a ditance of fo the a, we will divide the hell into a bunch of ing, and iply add up the foce between each ing and the a. Figue 3 how a ing on the phee and figue 4 how a ide view with cleae label. (The lette epeent the ae thing fo the peviou deivation.) Figue 3 θ α figue 4 Fo the peviou ection, we know that the attaction to a ing will be df = G M ing coα We need to find an expeion fo the a of each ing a it depend on θ. We define the uface denity of the phee a So the a of a ing would be σ = M phee A phee = M 4π 2 Side 3
ABHS PHYSICS (H) Gavitational Field of a Sphee denity cicufeence width d = σ (2π inθ )(dθ) d = 2π 2 σ inθdθ Now we have the athe unightly diffeential to integate fo θ = to 2π df = G 2π2 σ inθdθ coα Unfotunately, we cannot do thi integal yet. The vaiable and α vay with θ. While it would appea natual to iply find expeion fo and α that depend on θ (ince that i ou diffeential), that will give an integal that I do not know how to do. (At which point I would iply look it up in the CC handbook.) We will deive expeion fo θ and α that depend on. Fo the law of coine we can wite an expeion fo coα: 2 = + 2 coα coα = 2 + 2 2 To get id of the 2 te involving θ, we will again ue the law of coine: = + 2 2coθ Diffeentiating thi give u: Which we ewite a: 2d = 2inθdθ inθdθ = d Finally, we ae able to et up the integal, ake the ubtitution, and olve. df = G 2πσ (inθdθ )(coα) = G 2πσ d + 2 2 = G πσ + 2 d In changing vaiable, the liit of the integal alo change. Intead of integating fo θ = to 2π, we ae integating fo = - to +. So the integal i Side 4
ABHS PHYSICS (H) Gavitational Field of a Sphee F = + G πσ + 2 d = G πσ + 1+ 2 2 2 d Which becoe: F = G πσ 2 + = G πσ ( + ) 2 ( ) 2 ( + ) ( ) = G πσ [ 4] = G 4π2 σ Subtituting in ou definition of σ, we finally aive at: F = G M The above eult i fo an object outide the phee. To calculate the attaction to a point inide the phee, we have figue 5. θ α The integal i et up the exact ae way, with only one diffeence: the liit of the integation ae fo = - to +. (To ee thi, look at what happen to in figue 5 when we ake θ change fo to 2π.) The integal thu becoe: F = G πσ 2 + = G πσ ( + ) 2 ( + ) figue 5 ( ) 2 ( ) = G πσ [ ] = The gavitational foce on an object inide a unifo pheical hell i zeo! Side 5
ABHS PHYSICS (H) Unifo Sphee Gavitational Field of a Sphee To calculate the gavitational attaction to a unifo phee, we iply teat the phee a bunch of hell, and add up the foce fo all the hell. In figue 6, x i the adiu of a hell. All othe lette epeent the ae thing they did ealie. d = hell x figue 6 Defining the denity of the phee to be We have the a d of a hell: ρ = M phee = M 4 V phee 3 π 3 denity aea thickne d = ρ( 4πx 2 )dx Setting up the integal, and integating fo x = to give u F = df = G d ( ) = G 4πρx 2 dx = G 4πρ = G 4πρ x 2 dx 1 [ 3 x 3 ] = G 4 3π 3 ρ Finally, ubtituting in ou expeion fo the denity of the phee, we get: F = G M A wa the cae fo the hell, the gavitational attaction to a unifo phee i a if the whole phee wee one paticle at the cente of the phee and of the ae a a the phee. Side 6