Newton s Shell Theorem


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1 Newton Shell Theoem Abtact One of the pincipal eaon Iaac Newton wa motivated to invent the Calculu wa to how that in applying hi Law of Univeal Gavitation to pheicallyymmetic maive bodie (like planet, ta, and the like), one can egad thee bodie a ma point with all of thei ma concentated at a point. The key ingedient in howing thi i to how that fo a thin ma hell, the gavitational foce at a point outide thi hell i the ame a if all the ma of thi hell i concentated at it cente. Newton Law of Univeal Gavitation ay that if we have two point mae m and M epaated by a ditance, then the mutual foce exeted on each i given by F = G mm 2, whee the univeal contant i G ha appoximate value 1 G N m 2 /kg 2. Sometime, it moe convenient to meaue intead the gavitational field E eulting fom a point with ma M; meaued in unit of Newton pe kilogam it meaue the foce on a point ma (of 1 kg) placed in thi field. Theefoe, E will be diected adially inwad towad the initial point ma and have field tength E = E = GM 2, at a ditance (mete) away fom the point with ma M. Fo an extended maive object with ma M, not concentated at a point, the detemination of the eulting gavitational field at a given point equie that the contibution of each component paticle of ma dm be integated into a final anwe. Newton Shell Theoem tate eentially two thing, and ha a vey impotant conequence. Fit of all, it ay that the gavitational field outide a pheical hell having total ma M i the ame a if the entie ma M i concentated at it cente (cente of ma). Secondly, it ay that fo the ame phee the gavitational field inide the pheical hell i identically. Poving Newton Shell Theoem i the pimay objective of thi eay. 1 The value of G wa fit meaued in 1798 by Heny Cavendih; thi wa aleady 71 yea afte Newton death.
2 A a conequence of Newton hell method, one can conclude immediately that fo a pheical homogeneou olid having ma M, the eulting gavitational field i again the ame a if the entie ma wee concentated at a point. A omewhat moe eoteic conequence i that if the pheical homogeneou object ha adiu R, then the gavitational field inide the object a a ditance < R fom the cente i the ame a if total ma within a ditance fom the cente wee concentated at the object cente. (The ma outide the adiu can be ignoed.) In ode to pove the fit pat of Newton Shell Theoem we conide a pheical hell of total ma M and adiu R; we hall compute the magnitude of the gavitational field at a point whoe ditance i fom the cente of the pheical hell. We decompoe the hell into thin cicula ing, each at a (vaiable) ditance fom the point at Point outide the hell: E (field at point) Rdφ φ R Thin ma hell of denity σ which E i to be computed. Since the Figue 1: Point outide the hell ma denity of the hell i σ = M we ee that the total ma of the ing (ee 4πR2 Figue 1) i total ma of ing = σ aea of ing = σ 2πR in φ Rdφ = 1 M in φ dφ 2 Next, note that all of the ma i at a ditance fom the point in quetion; howeve, ince (by ymmety) the field diection i towad the cente of the pheical hell, the field tength contibution fom thi thin ing mut be de = GM co in φ dφ 2 2 = GM co d(co φ) 2 2. (1)
3 Uing the Law of Coine, we have Theefoe, R 2 = co, and 2 = R R co φ. co = R 2 2, and co φ = R , 2R and o d(co φ) = R d. Plugging into Equation (1) yield the field contibution fom the thin ing: de = GM(2 + 2 R 2 ) 4R 2 2 d. (2) Fom Equation (2) we conclude that the total gavitational field induced by the pheical hell i the integal of the contibution of all of the ing: E = =+R = R = GM 4R 2 =+R de = GM 4R ( 2 + R2 2 ) +R = R poving the fit pat of Newton Shell Theoem. To pove the econd pat, namely that the gavitational field inide the pheical ma hell i, note that the element of field tength de contibuted by a typical thin ing doe not change; ee Figue 2. The only change i that the limit of integation fo ae = R and = R +. Theefoe, E = =R+ =R = GM 4R 2 de = GM 4R ( 2 + R2 2 ) R+ =R+ =R R =, R 2 d 2 = GM GM 4R = R 4R2, 2 Point inide the hell: Rdφ E R 2 Thin ma hell of denity σ d 2 φ R Figue 2: Point inide the hell
4 exactly a pedicted. Finally, we hall detemine the gavitational field induced by a olid homogeneou pheical ma (total ma = M), both at point inide and outide the mae. If R, i.e., the point at which we ae to detemine the gavitational field i outide (o on the uface of) the pheical ma. Denote the ma denity by µ = 3M 4πR 3, and, a above, let be the ditance of the point at which the field i to be computed to the cente of the pheical ma. Next divide the phee into concentic thin ma hell, each of thickne dρ and adiu ρ, making the ma of each uch hell dm = 4πρ 2 µdρ = 3Mρ2 dρ. Fom the fit pat of Newton Shell Theoem, we have that the field tength contibution fom thi hell i R 3 de = 3GMρ2 dρ; 2 R 3 the total field tength i obtained a an integal: R R 3GMρ 2 E = de = dρ = GMρ3 R 2 R 3 2 R = GM 3, 2 in pefect ageement with ou oiginal contention. Finally, if the point at which we ae to compute the field tength i inide the homogeneou pheical ma ( < R), then by the econd pat of Newton Shell Theoem, we ee that the field contibution by the concentic ma hell of adiu ρ i given by de = 3GMρ2 dρ if ρ, 2 R 3 ρ R. Theefoe, the total field contibution i the integal E = de = 3GMρ 2 2 R 3 dρ = GM 3 R 3 = GM3 = G 2 R 3 2 total ma of phee of adiu,
5 and we e done! Ty thee: 1. Let P be a point and let l be an infinite line with ma denity µ kg/m (and o l ha infinite ma). Auming that P ha ditance fom thi line, compute the gavitational field tength at the point P Let P be a point and let Π be an infinite plane with ma denity µ kg/m 2. Compute the gavitational field tength at the point P. (Hint: note fit that by ymmety, the field vecto will point fom P towad to point on the plane Π cloet to P. Next, by Poblem #1 above, the contibution to the field in thi diection i 2Gµ co dx. 2 Now do an integation with epect to x.) 3 E Denity of plane = µ kg/m 2 x Denity of tip = µdx kg/m 2 dx Contibution to field fom tip = de= 2Gµdx 2 +x 2 N/kg 2 Show that E = Gµ dx [ 2 + x 2 = 2Gµ ] 3/2 In evaluating the integal, ue the tig ubtitution x = co. 3 You hould get E = 2Gπµ N/kg. dx 2Gµ [ 2 + x 2 = ] 3/2 N/kg.
6 3. Hee anothe appoach to Poblem #2, above. Thi time, tat with a a cicula dik of adiu R and ma M at a ditance fom the given point P. (a) Show that the field tength at the given point i E ρ Ma of dik = M kg Ma of thin ing = 2Mρdρ/R 2 kg dx dρ Contibution to field fom thin ing = de= 2GMρcodρ R 2 2 N/kg E = 2GM R 2 [ 1 ]. 2 + R 2 (b) Show how letting R ecove the eult of Poblem #2.
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