Newton s Shell Theoem vi Achimees s Ht Box n Single-Vible Clculus Pete McGth Pete McGth (pjmcgt@upenn.eu, MRID955520) eceive his Ph.D. fom Bown Univesity n is cuently Hns Remche Instucto t the Univesity of Pennsylvni. His mthemticl n esech inteests e elte to geomety n nlysis. Outsie of mthemtics, McGth is n vi tumpet plye n enjoys pefoming music fom viety of styles, incluing Boque, Clssicl, big-bn jzz, n michi. Rente wnts to she chocolte-covee onge with you, with the cvet tht you potion be selecte in the following wy: She will mke two pllel cuts septe by fixe istnce (which you o not get to pick) n will give you the piece of the onge in the slb in between you get to pick whee she mkes the top cut. Wht height shoul you pick to mximize the sufce e of chocolte on you piece? Allow youself time to pone this befoe eing on, becuse the nswe my chllenge you intuition. If you hve some knowlege of clculus, in pticul if you e fmili with the fomul fo the e of sufce of evolution, then you might ty to wok the nswe out fo youself (n compe you wok with the clcultion below!). In n effot to put moe spce between hee n the esolution of the poblem (in cse you wnt to think bout it, but you eyes wne) n to put the poblem on moe igoous footing, we mke the following efinition. Definition. We cll the subset of sphee S in slb between two pllel plnes P 1 n P 2,echofwhichintesectsS, spheiclsegment. Figue 1. Which segment hs lge e, the one t the pole o the one ne the equto? Colo vesions of one o moe of the figues in the ticle cn be foun online t www.tnfonline.com/ucmj. oi.og/10.1080/07468342.2018.1411655 MSC: 26A06 VOL. 49, NO. 2, MARCH 2018 THE COLLEGE MATHEMATICS JOURNAL 109
Using this nottion, we cn ephse ou poblem s follows: On sphee of ius,whichspheiclsegmentswithveticlseption hve the lgest sufce e? To suggest some subtlety to the poblem, consie the following: In the exteme cses, the segment coul be eithe bn oun the equto o cp t one of the poles, s shown in Figue 1. Ononehn,thespheeiswiestttheegionsuouningthe equto; on the othe, while the sphee becomes nowe ne the poles, it lso becomes fltte. Which competing fcto omintes? This poblem hs been consiee since ntiquity n ws esolve by Achimees in wht is known s his ht box theoem (we explin the nme below). In the context of the thought expeiment, it ens up tht Rente is evious you choice hs no effect on the piece of chocolte s e! In othe wos, ny two spheicl segments with the sme seption hve equl es. Thee is even simple expession fo the e. Theoem 1 (Achimees s ht box theoem). The e of ny spheicl segment fome by sphee of ius n two plnes septe by istnce is 2π. The conclusion is tht the competing fctos iscusse bove, the eltive beth n fltness of the segment chosen, exctly blnce. Pehps you cn see this cncelltion in the following poof. Poof. Fist, ecll tht the e of sufce given by otting gph y = f (x)oun the x-xis between x = n x = b is 2π b f (x) 1 + ( f (x)) 2 x. We cn elize sphee with ius centee t the oigin by otting the function f (x) = 2 x 2 efine on x bout the x-xis. Clculting f (x) n woking it into the e expession gives 2π b b 2 x 2 2 x x = 2π x= 2π(b ). 2 How i Achimees pove this, without clculus? His ingenious gument shows tht the e of spheicl segment fome by two plnes with seption is equl to the sufce e cut fom the cyline of ius enclosing the sphee whose xis is othogonl to the plnes. The theoem s nme ises fom imgining the cyline s box n the top hlf of the sphee s ht insie. See [1] fo n excellent exposition of this metho. Aseconthoughtexpeiment Next, suppose tht M. Z pilots spceship insie plnet-size, hollow spheicl shell with unifom mss ensity. How oes the net foce of the shell on him epen on his position? Wht if his ship is outsie the shell? Recll fom elementy physics tht the fomul fo the mgnitue of the foce F between two point pticles septe by istnce n with msses m 1 n m 2 is whee G is gvittionl constnt. F =G m 1m 2 2 (1) 110 C THE MATHEMATICAL ASSOCIATION OF AMERICA
This time, I will immeitely let the ct out of the bg (o the ht out of the box?) n evel the nswe, which ws iscovee by Newton. Theoem 2 (Newton s shell theoem). Let p be point pticle with mss m n Sbethinspheiclshellwithius,unifommssensity,ntotlmssM.LetR be the istnce between p n the cente of S. The mgnitue of the net gvittionl foce of S on p is { 0 if R <, F net = G Mm if R >. R 2 Let us ppecite, s we i with Theoem 1, howthisesultislsoinepenent of the geometic setup. In the lnguge of the thought expeiment bove, the fist pt of Newton s shell theoem ssets tht M. Z feels nothing fom the shell, egless of his position insie. As f s the net foce is concene, the shell my s well not exist! The secon pt, the cse whee p is outsie of S, ssetsthtthenetgvittionl foce is the sme s the sitution whee the entie shell S is eplce by point mss M t its cente. This pt of Theoem 2 mticlly simplifies clcultions in celestil mechnics becuse it llows plnets to be eplce by point msses. We begin with heuistic unestning of the competing fctos, s we i with Achimees s esult. Suppose p is insie S. If p is t the cente of S,thenthenetfoce is zeo by symmety. But suppose p is vey close to pt of S,whichwemythinkofs being compose of mny point msses. On one hn, thee e vey lge foces ising fom point pticles of S close to p ue to the vey smll ii in (1). On the othe hn, thee e oppositely iecte foces ising fom point pticles on the othe sie of S. These foces e smlle in mgnitue becuse the pticles e futhe fom p,but thee e mny moe of these secon type of points. The fist pt of Newton s esult shows tht thee is pefect net cncelltion. We cn mke the gument in the pevious pgph slightly moe convincing without much moe wok: Fix p insie S n subten fom p n infinitesiml cone C which intesects S in cps C 1 n C 2 on opposite sies of S s shown in Figue 2.Thepointson C 1 e fixe istnce 1 fom p,henceby(1), the mgnitue F 1 of the foce fom C 1 scles like 2 1.Ontheothehn, F 1 is popotionl to the e of C 1,whichscles like 1 2.Combiningtheseobsevtionsmens F 1 is inepenent of 1,nbyusing p C 2 C 1 Figue 2. The foces fom the cps C 1 n C 2 cncel. VOL. 49, NO. 2, MARCH 2018 THE COLLEGE MATHEMATICS JOURNAL 111
the sme gument to nlyze the mgnitue F 2 of the foce fom C 2,weseetht F 1 + F 2 = 0, since C 1 n C 2 e on opposite sies of p. Poving the shell theoem with the ht box theoem We hve obseve qulittive similities between Theoems 1 n 2, ech sseting tht pticul geometic quntity is inepenent of the initil setup. In fct, the connection uns eepe below, we use the ht box theoem to pove the shell theoem. AcommonpoofofTheoem 2 evlutes the foce fom the shell s tiple integl in spheicl coointes [2, p.40].bysymmetyconsietions,itispossibletowite the ltte s single, one-vible integl. In ou cse, Achimees s theoem emoves the necessity to set up this integl in ngul coointes, which woul equie invoking the lw of cosines n lte ppliction of chnge of vibles. We fin tht this mkes the entie eivtion moe tctble n intuitive. Poof of the shell theoem. Afte tnslting S n otting p bout S,wemysuppose S is centee t zeo n p is on the negtive x-xis s shown in Figue 3. Becuse of this, symmety implies tht the y- nz-components F y n F z of the net foce F net vnish. Theefoe, F net is iecte long the x-xis n stisfies F net = F x.wewill compute F net by integting the mgnitues of the foces fom spheicl segments of S with infinitesiml seption x. By (1), the mgnitue F of the foce between p n n infinitesiml piece S of S with mss M n istnce fom p is F = Gm 2 Since F y = F z = 0, we e inteeste only in the mgnitue F x of the x-components of such infinitesiml foces. Fom Figue 3 (emembe tht x is negtive thee), we see M. F x = Gm M. 2 We now focus on spheicl segments iecte long the x-xis with infinitesiml thickness x. SinceS hs constnt mss ensity, the infinitesiml mss M of such segment is popotionl to its e. By Theoem 1,llspheiclsegmentswiththickness x hve the sme infinitesiml e, so M = cx fo some constnt c. In fct, the constnt c = M/(2), since M = M = cx= c(2). p R x 0 Figue 3. Setup fo the shell theoem. 112 C THE MATHEMATICAL ASSOCIATION OF AMERICA
Integting the foces F x fom these segments on [, ], we fin F net = GmM 2 3 Using the Pythgoen theoem twice (o the lw of cosines), we hve 2 = R 2 + 2 + 2Rx, x. hence F net = GmM 2 (R 2 + 2 + 2Rx) 3 2 x. Tke eep beth; the inticte setup is one n the upshot is tht we hve euce the entie poblem to single one-vible integl. We integte by pts, tking u = n C: x x = (R 2 + 2 + 2Rx) 3 2 R(R 2 + 2 + 2Rx) 1 2 R(R 2 + 2 + 2Rx) 1 2 = R R 2 + 2 + 2Rx + R2 + 2 + 2Rx R 2 = 2 + Rx R 2 R 2 + 2 + 2Rx. Retuning to the oiginl integl n intoucing the ppopite bouns, we hve F net = GmM 2 [ ] + Rx 2 R 2 R 2 + 2 + 2Rx = GmM + R 2R + R. 2 ( + R) 2 ( R) 2 We now intepet this nswe in the two cses. If R <,sothtp is insie the shell, then the secon tem in the bckets is 1n F net =0. If R >, sothtp is outsie the shell, then the secon tem in the bckets is 1 n F net =GMm/R 2. Acknowlegment. The utho thnks M. Gulin fo n inspiing lunchtime convestion. Summy. Newton s shell theoem ssets tht the net gvittionl foce between point pticle n sphee with unifom mss ensity is the sme s the foce in the sitution whee the sphee is eplce by point pticle t its cente with the sme totl mss. We give n exposition of this theoem using only tools fom intouctoy one-vible clculus. A key simplifiction is esult of Achimees tht the e of the egion on sphee between two pllel plnes epens only on the seption between the plnes, not on thei position eltive to the sphee. Refeences [1] Apostol, T., Mntsknin, M. (2004). A fesh look t the metho of Achimees. Am. Mth. Monthly 111: 496 508. oi.og/10.2307/4145068. [2] Menzel, D. H. (1961). Mthemticl Physics. NewYok,NY:Dove. VOL. 49, NO. 2, MARCH 2018 THE COLLEGE MATHEMATICS JOURNAL 113