Dynamical variables in brachistochrone problem

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1 A. Tn, A. K. Chile M. Dokhnin Deptment of Phsics, Albm A & M Uniest, Noml, Albm 576, U.S.A. E-mil: un.tn@mu.edu (Receied Febu, ccepted 5 M ) Abstct The histoic Bchistochone poblem is widel discussed in the litetue. Howee, the discussion is pimil limited to the shpe of the cue long which pticle will descend, unde git, in the bsence of fiction, fom point to nothe not diectl below it, in the shotest mount of time. This stud exmines the ious dnmicl ibles ssocited with this motion, including the elocit, cceletion ek ectos, long with kinetic, potentil totl enegies, cutue centipetl foce, s the pticle undetkes its oune. The quntities e expessed s functions of the ngul pmete. The cceletion ek ectos e found to he constnt mgnitudes to otte counte-clockwise, with the fome tiling the ltte b 9 o. The elocit centipetl foce ectos lso otte counte-clockwise, but with hlf the ngul elocit, with the fome tiling the ltte b 9 o. This stud futhe exmines how the dnmicl ibles e ffected when kinetic fiction is pesent. Kewods: Bchistochone Poblem, Dnmicl Vibles, Jek ecto, Cutue. Resumen El poblem históico de l Bquistócon es mplimente discutido en l litetu. Sin embgo, l discusión se limit pinciplmente l fom de l cu lo lgo de l cul un ptícul descendeá, po gedd, en usenci de ficción, de un punto oto no diectmente debo de él, en el meno tiempo posible. Este estudio nliz ls difeentes ibles dinámics socids con este moimiento, incluendo los ectoes de elocidd, celeción de ek, unto con ls enegís cinétic, potencil enegís totles, l cutu fuez centípet, como l ptícul lle cbo su tectoi. Ls cntiddes son expesds como funciones del pámeto ngul. Los ectoes de celeción de ek son encontdos con mgnitudes constntes gin en sentido-ntihoio, con l pime finl de l segund po 9. L elocidd los ectoes de fuez centípet tmbién gin en sentido-ntihoio, peo con l mitd de l elocidd ngul, con l pime finl de l segund po 9. Este estudio demás exmin cómo ls ibles dinámics son fectds cuo l ficción cinétic está pesente. Plbs cle: Poblem de l Bquistócon, Vibles Dinámics, Vecto Jek, Cutu. PACS: 45..Dd, , 45.5.Dd, 45.4.A ISSN I. INTRODUCTION In 696, Johnn Benoulli posed the Bchistochone poblem s chllenge to the othe mthemticins of the d: To find the cue long which pticle will descend, unde git, fom point to nothe not diectl unde it, in the shotest mount of time. The poblem ws coectl soled b his elde bothe Jkob Benoulli, s well s b Newton, Leibniz L Hospitl, giing the segment of n ineted ccloid s the nswe [,,. It is this histoic poblem which ge ise to the new bnch of mthemtics clled the Clculus of Vitions. Mn textbooks he deoted pges to this fmous poblem, but inibl, the discussion ends buptl upon finding the cue. In this ppe, we exmine the ious dnmicl ibles ssocited with this motion, including the elocit, cceletion ek ectos, long with kinetic, potentil totl enegies, cutue centipetl foce, s the pticle undetkes its oune. Among the dnmicl ibles, we include the ek, which is the deitie of the cceletion ecto, o the thid deitie of the position ecto, with espect to time. The ek ecto hs ecentl been studied in poectile motion [4 motion of chged pticles [5. As bpoducts of the ek, one obtins the cutue tosion of the pth. If the fist thee deities of the position ecto in time, iz., the elocit, cceletion the ek ectos e,, espectiel, then the cutue κ tosion τ e gien b [5:, () o ( ) τ =. () Lt. Am. J. Phs. Educ. Vol. 6, No., June 96

2 A. Tn, A.K. Chile M. Dokhnin The ecipocl of the cutue funishes the dius of cutue, which in tun, gies the centipetl foce. II. THE BRACHISTOCHRONE PROBLEM Conside the poblem of pticle descending unde git fom point t the oigin (, ) to nothe (x, ) in the x- plne, not diectl unde the fist (Fig. ). Detemine the pth long which the pticle will slide, without fiction, in the shotest time. Fo the ske of lte conenience, eckon to be positie downwds. This is consetie sstem, fo which the totl eneg emins constnt. Futhe, if the pticle slides fom the est E = T + V =. In the usul notions, x = A ( θ ) ( ), () = A. () Remembeing tht is positie downwds, Eqs. () () e ecognized s the pmetic equtions of n ineted ccloid, i.e., cue tced out b point on cicle of dius A olling unde the positie x-xis (Fig. ). θ A x m mg =. () The time of pssge between the two points is: (x, ) ds dx + d ' τ = dt = = = dx, (4) g g with ' = d / dx. Fo τ to be minimum, the integ ' f = f (, ', x) =, (5) must stisf the Eule-Lgnge eqution: d dx f = ' f. (6) When f does not contin x explicitl, the Eule-Lgnge eqution educes to Beltmi s Identit: Thus, Squing enging: f f ' = const. (7) ' ' = const. (8) d + = A, (9) dx whee A is nothe constnt. One cn eif tht the following equtions constitute the solution to Eq. (9): FIGURE. Pth of descent of pticle in minimum time. III. DYNAMICAL VARIABLES IN THE BRACHISTOCHRONE PROBLEM Stting fom the position ecto, the elocit, cceletion ek ectos cn be clculted b successie diffeentition of the position ecto with espect to time. We he = A θ θ ( sin + A( )ˆ. () Letting θ = ωt, whee ω is the ngul elocit of the olling cicle, one obtins: = Aω θ + ω θ, () ( cos A sin ˆ A xˆ = ω + Aω ˆ, (4) A xˆ = ω Aω ˆ. (5) The mgnitudes of the boe quntities e: = = Aω, (6) = Aω, (7) = = Aω. (8) = Lt. Am. J. Phs. Educ. Vol. 6, No., June 97

3 Also, We he, futhe: = ω ( ) A ω =, (9) / A ω ( = ), () xˆ ˆ zˆ = Aω( ) Aω Aω Aω A ( θ) cos zˆ. xˆ ˆ zˆ = Aω Aω Aω Aω 5 = A ω zˆ. = A ω ( ) o ( ) = () (), (). (4) Eqs. (), () () gie the lue of the cutue κ tht of its ecipocl (the dius of cutue) R: A, (5) R = = A. (6) κ Eqs. (9) (6) funish the lue of the centipetl foce: F m R ma = = ω θ. (7) cos Eqs. () (4) indicte tht the tosion of the pth is zeo. This is to be expected s the motion tkes plce in eticl plne, the tosion is te of tuning of the tngent ecto out of the plne. The kinetic eneg of the pticle is obtined fom Eq. (9): T = m = ma ω ( ). (8) Likewise, the potentil eneg is [fom Eq. (): ( ) V = mg = mga. (9) The consetion of totl eneg ields the lue of the ngul elocit of the olling cicle: ω = A / g. In the boe equtions, the dnmicl quntities e conenientl expessed s functions of the ngle θ. If the motion spns the entie ccloid, then θ uns fom to π, the dution of the motion is π/ω. The hoizontl length of the ccloid is πa depth of the tecto, midws, t the lowest point (θ = π), is = A. The ngul dependences of the dnmicl ibles e shown in Tble I. TABLE I. Angul dependence of Dnmicl Vibles. Dnmicl ible Fomul θ-dependence Absciss Eq. () θ Odinte Eq. () cos θ x-component of elocit x = x& cos θ -component of elocit = & x-component of cceletion x = & x -component of cceletion = & x-component of ek x = && x& -component of ek = && & Speed = Mgnitude of cceletion = constnt Mgnitude of ek = constnt Momentum p m cos θ Kinetic eneg Eq. (8) cos θ Potentil eneg Eq. (9) cos θ Totl eneg E =T+V constnt Cutue Eq. (5) Tosion Eq. () Rdius of cutue Eq. (6) Centipetl foce Eq. (7) Eqs. (7) (8) indicte tht the cceletion ek ectos possess constnt mgnitudes thoughout the motion. The speed of the pticle, on the othe h, ies, being zeo t the onset of the motion, eching mximum t the middle of the ccloid, becoming zeo gin t the teminus. The ngles the elocit, cceletion ek ectos mke with the positie x-xis cn conenientl be expessed in tems of the ngle θ. Let the boe ngles be denoted b α, β γ, espectiel. Then, we he: θ π θ α = = = = () tn tn tn cot, x β = tn x = tn π ( cotθ ) = θ, () Lt. Am. J. Phs. Educ. Vol. 6, No., June 98

4 A. Tn, A.K. Chile M. Dokhnin γ = tn = tn tnθ = x ( ) θ. () Since is positie downwds, the ngles e eckoned positie if clockwise. Thus the cceletion ek ectos otte counte-clockwise with θ, with the fome tiling the ltte b 9 o. The elocit ecto, too, ottes counteclockwise, but with hlf the ngul speed. Also, since the centipetl foce is lws pependicul to the elocit, it too, ottes counte-clockwise with the sme ngul speed, leding the elocit ecto b 9 o. The elocit, cceletion ek ectos e depicted t intels of 9 o s in Fig.., A μ = -.5 μ =. - μ = x, A μ =. FIGURE. Bchistochone tectoies with ious coefficients of kinetic fiction. FIGURE. Velocit, cceletion ek ectos t intels of 9 o. IV. BRACHISTOCHRONE PROBLEM WITH KINETIC FRICTION When kinetic fiction is included, the Bchistochone poblem becomes much moe fomidble. A closed fom of solution ws obtined, but the solution is quite intctble, the eqution of motion could not be integted to gie the elocit s function of position [6. An ppoximte solution ws found, with neglect of the centipetl foce [7, which ws es to wok with. If μ is the coefficient of kinetic fiction, we he, insted of Eqs. () () (ide [7): [ x = A ( θ ) + μ( ), () = A ) + μ( θ + ). (4) Fig. displs the tectoies gien b Eqs. () (4) fo thee lues of μ equl to.,... x The dnmicl ibles e conenientl clculted using Eqs. () (4) following the usul pocedue. We he = Aω + + μ + ( + μ μ ) ˆ + μ + ( μ ) ˆ. (5) = Aω. (6) = Aω + μ ( μ ) ˆ μ ) ( μ ) μ zˆ μ ) ( μ ) + μ μ ) ( μ ) + μ / μ ) ( μ ) μ sin. (7) = A ω. (8) = A ω. (9) = A ω. (4) = A ω + θ. (4) A ( μ ) ( μ ) + μ. (4) = Aω μ. (4) = Aω μ. (44) ( μ ) ( μ ) + μ R = = A, (45) κ m F = = maω R ( μ ) ( μ ) + μ (46). Lt. Am. J. Phs. Educ. Vol. 6, No., June 99

5 The ngul dependences of these ibles e shown in Tble II. All quntities e ffected b fiction but the mgnitudes of the cceletion ek ectos emin constnts. It is es to eif tht the enties of Tble II educe to those of Tble I in the bsence of fiction. TABLE II. Angul dependences of Dnmicl Vibles with kinetic fiction. Dnmicl ible θ-dependence θ + μ + μ θ + sin ) + μsin + μ + μ μ μ + μ Absciss ( ) ( ) Odinte ( ) ( θ x-component of elocit ( ) θ -component of elocit ( ) x-component of cceletion -component of cceletion x-component of ek -component of ek ( ) Speed ( μ) ( μ) + μ Mgnitude of cceletion Mgnitude of ek Momentum constnt constnt ( μ) ( μ) + μ Kinetic eneg ( μ) ( μ) + μ Potentil eneg ( cos θ ) μ( θ + ) Cutue ( μ) ( μ) + μ Tosion Rdius of cutue μ μ + μ sin Centipetl foce μ μ + μ sin ( ) ( ) θ ( ) ( ) θ V. CONCLUSIONS The histoic Bchistochone poblem hs been ssocited with the getest mthemticl minds of tht time, including the Benoulli bothes, Newton, Leibniz L Hospitl, ll of whom cn be egded s foundes o co-foundes of Clculus. This poblem lso ge bith to the new bnch of mthemtics clled the Clculus of Vitions. Quite supisingl, the discussion of this poblem in the litetue is inibl elted to finding the shpe of the tecto, the dnmicl spects of the poblem e completel ignoed. It is hoped tht this stud fills n impotnt oid in the dnmicl spects of this fscinting poblem. REFERENCES [ Gdne, M., The Sixth Book of Mthemticl Gmes, (Uniesit of Chicgo Pess, Chicgo, 984), pp. -. [ Boe, C. B. Mezbch, U. C., A Histo of Mthemtics, (John Wile, USA, 99), pp. 45, 47. [ Count, R. Robbinds, H., Wht is Mthemtics? An Element Appoch to Ides Methods, (Oxfod Uniesit Pess, UK, 996), pp [4 Tn, A. Edwds, M. E., The ek ecto in poectile motion, Lt. Am. J. Phs. Educ. 5, (). [5 Tn, A. Dokhnin, M., Jek, cutue tosion in motion of chged pticle unde electic mgnetic fields, Lt. Am. J. Phs. Educ. 5, (). [6 Ashb, N., Bittin, W. E., Loe, W. F. Wss, W., Bchistochone with Coulomb fiction, Am. J. Phs. 4, 9-96 (975). [7 Weisstein, E. W., CRC Concise Encclopedi of Mthemtics, (Chpmn-Hll, USA, ), pp Lt. Am. J. Phs. Educ. Vol. 6, No., June

1. The sphere P travels in a straight line with speed

1. The sphere P travels in a straight line with speed 1. The sphee P tels in stight line with speed = 10 m/s. Fo the instnt depicted, detemine the coesponding lues of,,,,, s mesued eltie to the fixed Oxy coodinte system. (/134) + 38.66 1.34 51.34 10sin 3.639