Massachusetts Institute of TechnologyMathematical Methodsfor Materials Scientists and Engineers3.016 Fall 2005W. Craig CarterDepartment of Materials Science and EngineeringMassachusetts Institute of Technology77 Massachusetts Ave.Cambridge, MA 02139Problem Set 6: Due Wed. Dec. 7, Before 5PM: email to [email protected] following are this week’s randomly assigned homework groups. The first member of thegroup is the “Homework Jefe” who will be in charge of setting up work meetings and have respon-sibility for turning in the group’s homewo r k notebook. If some some reason, the first member inthe list is incapacitated, recalcitrant, or o therwise unavailable, then the second member shouldtake that position. Attention slackers: The Jefe should include a line at the top of your notebooklisting the group members t hat participated in the notebook’s production. Group names are bo ld-faced text.Brubu: Katherine Hartman (khartman), John Pavlish (jpavlish), Bryan Gortikov (bryho),Michele Dufalla (mdufalla),Richard Ramsaran (rickyr21), Eugene Settoon (geneset)Chubasco: Emily Gullotti (emgull), Kyle,Yazzie (keyazzie), Jill Rowehl (jillar), Allison Kunz(akunz), Saahil Mehra,(smehra), Jason Pelligrino (jpell19)Cordonazo: Kelsey Vandermeulen (kvander),Annika Larsson (alarsson), Omar Fabian (ofabian),Vladimir Tarasov (vtarasov), John Rogosic (jrogosic), Leanne Veldhuis (lveldhui)Haboob: Jina Kim,(jinakim), Maricel Delgadillo (maricela), Katrine Sivertsen (katsiv),Rene,Chen (rrchen), Lauren Oldja (oldja)Williwaw: Kimberly Kam (kimkam), Charles Cantrell (cantrell), Talia Gershon (tgershon), LisaWitmer (witmer), JinSuk Kim (jkim123)1Individual Exercise I6-1Kreyszig MathematicaRComputer Guide: pro blem 2.6, page 29Individual Exercise I6-2Kreyszig MathematicaRComputer Guide: pro blem 2.14, page 29Individual Exercise I6-3Kreyszig MathematicaRComputer Guide: pro blem 3.2, page 40Individual Exercise I6-4Kreyszig MathematicaRComputer Guide: pro blem 3.6, page 40Individual Exercise I6-5Kreyszig MathematicaRComputer Guide: pro blem 4.20, page 54Individual Exercise I6-6Kreyszig MathematicaRComputer Guide: pro blem 11.4, page 131Individual Exercise I6-7Kreyszig MathematicaRComputer Guide: pro blem 11.8, page 131Individual Exercise I6-8Kreyszig MathematicaRComputer Guide: pro blem 11.12, page 1322Group Exercise G6-1About how fast can you ride a bike on a level path? About how f ast can you ride a bike on agrade that increases 1 meter every 5 meters? About how fast can you ride a bike on a grade thatdecreases 1 meter every 5 meters? What is the maximum grade up which you could continuouslyride a bicycle?1. Write out a model that predicts a bicyclist’s speed as a function of the grade, S(m), andplot speed versus grade.2. The speed is the magnitude of the velocity vector ~v(m). Plot the vertical component ofvelocity ~v ·ˆk against the horizontal component of velocity,q(~v ·ˆi)2+ (~v ·ˆj)2, for severaldifferent values of grade m.3. Discuss whether your model f rom part 1 and 2 is a reasonable model for a continuouslychanging grade. For example, you may wish to consider whether your model for S(m) wouldpredict the total distance traveled from t = 0 to t = t0asdistance(~p(s)) =Zt00S(m)dtOr, if your model is inserted into the following equationaverage speed(~p(s)) =Rt00S(m)dtt0would it produce a good estimate for actual average speed? ~p(s) is a curve representing apath that a bicycle follows.4. Use your model to find the average speed on a path given by ~p(u) = (u, 0, A cos(ku)).Note that the ar clength element ds =pdx2+ dy2+ dz2; you may need to do numericalevaluations of elliptic integrals.Graphically represent average speed, as modeled by the equation above, as a function of theparameters A and k.3Group Exercise G6-2The first part of this problem was borrowed from Marc Spigelman,http://www.ldeo.columbia.edu/˜mspieg/Complexity/Problems/ who borrowed itfrom . H. Strogatz. Nonlinear Dynamics and Chaos: with applications to physics,biology, chemistry, and engineering, Addison-Wesley Publishing Co., Reading,MA, 1994.In this problem, analyze the complex relationship dynamics of young lovers.In this first case, it’s the “it isn’t me—it’s you” syndrome.Romeo tends to love Juliet, but suppose Juliet is a fickle lover: the more Romeo loves her,the more Juliet wants to find someone else who will treat her poorly. However, when Ro meo getsdiscouraged a nd begins to ignore Juliet when she seems uninter ested, Juliet begins to find himstrangely attractive. Romeo, on the other hand, is encouraged when encouragement encourages:he warms up when she loves him a nd grows cold when she doesn’t. Suppose R(t) is a measureof Romeo’s love of Juliet, when its positive he loves her and when it is negative he hates her.Similarly, J(t) is Juliet’s love or hate for Romeo at time t.1. Our model based on the above scenario isdRdt= JdJdt= −RAnalyze and illustrate their love affair.2. Suppose that, for Romeo, it “isn’t just you—but it’s also me.” Then a reasonable model is:dRdt= J + RdJdt= −RAnalyze and illustrate their love affair.3. Consider general linear romantic behavior (GLRB).dRdt= αJ + βRdJdt= γJ + ǫRCharacterize, with as creative prose as you can muster (warning, these may be published), thecharacteristics of the lovers and their relationship for (real) values of the GLRB parametersα, β, γ, and ǫ.4. Suppose, that Juliet thinks, “If Romeo’s love fo r me is only imaginary, t hen I would hatehim; but, if his love is real, I would love him.” But, Romeo thinks, “In my imagination,when I think of Juliet, my real love for her grows; but, the more I really love Juliet, themore my imaginatio n falters.” Now Romeo’s love has two components, real and imaginary.Let Rℜbe Romeo’s real love for Juliet and Rℑbe his imaginary part. Write a model for thisrelationship and analyze it. Witty descriptions will be