tags:*Lecture 011: Handling Tiny NumbersSteveSekula, 17 September 2010 (created 17 September 2010)Goals of this short lectureHeed my warning: be careful when neglecting small numbersCautionary Tale: GPS System and Special RelativityGlobal Positioning works by using satellites in orbit around the earth to (atleast) triangulate your position on the surface of the earth. A hand-heldGPS unit (in your phone, car, etc) relies on seeing signals from at least 3such sateliites. Military-grade GPS can locate a position on the earth towithin a few feet. Civilian GPS is good to within a few meters.QUESTION: Does anybody know what "The Special Theory ofRelativity" is, and/or what it tells us?ANSWER: it tells us that as you go faster, you observe clocks at restrunning more slowly (time runs slowly) and you observe that distancesbecome shorter (space contracts). This has huge effects on anythingthat relies on precision clocks, like GPS.But, did you know that Einstein's Special Theory of Relativity predicts avery, very tiny correction to the behavior of these satellites, and that if youneglect the correction then your location on earth will be wrong by 2 morekilometers every day? The satellites travel at about 14,000 km/h; that's0.001% of the speed of light. Normally, we only worry about specialrelativity when speeds get to about 5% of that of light, so you might betempted to ignore special relativity when engineering the GPS system.You'd made a potentially dangerous mistake in doing so.Why? Well, in the equations you would use to work the GPS problem andGeneral Physics - E&M (PHY 1308) LectureNotesGeneral Physics - E&M (PHY 1308) LectureNotesGeneral Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Documents/Notebooks/PHY1308...1 of 3 09/17/2010 08:11 AMdetermine the size of special relativistic effects, you encounter a formulalike this:where is a number, is also a number (whose value is much, muchsmaller than , ), and is some power to which the above sum israised ( can be positive, negative, rational, irrational, etc. - any realnumber).SInce 's value is SO much smaller than (in the GPS problem, it's value isjust 0.001% of ), your instinct might be to toss it. Your calculator'sinstinct IS to toss it, thus betraying you.But in the GPS problem, the system is required to be SO precise that youcannot just toss that number out and work the problem without it. It'ssmall value has huge, real consequences.Your Homework - the Dipole Problem (SS-11)Likewise, in your homework you have a problem involving two dipoles. Weknow that the dipole force between water molecules in the lungs, thougheach very tiny, is in fact non-zero and adds up to a big problem for infantswithout the pulmonary surfactant necessary to reduce the surface tensionof water (reduce the dipole force).Sine the size of the water dipole in SS-11 is SOOOOOO much tinier thanthe separation of the two dipoles, you might be tempted to neglect the sizeof the dipole. You'd make a fatal mistake. Be very careful when working thisproblem. Since you know that surface tension in the lungs is a realproblem, if you get "zero" for the answer to SS-11 you've obtained a resultin direct contradiction with Nature, and Nature always wins.Let me sketch the problem you'll encounter, and the solution.You're going to eventually encounter an integral like the following:(a ) + Îna Î a jÎj < < a n n Î a a (x)dx Zx +ÎAx +ÎBf = (a ) j+ xnx +ÎBx +ÎAGeneral Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Documents/Notebooks/PHY1308...2 of 3 09/17/2010 08:11 AMIn both limits of the integral, . You might be tempted to toss thesmall number and keep the big. But the problem is that the small numbermatters more than the big number, because at the end of you solution it'sthe only part of the two that survives, and it's the part that causes infantsall that trouble. That small part is the reason small insects can walk on thesurface of a pond. That small number matters.How do I deal with it? The Binomial ApproximationHere's how you handle it: the binomial approximation. This approximationlet's you simplify the nasty looking polynomial represented by:but ONLY when . The approximation (one version of this is inWolfson Appendix A) is:Some examples (an in every one, ):jÎj < ; < xAxB(a ) + ÎnjÎj < < a(a ) + ÎnÙ an1Ò+ nÎaÓjyj < < 1(1 ) 1 y) + y2Ù ( + 2(1 ) 1 y) + yÀ3Ù ( À 3(1 ) 1 y) À y4Ù ( À 4(1 ) 1 y) À yÀ2Ù ( + 2General Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Documents/Notebooks/PHY1308...3 of 3 09/17/2010 08:11
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