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Unformatted text preview: Using Small Quantities In many situations in physics, it is useful to make systematic approximations. Often, these involve identifying small quantities, and therefore being able to neglect, or throw away unimportant aspects of a problem or of an algebraic eqaution. Sometimes these approximations make it possible to solve an insoluble problem, other times they may lead to a clearer interpretation of a complicated result, and there are even situations where they are fundamental to a mathematical technique calculus. Equally, computer-generated solutions to problems in physics must almost always make use of approximations: this is the science of numerical analysis. In the course, we will often meet such situations, and each time I will make them as explicit as possible. However, it is useful to collect together the most useful approaches. What follows is based, in part, on Note 1.1 on pages 39-47 of Kleppner and Kolenkovs An Introduction to Mechanics, which is the textbook used for PHYS 251. Series Expansions The example given by K+K is a simple one: measuring g , the acceleration due to gravity, by dropping a small object, length , down inside an evacuated tube. The object falls through a height L , and then takes a small but finite time t to pass through a horizontal laser-beam. From the measurement of this time, g can be determined. Time t 6 ? L Time t + t 6 ? L + Laser beam 1 In terms of the quantities marked on the figure, the time t is t = s 2 g ( L +- L ) and g can be determined from g = 2 1 t ( L +- L ) 2 (1) No approximations yet. However, if , the height of the object, is very much smaller than L , the distance travelled, then t will be small, and the right-hand-side of this equation will...
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This note was uploaded on 04/29/2008 for the course PHYS 230 taught by Professor Harris during the Fall '07 term at McGill.
- Fall '07