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# W5.4 - Physics for Scientists Engineers 1 Spring Semester...

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February 5, 2008 Physics for Scientists&Engineers 1 1 Physics for Scientists & Physics for Scientists & Engineers 1 Engineers 1 Spring Semester 2008 Lecture 19

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February 5, 2008 Physics for Scientists&Engineers 1 2 Yesterday Yesterday s Main results: s Main results: Work done by a variable force, 1d case: Most general 3d case W = F ( x ') dx x 0 x ! ' W = ! F ! r ' ( ) d ! r ' ! r 0 ! r !
February 5, 2008 Physics for Scientists&Engineers 1 3 Integration Refresher (1) Integration Refresher (1) All indefinite integrals have an additive constant, c Definite integrals are taken by inserting upper and lower limits into the result from the indefinite integral, and taking the difference Integration is the inverse process to differentiation Some integrals Polynomials x n dx ! = 1 n + 1 x n + 1 + c for all n " 1 x ! 1 dx " = ln x + c

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February 5, 2008 Physics for Scientists&Engineers 1 4 Integration Refresher (2) Integration Refresher (2) Trigonometric functions: Exponentials: For more integrals, please see the “Math Primer” sin ax ( ) dx ! = " 1 a cos ax ( ) + c cos ax ( ) dx ! = 1 a sin ax ( ) + c e ax dx ! = 1 a e ax + c
February 5, 2008 Physics for Scientists&Engineers 1 5

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February 5, 2008 Physics for Scientists&Engineers 1 6
February 5, 2008 Physics for Scientists&Engineers 1 7 Spring Force Spring Force Let’s look at the force needed to stretch or compress a spring If you pull on a spring a little, it offers little resistance If you pull on a spring a lot, it offers a lot of resistance The more you pull, the more the spring pulls back

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