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Unformatted text preview: Homework 3 PHZ 5156 Due Tuesday, September 22, 2009 1. Consider the diffusion or heat flow equation in two spatial dimensions 2 x 2 + 2 y 2 u = 1 2 u t a) Use the method of separation of variables and take u ( x,y,t ) = F ( x,y ) T ( t )to show that this equation can be reduced to two ordinary differential equations 2 x 2 + 2 y 2 F ( x,y ) + k 2 F ( x,y ) = 0 and dT dt =- k 2 2 T b) Find the general set of solutions subject to the boundary conditions u ( x = ,y,t ) = 0, u ( x = L,y,t ) = 0, u ( x,y = 0 ,t ) = 0, u ( x,y = L,t ) = 0 (Hint: You might want to start by first applying separation of variables to the spatial equation, taking F ( x,y ) = X ( x ) Y ( y )). 2. The gravitational potential (potential energy per unit mass) at a distance r away from the center of the Earth is given by, ( r ) =- GM r where r = ( x 2 + y 2 + z 2 ) 1 2 , and G = 6 . 67 10- 11 Nm 2 kg 2 , and the mass of the Earth is M = 5 . 97 10 24 kg ....
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This note was uploaded on 08/08/2011 for the course PHZ 5156 taught by Professor Johnson,m during the Fall '08 term at University of Central Florida.
- Fall '08