20e-lecture21

20e-lecture21 - Lecture 21 - Wednesday May 20th...

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Unformatted text preview: Lecture 21 - Wednesday May 20th jacques@ucsd.edu Key words: Moments of inertia, center of mass, work 21.1 Moments of inertia and center of mass Let W be a solid with density ( x,y,z ) at point ( x,y,z ). Then the moments of inertia of W with respect to the co-ordinate axes are I x = ZZZ W ( y 2 + z 2 ) dV I y = ZZZ W ( x 2 + z 2 ) dV I z = ZZZ W ( x 2 + y 2 ) dV. These quantities measure the difficulty with which an object is rotated about the axes. As an exercise, one can determine whether it is harder to rotate a sphere x 2 + y 2 + z 2 = 1 with uniform density ( x ) = 1 around the z-axis or to rotate the cylinder x 2 + y 2 1 ,z 4 / 3 with uniform density ( x ) = 1 around the z-axis. Note that these objects clearly have the same mass. The center of mass of a solid object W in n dimensions is at the point w = ( w 1 ,w 2 ,...,w n ) defined by w i = Z Z Z W x i ( x ) m dV where m is the mass of the object. Example 1. Find the center of mass of H = { ( x,y,z ) : x 2 + y 2 + z 2 1 ,z } with uniform density....
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This note was uploaded on 02/12/2010 for the course MATH 20E taught by Professor Enright during the Spring '07 term at UCSD.

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20e-lecture21 - Lecture 21 - Wednesday May 20th...

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