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20e-lecture21

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

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Lecture 21 - Wednesday May 20th [email protected] 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 δ ( 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 δ ( x ) = 1 around the z -axis or to rotate the cylinder x 2 + y 2 1 , z 4 / 3 with uniform density δ ( 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 δ ( 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 0 } with uniform density. Solution. We can assume that δ ( x ) = 1 is the density function for the hemisphere, and the hemisphere is defined by x 2 + y 2 + z 2 1 where z 0. Then the mass of the

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

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