Chapter09-page2 - Physics 235 Chapter 09 1 rdm M Rcm = Example Problem 9.1 Find the center of mass of a hemispherical shell of constant density and

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Physics 235 Chapter 09 - 2 - R cm = 1 M rdm Example: Problem 9.1 Find the center of mass of a hemispherical shell of constant density and inner radius r 1 and outer radius r 2 . Put the shell in the z > 0 region, with the base in the x - y plane. By symmetry, x cm = y cm = 0 To find the z coordinate of the center-of-mass we divide the shell into thin slices, parallel to the xy plane. z cm = ρ zdV dV = zr 2 dr sin θ d d φ r = r 1 r 2 = 0 π 2 = 0 2 r 2 dr sin d d r = r 1 r 2 = 0 = 0 2 Using z = r cos and doing the integrals gives z cm = r 3 cos dr sin d d r = r 1 r 2 = 0 2 = 0 2 r 2 dr sin d d r = r 1 r 2 = 0 2 = 0 2 = 1 2 2 ( ) 1 4 r 2 4 r 1 4 ( ) 1 ( ) 2 ( ) 1 3 r 2 3 r 1 3 ( ) = 3 r 2 4 r 1 4 ( ) 8 r 2 3 r 1 3 ( ) Linear Momentum
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This note was uploaded on 01/08/2012 for the course PHY 235 taught by Professor Morgan during the Spring '09 term at SUNY Stony Brook.

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