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Chapter09-page2 - Physics 235 Chapter 09 1 rdm M Rcm =...

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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 ρ dV = ρ zr 2 dr sin θ d θ d φ r = r 1 r 2 θ = 0 π 2 φ = 0 2 π ρ r 2 dr sin θ d θ d φ r = r 1 r 2 θ = 0 π 2 φ = 0 2 π Using z = r cos θ and doing the integrals gives z cm = ρ r 3 cos θ dr sin θ d θ d φ r = r 1 r 2 θ = 0 π 2 φ = 0 2 π ρ r 2 dr sin θ d θ d φ r = r 1 r 2 θ = 0 π 2 φ = 0 2 π = ρ 1 2 2 π ( ) 1 4 r 2 4 r 1 4 ( ) ρ 1 ( ) 2 π
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