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Math Help - center of mass of a cone

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    center of mass of a cone

    how can i find the center of mass of a cone?
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  2. #2
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    A cone with radius R and height H has center of mass:

    M=\frac{k{\pi}R^{2}H}{3}

    M_{x=0}=M_{y=0}=0 by symmetry.

    M_{z=0}=\int_{0}^{2\pi}\int_{0}^{R}\int_{0}^{H-\frac{Hr}{R}}kzr \;\ dz \;\ dr \;\ d{\theta}

    =\int_{0}^{2\pi}\int_{0}^{R}\frac{krH^{2}}{2R^{2}}  (R-r)^{2}drd{\theta}

    =\int_{0}^{2\pi}\int_{0}^{R}\frac{kH^{2}}{2R^{2}}(  R^{2}r-2Rr^{2}+r^{3})drd{\theta}

    =2{\pi}\left(\frac{kH^{2}}{2R^{2}}\right)\left(\fr  ac{R^{4}}{2}-\frac{2R^{4}}{3}+\frac{R^{4}}{4}\right)

    =\boxed{\frac{k{\pi}R^{2}H^{2}}{12}}
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