# center of mass of a cone

• Jun 18th 2008, 11:52 AM
tahi
center of mass of a cone
how can i find the center of mass of a cone?
• Jun 18th 2008, 01:04 PM
galactus
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.

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

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

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

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

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