Using the result Пa2, determine the volume of a cone of height h and radius a. Find the position of its centre of mass.

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- May 13th 2008, 12:08 PMtulipVolume and centre of mass
Using the result Пa2, determine the volume of a cone of height h and radius a. Find the position of its centre of mass.

- May 13th 2008, 01:08 PMgalactus
Orient the cone with the vertex at the origin and the axis along the y-axis.

$\displaystyle M=\int_{0}^{h}{\rho}{\pi}\left(\frac{a}{h}y\right) ^{2}dy=\frac{1}{3}{\rho}{\pi}a^{2}h$

$\displaystyle M_{y=0}=\int_{0}^{h}{\rho}y{\pi}\left(\frac{a}{h}y \right)^{2}=\frac{1}{4}{\rho}{\pi}a^{2}h^{2}$

$\displaystyle \overline{y}=\frac{3h}{4}$

Center of mass is on axis of cone, 3/4 the length of the height from the vertex, or 1/4 of the length of the height from the base.