# Volume and centre of mass

• May 13th 2008, 12:08 PM
tulip
Volume 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 PM
galactus
Orient the cone with the vertex at the origin and the axis along the y-axis.

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

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

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

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.