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Math Help - Volume and centre of mass

  1. #1
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    Question 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.
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  2. #2
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    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.
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