# Find center of gravity.

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• Aug 31st 2007, 04:58 PM
kittycat
Find center of gravity.
Find the mass and the center of gravity of a cylindrical solid of height h and radius a , assuming that the density at each point is proportional to the distance between the point and the base of the solid.

Note : the top of the cylinder is z=h.
the bottom of the cylinder is x^2+y^2 =a .

Thank you very much.
• Aug 31st 2007, 06:38 PM
galactus
Since the density is proportional to the distance z from the base, the density function has the form $\delta(x,y,z)=kz$, where k is some unknown positive constant of proportionality.

$M=\int_{-a}^{a}\int_{-\sqrt{a^{2}-x^{2}}}^{\sqrt{a^{2}-x^{2}}}\int_{0}^{h}kz \;\ dzdydx$

The constant k does not affect the center of gravity.

Now, $\overline{z}=\frac{1}{M}\int_{-a}^{a}\int_{-\sqrt{a^{2}-x^{2}}}^{\sqrt{a^{2}-x^{2}}}\int_{0}^{h}z(kz) \;\ dzdydx$

The center of gravity is on the z-axis, because $\overline{x}=\overline{y}=0$
• Aug 31st 2007, 06:39 PM
Jhevon
Quote:

Originally Posted by kittycat
Find the mass and the center of gravity of a cylindrical solid of height h and radius a , assuming that the density at each point is proportional to the distance between the point and the base of the solid.

Note : the top of the cylinder is z=h.
the bottom of the cylinder is x^2+y^2 =a .

Thank you very much.

let the cylindrical solid be such that its base lies on the xy-plane, centered at the origin, and it extends vertically upwards until it hits the plane $z = h$. since the density at a point is proportional to the (vertical) distance of the point to the xy-plane, we have $\rho (x,y,z) = kz$ for some constant $k$. use that $\rho (x,y,z)$ in the formulas given here. i think you've proven that you need no help setting up and evaluating the integrals (i would probably use cylindrical coordinates to evaluate the integrals).

(again by symmetry, $\bar {x} = \bar {y} = 0$)

EDIT: thanks for beating me to it, galactus! ........well, at least i can take your answer as a confirmation i did the right thing
• Aug 31st 2007, 06:47 PM
TKHunny
It's symmetric about the x-axis and the y-axis.
The density function is z, which is independent of both x- and y-axis.
You've only the z-axis to worry about.