# Thread: Find center of gravity.

1. ## 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.

2. Since the density is proportional to the distance z from the base, the density function has the form $\displaystyle \delta(x,y,z)=kz$, where k is some unknown positive constant of proportionality.

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

The constant k does not affect the center of gravity.

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

The center of gravity is on the z-axis, because $\displaystyle \overline{x}=\overline{y}=0$

3. 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 $\displaystyle z = h$. since the density at a point is proportional to the (vertical) distance of the point to the xy-plane, we have $\displaystyle \rho (x,y,z) = kz$ for some constant $\displaystyle k$. use that $\displaystyle \rho (x,y,z)$ 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, $\displaystyle \bar {x} = \bar {y} = 0$)

EDIT: thanks for beating me to it, galactus! ........well, at least i can take your answer as a confirmation i did the right thing

4. 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.