Find Center of Mass of Hemisphere Homework

[JJfortherear]

Homework Statement

A hemisphere of radius R, covering +/- R in the x and y directions and 0 to R in the Z direction (only the top half of a sphere centered at the origin)
Find the center of mass.

Homework Equations

Cm=(1/m)\int(z dm)

The Attempt at a Solution

dm is sigma dA, and dA is x dz, where x = \sqrt{}R^2-z^2. The limits are 0 to R, and the answer should be 3/8 R, but when I evaluate the integral I get nothing like that.

 

[vik10622]

since you are trying to find for a hemisphere not for a hemispherical shell, your equation for 'dm' is wrong.
You should consider not the surface density, but instead the volume density ρ since it is a solid object
\rho = \frac{M}{\frac{2}{3} \pi R^3}

You then have

dm = \rho A dz where A is the area of the circular disc centered on the z-axis with radius r = \sqrt{R^2 - z^2}.

The final integral then becomes

CM (z-comp) = \frac{1}{M} \int_0^R \ dz\ \pi (R^2 - z^2) \rho\ z

which on solving gives the result \frac{3 R}{8}