Center of mass of a solid

**bazingasmile** · April 3rd 2011, 07:58 PM

Hi,

I'm confused on how to calculate the center of mass of a solid that's located out of one sphere and into another. The problem goes on like this:

Determine the center of mass of a solid located out of a sphere, of radius 1, centered at the origin, and inside a sphere, of radius 1, centered at point $\left(0,0,1\right)$ .

All I get from my textbook is the mass formula:
$m=\int\int\int_{E} \rho(x,y,z)\mathrm{d}V$

And then if the center of mass is actually the same thing as a center of inertia, I would have to calculate the moments related to each plan of coordinates.

How can I do that when I've got two spheres one above the other. I believe their equations are respectively:
$x^2+y^2+z^2=1$
and
$x^2+y^2+(z-1)^2=1$

As plotted below.

Can you please give me a hint? I'm blocked there.

Thanks a lot,

Bazinga

**running-gag** · April 4th 2011, 11:08 AM

Hi,

The center of mass G of a solid is given by the formula

$\int\int\int_{E} \rho(x,y,z) \vec{OM}\mathrm{d}V = \left(\int\int\int_{E} \rho(x,y,z) \mathrm{d}V\right) \vec{OG}$

If the solid is homogeneous, $\rho(x,y,z)$ is constant (independent with respect to x, y and z) and therefore can be pulled out of the integral

$\int\int\int_{E}\vec{OM}\mathrm{d}V = \left(\int\int\int_{E} \mathrm{d}V\right) \vec{OG}$

In your example, as far as I have understood, the solid is axysimmetric with respect to the z axis therefore G is on the z axis

The projection of the formula on the z axis gives :

$\int\int\int_{E} z r \mathrm{d}r \mathrm{d}\theta \mathrm{d}z = \left(\int\int\int_{E} r \mathrm{d}r \mathrm{d}\theta \mathrm{d}z) z_G$

Thread: Center of mass of a solid

LinkBack

Thread Tools

Display

Center of mass of a solid

Similar Math Help Forum Discussions

Center of Mass

please help with using double integrals to solve for mass and center of mass

Find mass and center of mass of the solid S....

center of mass of a solid

Center of Mass

Search Tags