# Thread: How do I find moments of 3d object?

1. ## How do I find moments of 3d object?

If lets say I have $\displaystyle y=x^{2}$ and $\displaystyle x=2$ as the area (top view) of a shape and the height is 10 units.

How would I find the x,y and z moments of this 3d object? Also how would I find the center of mass?

2. The (first) moment of inertia in space is.

$\displaystyle M_{xy} = \iiint_V \delta(x,y,z)z dV$

$\displaystyle M_{yz} = \iiint_V \delta(x,y,z)xdV$

$\displaystyle M_{xz} = \iiint_V \delta(x,y,z) ydV$

Where $\displaystyle \delta(x,y,z)$ is the density of the region.

3. Originally Posted by ThePerfectHacker
The (first) moment of inertia in space is.

$\displaystyle M_{xy} = \iiint_V \delta(x,y,z)z dV$

$\displaystyle M_{yz} = \iiint_V \delta(x,y,z)xdV$

$\displaystyle M_{xz} = \iiint_V \delta(x,y,z) ydV$

Where $\displaystyle \delta(x,y,z)$ is the density of the region.
I'm really lost... $\displaystyle \delta(x,y,z)$ so is this like x times y times z or what? Like where is the original equation of y=x^2 and x=2?

4. Originally Posted by circuscircus
I'm really lost... $\displaystyle \delta(x,y,z)$ so is this like x times y times z or what? Like where is the original equation of y=x^2 and x=2?
no, $\displaystyle \delta (x,y,z)$ is the density function. it depends on what the question says it is. the equations you were given describe the solid over which you are integrating, not the density of the solid, we need additional conditions to find $\displaystyle \delta (x,y,z)$

also, define $\displaystyle m = \iiint_V \delta(x,y,z) dV$

the center of mass is the point $\displaystyle (\bar {x}, \bar {y}, \bar {z})$

where $\displaystyle \bar {x} = \frac {M_{yz}}{m}$, $\displaystyle \bar {y} = \frac {M_{xz}}{m}$, and $\displaystyle \bar {z} = \frac {M_{xy}}{m}$