# Moment of Inertia

1. Apr 30, 2007

### r16

1. The problem statement, all variables and given/known data
There is a rectangular prismof uniform mass distribution with lengths of $a$, $b$, and $c$ (b>a>c). Calculate it's rotational inertia about an axis through one cornet and perpendicular to the large faces.

2. Relevant equations
$$I = \int r^2 dm$$
$$r^2 = x^2 + y^2 + z^2$$
$$\rho = \frac{M}{V}$$
$$V = abc$$

3. The attempt at a solution

I am examining a cubic differential mass of $dm$'s contribution on the moment of inertia about the axis of rotation. The radius between $dm$ and the axis of rotation is $r^2 = x^2 + y^2 + z^2$. The density, $\rho$, is constant which is $\frac{M}{V}$, so $dm = \rho dV$.

$$I = \int r^2 dm = \int (x^2 + y^2 + z^2) \rho dV$$
$$I = \rho \iiint_V x^2 dV + y^2 dV + z^2 dV = \int^a_0 \int^b_0 \int^c_0 x^2 dzdydx + \int^a_0 \int^b_0 \int^c_0 y^2 dzdydx + \int^a_0 \int^b_0 \int^c_0 z^2 dzdydx [/itex] [tex] I = \frac{\rho}{3} ( a^3 bc + ab^3 c + abc^3)$$
$$I = \frac{M}{3abc} ( a^3 bc + ab^3 c + abc^3)$$
$$I = \frac{M}{3} (a^2 + b^2 + c^2)$$

This problem looked cool so I did it, but it was an even one so there is no answer in the back of the book. Does this look right?

Last edited: Apr 30, 2007