Parallel axis theorem
From QED
The parallel axis theorem states that if ICM is the moment of inertia of an object about an axis through the center of mass, then the moment of inertia about a parallel axis displaced by a distance of R is:
- I = mobjectR2 + ICM
In 3 space, using cylindrical coordinates
![I_{CM} = \int\int\int_{V} \rho(\mathbf{r}) [\mathbf{r}-\mathbf{r_{CM}}]^2 dr d\theta dz](../../images/math/f/9/b/f9b25212f48736cea2f0aea1a9d247ec.png)
and
![I = \int\int\int_{V} \rho(\mathbf{r}) [\mathbf{r}-\mathbf{r_0}]^2 dr d\theta dz](../../images/math/a/b/b/abbc3a94db51eba371ae971aeda2d95e.png)
![I = \int\int\int_{V} \rho(\mathbf{r}) [(\mathbf{r}-\mathbf{r_{CM}})+(\mathbf{r_{CM}}-\mathbf{r_0})]^2 dr d\theta dz](../../images/math/6/2/e/62e2912a407e2b42270a92d984d125ae.png)
![I = \int\int\int_{V} \rho(\mathbf{r}) [(\mathbf{r}-\mathbf{r_{CM}})^2 + 2 (\mathbf{r}-\mathbf{r_{CM}})(\mathbf{r_{CM}}-\mathbf{r_0}) + (\mathbf{r_{CM}}-\mathbf{r_0})^2] dr d\theta dz](../../images/math/a/b/1/ab1f6594287af4f72e9b85e63174cca3.png)
![I = m_{object} R^2 + \int\int\int_{V} \rho(\mathbf{r}) [(\mathbf{r}-\mathbf{r_{CM}})^2 + 2 (\mathbf{r}-\mathbf{r_{CM}})(\mathbf{r_{CM}}-\mathbf{r_0})] dr d\theta dz](../../images/math/7/7/8/778d79c120a26affd64efee5f92894ba.png)
But since 
![I = m_{object} R^2 + \int\int\int_{V} \rho(\mathbf{r}) [(\mathbf{r}-\mathbf{r_{CM}})^2] dr d\theta dz](../../images/math/9/7/e/97e386c08c85d09c3438392bad9981d6.png)
I = mobjectR2 + ICM
This page was recovered in October 2009 from the Plasmagicians page on Parallel_axis_theorem dated 03:12, 16 October 2006.
