Parallel Axis Theorem and interia tensors

[pimpalicous]

Is the parallel axis theorem always valid for inertia tensors? We have only seen examples with flat (2d) objects and was wondering if it would also be valid for 3d objects, like a h emisphere, for example. Thanks.

 

[aerospaceut10]

Yep, just use the distance projected from the axis that it's being measured around.

 

[D H]

The general form for the parallel axis theorem is

$$\mathbf{J} = \mathbf{I} + m(r^2\mathbf{1} - \boldsymbol{r}\boldsymbol{r}^T)$$

where

$\mathbf{J}$ is the inertia tensor of some object about some point removed from the center of mass,
$\mathbf{I}$ is the inertia tensor of the object about its center of mass
$m$ is the object's mass
$\boldsymbol{r}$ is the displacement of the point in question from the center of mass, expressed as a column vector
$\mathbf{1}$ is the identity matrix.

 

[adriank]

I think you meant $$\mathbf{J} = \mathbf{I} + m(r^2\mathbf{1} - \mathbf{r}\mathbf{r}^T)$$, because $$\mathbf{r}^T \mathbf{r} = r^2$$, which is not even the right type of tensor.

To show that this gives the correct moment of inertia about a certain axis, let e be a unit vector giving the direction of this axis. Then the moment of inertia about this axis is
$$\mathbf{e}^T\mathbf{Je}<br /> = \mathbf{e}^T \mathbf{Ie} + m(r^2 \mathbf{e}^T\mathbf{1e} - \mathbf{e}^T\mathbf{rr}^T\mathbf{e})$$
$$= \mathbf{e}^T \mathbf{Ie} + mr^2 - m(\mathbf{r} \cdot \mathbf{e})^2$$
$$= \mathbf{e}^T \mathbf{Ie} + mr_\perp^2$$, where $$r_\perp$$ is the perpendicular distance from the centre of mass to the axis.

 

[D H]

I think you meant $$\mathbf{J} = \mathbf{I} + m(r^2\mathbf{1} - \mathbf{r}\mathbf{r}^T)$$

Oops. I corrected my post, thanks.

 

[pimpalicous]

so does that mean if we find the inertia tensor at the center of the base of a hemisphere we can use the parallel axis theorem to find the tensor for the center of mass

 

[adriank]

Yes, but in reverse. You know J, and you must find $$\mathbf{I} = \mathbf{J} - m(r^2 \mathbf{1} - \mathbf{rr}^T)$$.