Parallel Axis Theorem and interia tensors

In summary, the parallel axis theorem can be used to find the inertia tensor of a 3d object, such as a hemisphere, about its center of mass by subtracting the mass times the distance projected from the axis from the initial inertia tensor.
  • #1
pimpalicous
16
0
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.
 
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  • #2
Yep, just use the distance projected from the axis that it's being measured around.
 
  • #3
The general form for the parallel axis theorem is

[tex]\mathbf{J} = \mathbf{I} + m(r^2\mathbf{1} - \boldsymbol{r}\boldsymbol{r}^T)[/tex]

where

[itex]\mathbf{J}[/itex] is the inertia tensor of some object about some point removed from the center of mass,
[itex]\mathbf{I}[/itex] is the inertia tensor of the object about its center of mass
[itex]m[/itex] is the object's mass
[itex]\boldsymbol{r}[/itex] is the displacement of the point in question from the center of mass, expressed as a column vector
[itex]\mathbf{1}[/itex] is the identity matrix.
 
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  • #4
I think you meant [tex]\mathbf{J} = \mathbf{I} + m(r^2\mathbf{1} - \mathbf{r}\mathbf{r}^T)[/tex], because [tex]\mathbf{r}^T \mathbf{r} = r^2[/tex], 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
[tex]\mathbf{e}^T\mathbf{Je}
= \mathbf{e}^T \mathbf{Ie} + m(r^2 \mathbf{e}^T\mathbf{1e} - \mathbf{e}^T\mathbf{rr}^T\mathbf{e})[/tex]
[tex]= \mathbf{e}^T \mathbf{Ie} + mr^2 - m(\mathbf{r} \cdot \mathbf{e})^2[/tex]
[tex]= \mathbf{e}^T \mathbf{Ie} + mr_\perp^2[/tex], where [tex]r_\perp[/tex] is the perpendicular distance from the centre of mass to the axis.
 
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  • #5
adriank said:
I think you meant [tex]\mathbf{J} = \mathbf{I} + m(r^2\mathbf{1} - \mathbf{r}\mathbf{r}^T)[/tex]
:blushing:
Oops. I corrected my post, thanks.
 
  • #6
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
 
  • #7
Yes, but in reverse. You know J, and you must find [tex]\mathbf{I} = \mathbf{J} - m(r^2 \mathbf{1} - \mathbf{rr}^T)[/tex].
 

1. What is the Parallel Axis Theorem?

The Parallel Axis Theorem is a physical law that states that the moment of inertia of a rigid body about an axis parallel to its center of mass is equal to the moment of inertia about an axis through its center of mass plus the product of the body's mass and the square of the distance between the two axes.

2. How is the Parallel Axis Theorem used in rotational motion?

The Parallel Axis Theorem is used to calculate the moment of inertia of an object about an axis that is not through its center of mass. This is important in rotational motion, as the moment of inertia determines how difficult it is to change the rotational motion of an object.

3. What is an inertia tensor?

An inertia tensor is a mathematical representation of the distribution of mass within a rigid body. It is a 3x3 matrix that can be used to calculate the moment of inertia of an object about any given axis.

4. How is the inertia tensor related to the Parallel Axis Theorem?

The inertia tensor is used in the calculation of the moment of inertia in the Parallel Axis Theorem. The tensor contains information about the mass distribution of an object, which is necessary for calculating the moment of inertia about a specific axis.

5. Can the Parallel Axis Theorem be applied to all objects?

Yes, the Parallel Axis Theorem can be applied to all objects, as long as they are rigid bodies. It is a fundamental law of physics that applies to rotational motion and is used in many practical applications, such as engineering and mechanics.

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