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But when we talk about inertia tensor, we calculate about a point. Is there a reason for this difference? Am I missing something?

I am new to tensors.

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- Thread starter dpa
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But when we talk about inertia tensor, we calculate about a point. Is there a reason for this difference? Am I missing something?

I am new to tensors.

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SteamKing

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arildno

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Do remember though, that ESSENTIALLY, all torques are computed relative to a POINT, not relative to an axis.

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Very. One simple example: Perform rotation A followed by rotation B. Then do it again, but this time perform rotation B first. It doesn't make any difference with 2-D rotations. Rotations in 2-D are commutative. That's no longer true in three dimensions (or higher).3-D rotations are nasty.

Another example is the inescapable fact that angular velocity and angular momentum do not necessarily point in the same direction with 3-D rotations. A rigid body set into rotation in empty space generally will tumble. Angular momentum is a conserved quantity but angular velocity is not.

Rotations in 4-D space or higher are nastier yet. The concept that rotation is about an axis is something that pertains to 3-D space only. On the other hand, the 2-D concept of rotation about a point (or parallel to a plane) does generalize to higher dimensions.

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arildno

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However, for FREE rotations in 3-D, a beatiful result is still present, as my new signature shows.

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Yep. Love that expression. It's so nonsensical (or at least counterintuitive), but then again, nonsensical (or at least counterintuitive) is a perfect description of free rotations in 3-D space.However, for FREE rotations in 3-D, a beatiful result is still present, as my new signature shows.

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arildno

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Yes. You are missing something. You were taught in freshman physics that ##T=I\alpha## is the rotational equivalent of ##F=ma##. That's only true for those special cases where you can ignore the tensorial nature of the inertia tensor, or where motion is constrained to two dimensions. That was a "lie to children" (http://en.wikipedia.org/wiki/Lie-to-children). It's more complicated in general, and you are now deemed to have adequate mathematical understanding to *start* taking the next step toward understanding rotation.But when we talk about inertia tensor, we calculate about a point. Is there a reason for this difference? Am I missing something?

Filling in that "missing something" starts with understanding Euler's equations. Euler didn't use tensors, so his notation is a bit verbose. It becomes nice and simple with a tensorial notation. To really understand rotations you need to know a bit about group theory (the group SO(3), in particular) and you need to know a bit about Lie groups / Lie algebras.

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"we talk about rotation and inherently, we talk about moment of inertia about an axis."

As DH points out, freshman courses speedily goes over to rotation about a fixed axis.

This means it is very easy to forget what was breezily derived at the very start of the course, namely that when we compute the torques about a (fixed) point, we gain the formula:

[tex]\vec{\tau}=\frac{d\vec{L}}{dt}[/tex]

where [itex]\vec{\tau}[/itex] are the externally applied torque, and [itex]\vec{L}[/itex] the quantity called "angular momentum".

THAT equation is (almost*) perfectly general for torques about a fixed point in classical mechanics, but it is a wolf in sheep's clothing.

In freshman courses, most of the wolf's teeth are pulled out to begin with (DH has given you the names of a few of those teeth), by saying we limit ourselves to cases of rotation about a fixed axis.

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*It holds perfectly for the ideal, perfectly rigid body. For other types of objects, the left hand side of the equation might get a lot nastier, not just RHS (which is, in 3-D, already exceedingly nasty for the perfectly rigid body as well).

As DH points out, freshman courses speedily goes over to rotation about a fixed axis.

This means it is very easy to forget what was breezily derived at the very start of the course, namely that when we compute the torques about a (fixed) point, we gain the formula:

[tex]\vec{\tau}=\frac{d\vec{L}}{dt}[/tex]

where [itex]\vec{\tau}[/itex] are the externally applied torque, and [itex]\vec{L}[/itex] the quantity called "angular momentum".

THAT equation is (almost*) perfectly general for torques about a fixed point in classical mechanics, but it is a wolf in sheep's clothing.

In freshman courses, most of the wolf's teeth are pulled out to begin with (DH has given you the names of a few of those teeth), by saying we limit ourselves to cases of rotation about a fixed axis.

-----------------------------------------------------------------------

*It holds perfectly for the ideal, perfectly rigid body. For other types of objects, the left hand side of the equation might get a lot nastier, not just RHS (which is, in 3-D, already exceedingly nasty for the perfectly rigid body as well).

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