Inertia tensor of a body rotating about 3 axes

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Discussion Overview

The discussion revolves around the inertia tensor of a rigid body, specifically addressing how it behaves when the body rotates about multiple axes, such as in the case of a spacecraft. Participants explore the implications of this multi-axis rotation on the inertia tensor's representation and its components.

Discussion Character

  • Homework-related
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about the inertia tensor for a body rotating around three axes, seeking clarification on its representation.
  • Another participant asserts that the inertia tensor can be represented as a symmetric 3x3 matrix regardless of the axis orientations, emphasizing that the matrix remains full and symmetric.
  • A participant provides an example involving a cube and questions whether its inertia tensor differs when rotating around multiple axes compared to just one axis.
  • It is noted that the mass moment of inertia tensor is defined by the mass distribution within the body and does not change with the axis of rotation, although some components may become irrelevant for specific rotations.
  • One participant clarifies that a body cannot rotate about multiple axes simultaneously in the context of angular momentum representation, which is defined by a single vector direction.
  • Another participant counters this by stating that the angular velocity vector can have components along all three axes, suggesting a simultaneous rotation interpretation.

Areas of Agreement / Disagreement

Participants express differing views on whether a body can rotate about multiple axes simultaneously, leading to a lack of consensus on this aspect of angular momentum and inertia tensor representation.

Contextual Notes

Some assumptions regarding the definitions of rotation and inertia tensor components may not be fully articulated, and the discussion does not resolve the implications of multi-axis rotation on the inertia tensor.

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Homework Statement


Hello,
I know about the inertia tensor about one axis, but how about a body that rotates around 3 axis x,y and z such as a spacecraft with changes in the attitude.

Thanks for you help.

Homework Equations

The Attempt at a Solution

 
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What do you mean when you say, "How about a body that rotates around 3 axes ..."

The inertia tensor remains representable as a 3x3 matrix, just as before. For arbitrary axis orientations, the matrix is symmetric and full (no zero elements).
 
Hello, thank you for your response.

let's say we have a cube with coordinate frame at its centre. The body can rotate about z, y and x. In that case would the inertia tensor be different from the common inertia tensor of a cube that rotates only around z?

Thank you again
 
For a three dimensional rigid body, the mass moment of inertia tensor can be fully represented by a symmetric 3x3 matrix. If you look at the definition of each of the elements, they each depend only on the distribution of mass within the body. They do not depend on the axis of rotation; there may not be any axis of rotation defined.

If you now restrict rotation to one axis, most of the components of the inertia matrix become irrelevant, but that does not mean that they are changed. The just no longer contribute to the angular momentum or the kinetic energy.
 
The body can rotate about z, y and x.

It can't do that simultaneously, if that's what you're thinking. The body's angular momentum is represented by a vector--and that direction of that vector is the axis about which it rotates. The axis may not coincide with any of the coordinate axes, but it is a single axis.
 
Thank you all for your explanations. it is clear in my head now :)
 
John Park's statement is entirely correct, but I would like to add that the angular velocity vector can have components in all three axes. This can be understood as saying that it is rotating about all three axes simultaneously.
 

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