Inertia tensor of a body rotating about 3 axes

[Sed]

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

 

[Dr.D]

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).

 

[Sed]

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

 

[Dr.D]

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.

 

[John Park]

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.

 

[Sed]

Thank you all for your explanations. it is clear in my head now :)

 

[Dr.D]

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.