- #1

In a previous thread it was concluded that both torque and angular momentum are taken about a chosen origin, and these quantities are generally not invariant under translations of the origin (since ##\vec{r}## changes but ##\vec{v}## does not, etc.). The moment of inertia tensor doesn't implicitly refer to an origin, but it's value depends on the position in space (i.e. the MOI tensor at the centre of mass is different to that at a point on the edge of the body).

However, I did some more reading and found that the angular velocity vector ##\vec{\omega}## as well as the angular acceleration ##\vec{\alpha}## are invariant under translations of the origin. This doesn't make much sense, because if for instance ##\vec{v} = \vec{\omega} \times \vec{r}##, and ##\vec{v}## is invariant under origin translations but ##\vec{r}## definitely is not, then ##\vec{\omega}## has to change also!

I tried looking in Morin but he just mentions that ##\vec{\omega}## is parallel to the axis of rotation. On the one hand, it makes sense to speak of the ##\vec{\omega}## of a rigid body about it's centre of mass, which is surely origin invariant (as long as the basis vectors are still oriented in the same directions). However, if now choose an origin in the corner of the lab in which the body is spinning, wouldn't ##\vec{\omega}## be different for every point on the body?

However, I did some more reading and found that the angular velocity vector ##\vec{\omega}## as well as the angular acceleration ##\vec{\alpha}## are invariant under translations of the origin. This doesn't make much sense, because if for instance ##\vec{v} = \vec{\omega} \times \vec{r}##, and ##\vec{v}## is invariant under origin translations but ##\vec{r}## definitely is not, then ##\vec{\omega}## has to change also!

I tried looking in Morin but he just mentions that ##\vec{\omega}## is parallel to the axis of rotation. On the one hand, it makes sense to speak of the ##\vec{\omega}## of a rigid body about it's centre of mass, which is surely origin invariant (as long as the basis vectors are still oriented in the same directions). However, if now choose an origin in the corner of the lab in which the body is spinning, wouldn't ##\vec{\omega}## be different for every point on the body?

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