Angular motion about a point

Question

If a rigid body is undergoing pure rotational motion about its centre of mass, we know that angular speed of all points on the rigid body is same with respect to the centre of mass. Then let's say if we take another arbitrary point, P outside the body and calculate angular velocity of points on the rigid body about P, will all the points on the rigid body have same angular velocity with respect to point P?

Answer 1

The angular velocity $\boldsymbol{\omega}$ of a rigid body is not defined w.r.t. any point, but it can be defined as the (pseudo)vector that relates the difference of velocity of 2 points with their difference in position,

$\mathbf{v}_B - \mathbf{v}_A = \boldsymbol{\omega} \times (\mathbf{r}_B - \mathbf{r}_A)$,

for every pair of material points $A$, $B$ of the rigid body.

Answer 2

Each point of a rigid body (not only the COM) can be taken as a centre of rotation. All the other points have the same instantaneous angular velocity with respect to it. The angular velocity is defined as the vector $\boldsymbol \omega$ such that: $\mathbf v_i = \boldsymbol \omega \times \mathbf r_i$

But if the point is not in the body, so that the its distance to each of its point is not constant, then $\boldsymbol \omega$ is not the same for the $\mathbf r_i$'s.

Answer 3

I don't think your original statement (that all points in the rigid body have equal angular velocity) is even true. A point exactly at the center of mass has $\omega=0$, where $\omega$ is the angular velocity, while the angular velocity gets larger and larger as the distance from the COM increases. Thus, this identity does not hold for a point $P$ outside of the body either.

What might be more useful is to think about angular velocity in terms of the moment of inertia of a body, which is defined as $\int\rho r^2dV$, where $\rho$ is the density, $r$ is the distance from the COM, and the integral is over all points in the body. Then, the Parallel Axis Theorem can be used to find the moment of inertia about points other than the COM.