If the axis of rotation is not passing through the centre of mass but some other point of a rigid body then how do we define the direction of angular parameters that is angular velocity, angular acceleration and angular displacement.

For example :enter image description here

This is a sphere with a void in between rolling on a flat surface.C is the centre of mass and O is the axis of rotation.

So basically my question is, will the angular velocity of the point A be perpendicular to OA or CA? Also will the angular acceleration of A be parallel or antiparallel to the angular velocity and correspondingly the centripetal acceleration would be perpendicular to the angular velocity right? I am confused about this.


2 Answers 2


If the body is rigid the the angular velocity is the same about any point on the body!

Imagine a line between $C$ and $O$ and relative to $O, \, C$ rotates by an angle $\theta$ clockwise ie the body has moved to the right.

Now consider that line from the point of view of position $C$.
$O$ has rotated by an angle $\theta$ clockwise whilst the body has moved to the right.

The webpage Rigid Bodies has some nice animations to illustrate this.

If there is no slipping then every point on the body is has a horizontal component of translational velocity $v = R\,\omega$ to the right as well as a component of translational velocity due to rotation about position $O$ which for position $C$ would be $\frac R2\,\omega$ vertically upwards.


The angular velocity, is perpendicular to OA, therefore the centripetal a ist in direction AO, in the moment of your sketch, the Point C is left of A, so the weight is slowing down the rotation, if C is right of A it is increasing the rotation in your indicated direction of angular velocity


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