Consistency
We have supposed that the rigid body rotates around an arbitrary point. We should prove that the angular velocity previously defined is independent from the choice of origin, which means that the angular velocity is an intrinsic property of the spinning rigid body.
See the graph to the right: The origin of lab frame is O, while O1 and O2 are two fixed points on the rigid body, whose velocity is and respectively. Suppose the angular velocity with respect to O1 and O2 is and respectively. Since point P and O2 have only one velocity,
The above two yields that
Since the point P (and thus ) is arbitrary, it follows that
If the reference point is the instantaneous axis of rotation the expression of velocity of a point in the rigid body will have just the angular velocity term. This is because the velocity of instantaneous axis of rotation is zero. An example of instantaneous axis of rotation is the hinge of a door. Another example is the point of contact of a pure rolling spherical rigid body.
Read more about this topic: Angular Velocity, Rigid Body Considerations
Famous quotes containing the word consistency:
“The lawyer’s truth is not Truth, but consistency or a consistent expediency.”
—Henry David Thoreau (1817–1862)
“All religions have honored the beggar. For he proves that in a matter at the same time as prosaic and holy, banal and regenerative as the giving of alms, intellect and morality, consistency and principles are miserably inadequate.”
—Walter Benjamin (1892–1940)