page_title: Angular velocity
section_title: Consistency
hierarchical_section_title: Angular velocity / Rigid body considerations / Consistency
context_page_description: In physics, angular velocity refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time. There are two types of angular velocity: orbital angular velocity and spin angular velocity. Spin angular velocity refers to how fast a rigid body rotates with respect to its centre of rotation. Orbital angular velocity refers to how fast a point object revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. Spin angular velocity is independent of the choice of origin, in contrast to orbital angular velocity which depends on the choice of origin.
  In general, angular velocity is measured in angle per unit time, e.g. radians per second. The SI unit of angular velocity is expressed as radians per second with the radian having a dimensionless value of unity, and thus the SI units of angular velocity are oftentimes listed as simply 1/s or s⁻¹. Angular velocity is usually represented by the symbol omega. By convention, positive angular velocity indicates counter-clockwise rotation, while negative is clockwise.
context_section_description: We have supposed that the rigid body rotates around an arbitrary point. We should prove that the spin angular velocity previously defined is independent of the choice of origin, which means that the spin angular velocity is an intrinsic property of the spinning rigid body. (Note the marked contrast of this with the orbital angular velocity of a point particle, which certainly does depend on the choice of origin.)
  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 the velocity of a point in the rigid body will have just the angular velocity term. This is because the velocity of the instantaneous axis of rotation is zero. An example of the instantaneous axis of rotation is the hinge of a door. Another example is the point of contact of a purely rolling spherical (or, more generally, convex) rigid body.
caption: Proving the independence of spin angular velocity from choice of origin