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ANGULAR VELOCITY

Angular velocity is a vector quantity which is defined for a rigid body or a frame of reference. Let b1, b2 and b3 be three perpendicular unit vectors and form a right-handed frame of reference which is fixed on rigid body B moving in a reference frame A. Angular velocity of the body B is defined as
(1) ωB ⁄ A = b1b2̇b3 +  b2b3̇b1 +  b3b1̇b2
Using this definition, if β is a unit vector that is fixed in the rotating frame (or attached to the rigid body) B with respect to A, then the time derivation of β (β̇) is defined as
(2) β̇ = ωB ⁄ A × β
2D OR PLANAR ANGULAR VELOCITY
Let frame B (with unit vectors b1, b2 and b3) rotate with respect to frame A (with unit vectors a1, a2 and a3). Figure 1↓ shows a special case when B rotates about b3, which is also equal to a3 . (This is a rotation about an the axis normal to the page.) At any point in time, the unit vectors b are oriented at an angle θ with respect to the unit vectors a. The time rate of change of θ is the magnitude of angular velocity. The direction of angular velocity in this case is b3 =  a3.
figure images/c12.png
Figure 1 Planar angular velocity-- a1 −  a2 plane

(3) ωB ⁄ A = θ̇b3 = θ̇a3
Similarly in the case shown in Fig. 2↓, angular velocity is
(4) ωB ⁄ A = θ̇b1 = θ̇a1
figure images/c23.png
Figure 2 Planar angular velocity-- a2 −  a3 plane

and in the case shown in Fig. 3↓, angular velocity is
(5) ωB ⁄ A = θ̇b2 = θ̇a2
figure images/c31.png
Figure 3 Planar angular velocity-- a3 −  a1 plane

These cases are called simple rotation.
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