takes smaller and smaller values), the ratio ∆ θ / ∆ t approaches a limit which is the instantaneous angular velocity d θ /d t of the particle at the position P. We denote the instantaneous angular velocity by ω (the Greek letter omega). We know from our study of circular motion that the magnitude of linear velocity v of a particle moving in a circle is related to the angular velocity of the particle ω by the simple relation υ , where r is the radius of the circle. We observe that at any given instant the relation v applies to all particles of the rigid body.
Thus for a particle at a perpendicular distance r i from the fixed axis, the linear velocity at a given instant v i is given by ( . ) The index i runs from to n , where n is the total number of particles of the body. For particles on the axis, , and hence v = ω r = . Thus, particles on the axis are stationary.
This verifies that the axis is fixed . Note that we use the same angular velocity ω for all the particles. We therefore, refer to ωωωωω as the angular velocity of the whole body . We have characterised pure translation of a body by all parts of the body having the same velocity at any instant of time.
Similarly, we may characterise pure rotation by all parts of the body having the same angular velocity at any instant of time . Note that this characterisation of the rotation of a rigid body about a fixed axis is just another way of saying as in Sec. . that each particle of the body moves in a circle, which lies in a plane perpendicular to the axis and has the centre on the axis.
In our discussion so far the angular velocity appears to be a scalar. In fact, it is a vector. We shall not justify this fact, but we shall accept it. For rotation about a fixed axis, the angular velocity vector lies along