ω r ⊥ and is along the tangent to the circle described by the particle at P. The linear velocity vector v at P has the same magnitude and direction. Thus, v = ω ω × r ( . ) In fact, the relation, Eq.
( . ), holds good even for rotation of a rigid body with one point fixed, such as the rotation of the top [Fig. . (a)].
In this case r represents the position vector of the particle with respect to the fixed point taken as the origin. We note that for rotation about a fixed axis, the direction of the vector ωωωωω does not change with time. Its magnitude may, however, change from instant to instant. For the more general rotation, both the magnitude and the direction of ω ω may change from instant to instant.
. . Angular acceleration You may have noticed that we are developing the study of rotational motion along the lines of the study of translational motion with which we are already familiar. Analogous to the kinetic variables of linear displacement (s) and velocity ( v ) in translational motion, we have angular displacement ( θθθθθ ) and angular velocity ( ωωωωω ) in rotational motion.
It is then natural to define in rotational motion the concept of angular acceleration in analogy with linear acceleration defined as the time rate of change of velocity in translational motion. We define angular acceleration ααααα as the time rate of change of angular velocity. Thus, d t ( . ) If the axis of rotation is fixed, the direction of ω ω and hence, that of ααααα is fixed.
In this case the vector equation reduces to a scalar equation d t α = ( . ) . TORQUE AND ANGULAR MOMENTUM In this section, we shall acquaint ourselves with two physical quantities (torque and angular momentum) which are defined as vector products of two vectors. These as we shall see, are especially important in the discussion of motion of systems of particles, particularly rigid bodies.
. . Moment of force (Torque) We have learnt that the motion of a rigid body, in general,