Consider the motion of a single particle P of mass m in a circular path of radius r. suppose this motion is taking place by attaching the particle P at the end of a mass-less rigid rod of length r whose other end is pivoted at the center O of the circular path, as shown in Fig. 5.1 (a). As the particle is moving on the circular path, the rod OP rotates in the plane of the circle.

The axis of rotation passes through the pivot O and is normal to the plane of rotation. Consider a system of axes as shown in fig. 5.1 (b).

The z-axis is taken along the axis of rotation with the pivot O as origin of coordinates axes x and y are taken in the plane of rotation. While OP is rotating, suppose at any instant t, its position is OP1, making angle ϴ with x-axis. At later time t + ∆t, let its position be OP2 making angle Ө + ∆Ө with x-axis (Fig. 5.1c).

 circular-path-1 circular-path-2 circular-path-3  circular-path-4

Angle ∆ϴ defines the angular displacement of OP during the time interval t.

For very small values of ∆Ө, the angular displacement is a vector quantity.

The angular displacement ∆ϴ is assigned a positive sign when the sense of rotation of OP is counter clock wise.

The direction associated with ∆Ө is along the axis of rotation and is given by right hand rule which states that

Grasp the axis of rotation in right hand with fingers curling in the direction of rotation; the thumb points in the direction of angular displacement, as shown in Fig 5.1 (d).

Three units are generally used to express angular displacement, namely degrees, revolution and radian. We are already familiar with the first two. As regards radian which is SL unit, consider an arc of length S of a circle of radius r (fig 5.2) which subtends an angle 0 at the centre of the circle. Its value in radians (rad) is given as

circular-path-5

Fig. 5.2

angular-displacement-equation-01

Or                        S = rӨ           (where Ө is in radian)   ………       (5.1)

If OP is rotating, the point P covers a distance s = πr in one revolution of P. in radian it would be

angular-displacement-equation-02