Circular Motion

Introduction:

                  Many objects move in circular paths and their direction is continually changing. Since acceleration is a vector quantity, this change of direction means that their acceleration are not constant. A stone whirled around by a string, a car turning around a corner and satellites in orbits around the Earth are all examples of Circular Motion.

Angular Displacement:

                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 (a). As the particle is moving on the circular path, the rod OP rotates in the plane of circle. The axis of rotation passes through the pivoted O and is normal to the plane of rotation. Consider a system of axes as shown in fig (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 , making angle with x-axis. At later time ,let its position be making angle with x-axis fig (c).

Angle defines the angular displacement of OP during the time interval . 

    The angular displacement ∆Ѳ is assigned a positive sign when the sense of rotation of OP is counter clockwise.

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

    The direction associated with ∆Ѳ 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 figure-(d)