# Circular Motion

In many natural events the motion is circular or nearly circular.A particle moving in a circular path is said to be in circular motion. It is another example of two-dimensional motion (Figure 4.8).

**Figure 4.8 Examples of Circular Motion**

*Motion of a particle along a circle with constant speed is called uniform circular motion*. In this case, the particle moves with a constant speed along the circular path and has an acceleration. This acceleration is due to the

*change in the direction of velocity*.

*Motion of a particle moving along a circle with varying speed is known as non-uniform circular motion*. In this case, the velocity of the particle changes both in magnitude and direction.

# Some Terms Related to Circular Motion

â€‹Consider a particle moving in a circle of radius

*r*

*with a constant speed*

*v*. Let us choose the centre of the circle

*O*

*as the origin, and the*

*XY*

*plane as the plane of motion as shown in Figure 4.9. Let the particle starts from point*

*A*

*on the*

*X*-axis. As the particle moves along the circle, the radius vector rotates in the plane.

**Figure 4.9 Circular Motion**

**Angular Displacement (**

*Î¸***)**The angle swept by the radius vector in a given time is called the angular displacement of the particle. If the particle is at

*A*

*at time*

*t*

*= 0 and at*

*B*

*at a later instant of time*

*t*, then the angular displacement of the particle is

*âˆ*

*AOB*

*=*

*Î¸*. It is associated with linear displacement

*AB*.

*Î¸*

*is the angle subtended at the centre of a circle of radius*

*r*

*by an arc of length*

*l*, then

*rad.*

*â‰ˆ*57.3Â°.

**Angular Speed (**

*Ï‰***)**The angular displacement of a particle per unit time is called the angular speed. It is denoted by

*Ï‰*

*and its unit is rad s*

^{âˆ’}

^{1}.

*Î¸*

*is the angular displacement in time*

*t*, then the angular speed

*Ï‰*

*=*

*rad s*

^{âˆ’}

^{1}.

**Period (T)**It is the time taken by the particle to complete one rotation. For one complete rotation, the angular displacement

*Î¸*

*= 2*

*Ï€*

*and the corresponding time taken is a period*

*T*. Hence,

*Ï‰*

*=*

*rad s*

^{âˆ’}

^{1}.

**Frequency (f)**It is the number of rotations completed by a particle in a circular motion in one second. The unit of frequency is hertz (Hz). Therefore,

**Relation between Linear Velocity and Angular Velocity**Let a particle in motion describe a circular path of radius

*r*, with a uniform speed

*v*. The velocity of the particle is directed tangentially at every point on the circular path. Let the particle move from a point

*A*

*to another point*

*B*

*in time*

*t*.

*AB*

*=*

*s*

*= Distance covered and*

*=*

*Î¸*

*= Angular displacement.*

*Ã—*

*Angle subtended at the centre of curvature*