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Circular Motion

In many natural events the motion is circular or nearly circular.
For example, motion of electrons around the nucleus, motion of earth around the sun, etc., are 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).
Description: Description: FIG-4.8.tif
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.
Presently, we discuss uniform circular motion only.

Some Terms Related to Circular Motion

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.
Description: Description: 58386.png
Figure 4.9 Circular Motion
Angular Displacement (θ) 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 AOB = θ. It is associated with linear displacement AB.
The SI unit of angular displacement is radian (rad).
If θ is the angle subtended at the centre of a circle of radius r by an arc of length l, then Description: 56657.png rad.
1 rad ≈ 57.3°.
Angular Speed (ω) The angular displacement of a particle per unit time is called the angular speed. It is denoted by ω and its unit is rad s1.
If θ is the angular displacement in time t, then the angular speed ω = Description: 56673.png rad s1.
Period (T) It is the time taken by the particle to complete one rotation. For one complete rotation, the angular displacement θ = 2π and the corresponding time taken is a period T. Hence, ω = Description: 56698.png rad s1.
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.
Let AB = s = Distance covered and Description: 56711.png = θ = Angular displacement.
Angular velocity
Arc length = Radius × Angle subtended at the centre of curvature

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