Circular Motion
To create circular motion in a body it must be given some initial velocity and a force must then act on the body which is always directed at right angles to instantaneous velocity.
Since this force is always at right angles to the displacement due to the initial velocity, therefore no work is done by the force on the particle. Hence, its kinetic energy and thus speed is unaffected.
Variables of circular motion
Displacement The change of position vector or the displacement of the particle from position A to the position B is given by referring Fig 12.
â‡’
Fig. 12
Putting r_{1} = r_{2} = r we obtain
â‡’
Distance The distanced covered by the particle during the time t is given as
d = length of the arc AB = rÎ¸
Angular displacement (Î¸) The angle turned by a body moving on a circle from some reference line is called angular displacement. Angular displacement is an axial vector quantity.
Relation between linear displacement and angular displacement,
or s = rÎ¸
Angular velocity (Ï‰) Angular velocity of an object in circular motion is defined as the time rate of change of its angular displacement, i.e.
 Angular velocity is an axial vector.
 Relation between angular velocity and linear velocity:
Note: It is important to note that nothing actually moves in the direction of the angular velocity vector . The direction of simply represents that the rotational motion is taking place in a plane perpendicular to it.
 For uniform circular motion, Ï‰ remains constant, whereas for nonuniform motion, Ï‰ varies with respect to time.
Change in velocity A particle is performing uniform circular motion as it moves from A to B during time t, as shown in Fig. 13(a). The change in velocity vector is given as
Fig. 13
or â‡’
For uniform circular motion, v_{1} = v_{2} = v.
So
Relation between linear velocity and angular velocity in vector form,
Time period (T) In circular motion, time period is defined as the time taken by the object to complete one revolution on its circular path.
Frequency (n) In circular motion, frequency is defined as the number of revolutions completed by the object on its circular path in a unit time.
Notes:
 Relation between angular velocity, frequency, and time period:
 If two particles are moving on same circle or different coplanar concentric circles in same direction with different uniform angular speeds Ï‰_{A} and Ï‰_{B }respectively, the angular velocity of B relative to A will be
Special case: If Ï‰_{B} = Ï‰_{A}, Ï‰_{rel} = 0, and so T= âˆž., particles will maintain their position relative to each other. This is what actually happens in case of geostationary satellite (Ï‰_{1} = Ï‰_{2} = constant).
Angular acceleration (Î±) Angular acceleration of an object in circular motion is defined as the time rate of change of its angular velocity.
 Units: rad. s^{â€“2}
 Dimension: [M^{0}L^{0}T^{â€“2}]
 Relation between linear acceleration and angular acceleration
Centripetal acceleration
 Acceleration acting on the object undergoing uniform circular motion is called centripetal acceleration.
 It always acts on the object along the radius towards the center of the circular path (Fig. 14).
 Magnitude of centripetal acceleration
Fig. 14
 The centripetal acceleration vector acts along the radius of the circular path at that point and is directed towards the center of the circular path.
Equations of circular motion
For accelerated motion 
For retarded motion 








where
Ï‰_{1} = Initial angular velocity of particle
Ï‰_{2} = Final angular velocity of particle
Î± = Angular acceleration of particle
Î¸ = Angle covered by the particle in time t
Î¸_{n} = Angle covered by the particle in nth second