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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 32350.pngof the particle from position A to the position B is given by referring Fig 12.
⇒ 32337.png
Fig. 12
Putting r1 = r2 = r we obtain
Distance The distanced covered by the particle during the time t is given as
d = length of the arc AB
Angular displacement (θ) 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, 32313.png
or    s = 
Angular velocity (ωAngular 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: 32295.png
Note: It is important to note that nothing actually moves in the direction of the angular velocity vector 32935.png. The direction of 32929.png simply represents that the rotational motion is taking place in a plane perpendicular to it.
  • For uniform circular motion, ω remains constant, whereas for non-uniform motion, ω 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 32252.png  32245.png
For uniform circular motion, v1 = v2 = v.
So 32239.png
Relation between linear velocity and angular velocity in vector form, 32233.png 
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.


  • 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 ωA and ωB respectively, the angular velocity of B relative to A will be
    So the time taken by one to complete one revolution around O with respect to the other (i.e., time in which B completes one revolution around Owith respect to the other),
    32906.png 32900.png
Special case: If ωB = ωAωrel = 0, and so T= ∞., particles will maintain their position relative to each other. This is what actually happens in case of geostationary satellite (ω1 = ω2 = constant).


Angular acceleration (α) Angular acceleration of an object in circular motion is defined as the time rate of change of its angular velocity.
  • Units: rads–2
  • Dimension: [M0L0T–2]
  • Relation between linear acceleration and angular acceleration 32196.png
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









ω1 = Initial angular velocity of particle
ω2 = Final angular velocity of particle
α = Angular acceleration of particle
θ = Angle covered by the particle in time t
θn = Angle covered by the particle in nth second

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