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Motion in a Circular Path

Bodies moving in a circular path are said to be in circular motion. In everyday life, in astronomy, in atomic physics etc., we have a number of examples of objects moving in almost nearly circular paths.

Examples of Circular Motion
  • Rotation of rigid body about an axis
  • Rotation of automobiles wheels the spinning of earth about its axis
  • The spinning of earth about its axis
  • Motion of planets around the sun
  • Motion of electrons around the nucleus of an atom etc.
There are two types of circular motion
  1. Uniform circular motion and
  2. Non-uniform circular motion.
In uniform circular motion the particles in circular path moves with a uniform speed. The particle covers equal distance in equal intervals of time. But in non-uniform circular motion, the speed of the particle in circular motion is different at different points along the circular path.

Angular Variables

  1. Angular displacement
    Consider a body moving along a circular path or radius r. At any instant the particle is at A. After some time the particle reaches the point B. We say there is a change in its ‘angular position’. In fig. 1(a) OA is the initial position vector and OB the final position vector.

    The angular displacement is the angle swept out by the radius vector in a given time interval AOB = θ is the angular displacement.

    If the angular displacement is small, it is a vector quantity. It is directed along the axis of rotation and its sense (Upward or downward) is given by the right hand screw rule as shown in figure 1(c).

    The angular displacement is measured in radian. In figure 1(a) let the length of the arc AB be s.

    Then the angle θ in radian =
    θ =

    when θ = 3600, the length of the arc, which is the circumference of the circle is 2π r.
    3600 = 2 π radian

    The relation connecting angle in degree and angle in radian is,
    3600 = 2π radian
    10 = radian
    1 radian =
    1 radian = 57.270
    Note: If we use π = 3.14, 1 radian = 57.270
  2. Angular velocity
    The magnitude of the angular velocity is the rate of change of angular displacement with time or it is the angle swept out by the radius vector is one second. Angular velocity is denoted by ω.

    Let θ be the angle subtended at the centre of the circle in t seconds by a body in uniform circular motion then ω=

    Suppose Δ Q is a very small angular displacement made in an interval of time Δ t. Then the instantaneous angular velocity,
    ω =
    If θ is in radian, and t in second then the angular velocity is in radian/s.
    Dimensional formula of ω is [T-1].

Angular velocity ω is a vector

Its direction is taken along the axis of rotation ω is directed out of the page, parallel to the rotation axis, if the rotation is anticlockwise (counterclockwise).

If the rotation is clockwise ω is directed into the page as shown in figure 2b, the direction of the angular velocity ω can be identified by curling the fingers of the right hand around the axis of rotation in the direction of rotation. Then the right thumb will point in the direction of ω. (See figure 2c) [known as right hand thumb rule].

###SUB-TOPIC###Linear Displacement in terms of Angular Displacement###
We have seen that the angular displacement in radian is θ = where s is the length of the arc AB. When the angle θ is very small s can be assumed to be almost straight line s= rθ . If is a small angular displacement corresponding to the linear displacement then =.

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