Uniform Circular Motion

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  • The motion of a particle along the circumference of a circle with a constant speed is called uniform circular motion (U.C.M.).
  • Examples: The motion of the earth around the sun, The motion of an electron around the nucleus.

The Period of Revolution of Uniform Circular Motion:

  • The time taken by a particle performing uniform circular motion to complete one revolution is called as the period of revolution or periodic time or simply period (T).
  • It is denoted by ‘T’. The S. I. Unit of the period is second (s). Its dimensions are[MºLºT 1].

The Frequency of Revolution of Uniform Circular Motion:

  • The number of revolutions by the particle performing uniform circular motion in unit time is called as frequency (n) of revolution.
  • The frequency is denoted by letter ‘n’ or ‘f’. The S. I. Unit of frequency is hertz (Hz). Its dimensions are [MºLºT-1].
  • In time T the particles complete one revolution. Thus the particle completes 1/T revolutions in unit time. Thus n = 1/T.

Relation Between Linear Velocity and Angular Velocity:

  • Consider a particle performing uniform circular motion, along the circumference of the circle of radius ‘r’ with constant linear velocity ‘v’ and constant angular speed ‘ω’ moving in the anticlockwise sense as shown in the figure.
  • Suppose the particle moves from point P to point Q through a distance ‘δx’along the circumference of the circular path and subtends the angle ‘δθ’ at the centre O of the circle in a small interval of time ‘δt’. By geometry

δx = r . δθ

  • If the time interval is very very small then arc PQ can be considered to be almost a straight line. Therefore magnitude of linear velocity is given by

  • Thus the linear velocity of a particle performing uniform circular motion is radius times its angular velocity. In vector form above equation can be written as

  • The linear velocity can be expressed as the vector product of angular velocity and radius vector.
  • The following figure shows relative positions of the linear velocity vector, angular velocity vector, and radius or position vector.

Uniform Circular Motion

Proof of  

  • For smaller magnitudes angular displacement, angular velocity are vector quantities. Let rbe the position vector of the particle at some instant. Let the angular displacement in small time δt be (dq). Let the corresponding linear displacement (arc length) be ( ds). By geometry

Dividing both sides of the equation by δt and taking the limit

Expression for Period of Revolution of Uniform Circular Motion:

  • Let us consider particle performing a uniform circular motion. Let ‘T’ be its period of revolution. During the periodic time (T), particle covers a distance equal to the circumference 2pr of the circle with linear velocity v.

This is an expression for the period of revolution for particle performing the uniform circular motion.

The Expression for Angular velocity:

  • The angular velocity of a body performing uniform circular motion is given by  ω = θ / t
  • In one period i.e. in time T seconds, the body performing uniform circular motion traces an angle of 2p radians.

Where ‘n’ is the frequency of U.C.M. and ‘N’ is the angular speed of the body in r.p.m.

The Expression for Angular Acceleration:

  • When a body is performing non-uniform circular motion, its angular velocity changes. Hence the body possesses angular acceleration.
    The rate of change of angular velocity w.r.t. time is called as the angular acceleration.
  • We know that acceleration is the rate of change of velocity with respect to time.

r = radius of circular path = constant.

ω = angular velocity of the particle performing a circular motion

Circular Motion

  • Where ‘α’ is angular acceleration. Hence, linear acceleration = radius x angular acceleration.
  • If speed is increasing linear acceleration is in the same direction as that of linear velocity. If speed is decreasing linear acceleration is in the opposite direction to that of linear velocity. It is also referred as tangential acceleration. For uniform circular motion α = 0.
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