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Motion in a plane


7. Uniform circular motion

  • When an object moves in a circular path at a constant speed then motion of the object is called uniform circular motion.
  • In our every day life ,we came across many examples of circular motion for example cars going round the circular track and many more .Also earth and other planets revolve around the sun in a roughly circular orbits
  • Here in this section we will mainly consider the circular motion with constant speed
  • if the speed of motion is constant for a particle moving in a circular motion still the particles accelerates becuase of costantly changing direction of the velocity.
  • Here in circular motion ,we use angular velocity in place of velocity we used while studying linear motion


(A) Angular velocity
  • Consider an object moving in a circle with uniform velocity v as shown below in the figure


    Uniform circular motion

    The velocity v at any point of the motion is tangential to the circle at that point.Let the particle moves from point A to point Balong the circumference of the circle .The distance along the circumference from A to B is
    s=Rθ                                          (23)
    Where R is the radius of the circle and θ is the angle moved in radian's
  • Magnitude of velocity is
    v=ds/dt=Rdθ/dt                                          (24)
    Since radius of the circle remains constant quantity,
    ω=dθ/dt                                          (25)
    is called the angular velocity defined as the rate of change of angle swept by radius with time.
  • Angular velolcity is expressed in radians per second (rads-1)
  • From equation 24 and 25,we find the following
    v=ωR                                          (26)
  • Thus for a particle moving ain circular motion ,velocity is directly proportional to radius for a given angular velocity
  • For uniform circular motion i.e, for motion with constant angular velocity the motion would be periodic which means particle passes through each point of circle at equal intervals of time
  • Time period of motion is given by
    T=2π/ω                                          (27)
    Since 2π radians is the angle θ in one revolution
  • If angular velocity ω is constant then integrating equation (25) with in limits θ0 to θ,we find


    where θ0 is the angular position at time t0 and θ is the angular position at time t .The above equation is similar to rectilinear motion result x-x0=v(t-t0)
(B) Angular acceleration
  • Angular acceleration is defined as the rate of change of angular velocity moving in circular motion with time.
    Thus
    α=dω/dt=d2θ/dt2                                          (29)
    Unit of angular acceleration is rads-2
  • For motion with constant angular acceleration


    or
    ω=ω0+α(t-t0)                                          (30)
    where ω0 is the angular velocity at time t0
    Again since
    ω=dθ/dt
    or dθ=ωdt then from equation 30

    If in the begining t0=0 and θ0=0 the angular position at any time t is given by
    θ=ωt+(1/2)αt2
    This result is of the form similar to what we find in case of uniformly accelerated motion while studying rectilinear motion




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