Circular Motion

Uniform Circular Motion:

When an object moving with constant speed has a path along a circle with radius r.

Period and Frequency

  • Period (T): the time taken for one complete revolution, measured in seconds (s)
  • Frequency (f): the amount of complete revolutions done in one second, measured in Hertz (Hz)
  • The period and frequency of rotations are inversely proportional: T = 1/f

Position and Speed

    Distance on a Circle

  • There are two ways to find the distance travelled by an object moving along a circular path:
    • The angle the body has covered, θ, starting from a given point (could be degrees or radians)
    • The length of the circular path travelled, s, starting from a given point: s = θr (where θ is the angle in radians)

    Velocity (v):

  • The rate of change of position on the circle (length of the arc covered in a specific amount of time)
  • v = 2πr/T, because v = s/T where s = 2πr (when the object completes one circle)
  • The direction of velocity is always tangent to the circle at the object's location

    Angular Velocity (ω):

  • The rate of change of the angle covered by the object (measured in radians per second)
  • ω = Δθ/Δt, or for one complete circle: ω = 2π/T
  • Hence, v = ωr

Centripetal Acceleration

  • As the direction of velocity is constantly changing for a body moving around a circle, it has acceleration
  • This acceleration is called the centripetal acceleration (a):
    • a = v2/r
    • a = ω2r
    • a = 4π2r/T2
  • The change in velocity points towards the center of the circle, hence the centripetal acceleration also points towards the center of the circle

Centripetal Force

  • "Force" responsible for the centripetal acceleration, F = mv2/r
  • Acts perpendicular to the velocity
  • Causes the body to change direction and accelerate even though the speed is constant
  • Represents a net force and itself is not a real force (i.e. it is always provided by other forces, such as tension forces, frictional forces, normal forces, etc.)