Angular Kinematics Notes

Angular Distance vs. Angular Displacement

  • Angular Distance
    • Total length of angular path.
  • Angular Displacement
    • Equal in magnitude to the angle difference between initial and final position of an object.
  • Units
    • Example: Total from one swing is 85 degrees, and the finish is 30 degrees back.

Angular Kinematic Quantities

  • The angular displacement at the elbow is zero when the forearm returns to its original position upon completing a curl exercise.

Units of Angular Measure

  • 90 degrees = \frac{π}{2} radians = \frac{1}{4} revolution
  • 180 degrees = π radians = \frac{1}{2} revolution
  • 270 degrees = \frac{3π}{2} radians = \frac{3}{4} revolution
  • 360 degrees = 2π radians = 1 revolution

Angular Speed vs. Angular Velocity

  • Angular Speed
    • Angular distance / change in time.
  • Angular Velocity
    • Angular displacement / change in time.
    • \frac{Af – Ai}{t2 – t1}
  • Units
    • degrees/s or radians/s

Angular Acceleration

  • Rate of angular velocity change with respect to time.
  • \frac{Avf – Avi}{t2 – t1}
  • Units
    • Degrees/s^2 or rad/s^2
  • Average and instantaneous Angular Acceleration

Relationship between Linear and Angular Displacement

  • Given a point on a rotating body and the axis of rotation:
    • linear displacement
  • Curvilinear distance = radius of rotation * angular displacement
  • Two conditions must be met (not specified in the transcript).

Linear and Angular Velocity

  • Linear Velocity = Radius of rotation x angular velocity
  • Two Components (not specified in the transcript)
  • If angular velocity is constant:
    • RR = Linear Velocity
  • If linear velocity is constant:
    • RR < angular velocity

Linear and Angular Acceleration Combined

  • When an object accelerates along a curved path, it experiences both linear and angular acceleration simultaneously.
  • Two components to consider:
    • Tangential acceleration
    • Radial Acceleration

Linear and Angular Acceleration

  • Tangential Acceleration

    • Time rate change in linear velocity as it accelerates along a curved path.
    • Calculations
      • Tangential Acceleration = \frac{linear velocityf – linear velocityi}{change in time}
    • Increasing Linear (Tangential) Acceleration = radius of rotation x angular acceleration
      • Should be max just before release
      • Holding affect creates a radial acceleration tangentile point

  • Radial Acceleration (AKA Centripetal Acceleration)

    • Rate of change in direction of a rotating body as it moves along a curved path.
    • Restraining Force
      • Centripetal Force (force towards the center)
      • Centrifugal Force (apparent outward force)
    • Radial acceleration Formula
      • \frac{(linear velocity)^2}{radius of rotation}
    • Gforce ride pulling force out restraining force to the center of axis restrain force that pulls in

  • Small axis rotation greater centrifugal force larger axis smaller centrifugal force i.e. on a track to counter centrifugal force lean in to create centripetal force

  • 180 Daytona Ave speed = 210.30 mph Michigan Top Speed = 196.23 mph Banking is higher increasing centripetal force