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.
  • $\frac{A<em>f – A</em>i}{t<em>2 – t</em>1}$
• Units
  • degrees/s or radians/s

Angular Acceleration

• Rate of angular velocity change with respect to time.
• $\frac{Av<em>f – Av</em>i}{t<em>2 – t</em>1}$
• 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 = $\frac{linear velocity<em>f – linear velocity</em>i}{change in time}$
  • 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