Angular Motion

Measuring a Circle  We use degrees to measure position around the circle.  There are 2  radians in the circle. This matches 360°This matches 360°  The distance around a circle is s = r , where  is in radians.  r  The angular displacement is 

Angular Velocity  For circular motion, only the time rate of change of the angle matters.  The time rate of change of the angle is called the angular velocity. Symbol (  )Symbol (  ) Units (rad/s or 1/s = s -1 )Units (rad/s or 1/s = s -1 )

Velocity and Angular Velocity  Velocity has an angular equivalent. Linear velocity (v)Linear velocity (v) Angular velocity (  )Angular velocity (  )  They are related, since the displacement is related to the angle.

Cycles or Radians  Frequency is measured in cycles per second.  There is one cycle per period.  Frequency is the inverse of the period, f =1/T.  Angular velocity is measured in radians per second.  There are 2  radians per period.  Angular velocity,  = 2  / T.  Angular velocity,  = 2  f.

Direction of Angular Velocity  Angular velocity can be clockwise or counterclockwise around the axis of rotation.  It has magnitude and direction – it must be a vector. Only two directions are possible for a fixed axisOnly two directions are possible for a fixed axis

Right-hand Rule  Along the axis of rotation there are two directions.  The angular velocity can point either way along the axis of rotation.  By convention the direction follows the thumb if the rotation follows the curve of the right hand.

Angular Acceleration  In uniform circular motion there is a constant radial acceleration. a r = v 2 / r = r  2  If the angular velocity changes there is acceleration tangent to the circle as well as radially. The angular acceleration is 

Angular Acceleration Vector  The angular acceleration vector is the time derivative of the angular velocity vector. Along the axis if the angular velocity only changes magnitudeAlong the axis if the angular velocity only changes magnitude In other directions if the axis changes directionIn other directions if the axis changes direction next