Angular Variables

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.

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 

Uniform or Nonuniform  Centripetal acceleration is constant for uniform circular motion.  It changes for nonuniform circular motion. The magnitude increases or decreases.The magnitude increases or decreases. There is a tangential acceleration.There is a tangential acceleration. Net vector is not antiparallel to radius.Net vector is not antiparallel to radius.

Rotational Motion  Kinematic equations with constant linear acceleration were defined. v av = ½ (v 0 + v)v av = ½ (v 0 + v) v = v 0 + atv = v 0 + at x = x 0 + v 0 t + ½at 2x = x 0 + v 0 t + ½at 2 v 2 = v a(x - x 0 )v 2 = v a(x - x 0 )  Kinematic equations with constant angular acceleration are similar.  av = ½ (  0 +  )  =  0 +  t  =  0 +  0 t + ½  t 2  2 =   (  -  0 ) next