Angular MotionMeasuring a CircleAngular VelocityVelocity and Angular VelocityCycles or RadiansDirection of Angular VelocityRight-hand RuleAngular AccelerationAngular Acceleration VectorAngular MotionAngular MotionMeasuring a CircleMeasuring a CircleWe use degrees to measure We use degrees to measure position around the circle.position around the circle.There are 2There are 2 radians in the radians in the circle.circle.•This matches 360This matches 360°°The distance around a circle The distance around a circle is is ss = = rr , where , where  is in is in radians.radians.r12The angular displacement is Angular VelocityAngular VelocityFor circular motion, only the time rate of change of For circular motion, only the time rate of change of the angle matters.the angle matters.The time rate of change of the angle is called the The time rate of change of the angle is called the angular velocity.angular velocity.•Symbol (Symbol ())•Units (rad/s or 1/s = sUnits (rad/s or 1/s = s-1-1))rdtdrv Velocity and Angular Velocity and Angular VelocityVelocityVelocity has an angular Velocity has an angular equivalent.equivalent.•Linear velocity (Linear velocity (vv))•Angular velocity (Angular velocity ())They are related, since the They are related, since the displacement is related to displacement is related to the angle.the angle.dtdxtxvt 0limrdtdrdtrddtdsvrs)(dtdtt 0limvCycles or RadiansCycles or RadiansFrequency is measured in Frequency is measured in cycles per second.cycles per second.There is one cycle per There is one cycle per period.period.Frequency is the inverse of Frequency is the inverse of the period, the period, ff =1/ =1/TT..Angular velocity is measured Angular velocity is measured in radians per second.in radians per second.There are 2There are 2 radians per radians per period.period.Angular velocity, Angular velocity,  = 2 = 2//TT..Angular velocity, Angular velocity,  = 2 = 2ff..Direction of Angular Direction of Angular VelocityVelocityAngular velocity can be clockwise or Angular velocity can be clockwise or counterclockwise around the axis of rotation.counterclockwise around the axis of rotation.It has magnitude and direction – it must be a vector.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 axisRight-hand Rule Right-hand Rule Along the axis of rotation there are two directions.Along the axis of rotation there are two directions.The angular velocity can point either way along the The angular velocity can point either way along the axis of rotation.axis of rotation.By convention the direction follows the thumb if the By convention the direction follows the thumb if the rotation follows the curve of the right hand.rotation follows the curve of the right hand.Angular AccelerationAngular AccelerationIn uniform circular motion In uniform circular motion there is a constant radial there is a constant radial acceleration.acceleration.•aarr = = vv22 / / rr = = rr22If the angular velocity If the angular velocity changes there is changes there is acceleration tangent to the acceleration tangent to the circle as well as radially.circle as well as radially.rdtdrdtdvatratavThe angular acceleration is Angular Acceleration Angular Acceleration VectorVectorThe angular acceleration The angular acceleration vector is the time derivative vector is the time derivative of the angular velocity of the angular velocity vector.vector.•Along the axis if the angular Along the axis if the angular velocity only changes velocity only changes magnitudemagnitude•In other directions if the axis In other directions if the axis changes directionchanges