Angular Kinematics
• Linear kinematics does not handle curving
trajectories well
– Always have acceleration due to changing
direction
– Need to know radius of turn
• Want to isolate the motion of curving
trajectories
• A straight line has an always changing
“radius” from a single point
• Can think of “rolling up” a line so the
radius is constant
Definition of angular quantities
• Angle corresponds to displacement
– q=x/r
• x is distance along curve, r is radius of curve
• Angular velocity corresponds to velocity
– w = Dq/Dt = v / r
• Angular acceleration corresponds to tangential
acceleration
– a = Dw/Dt = at / r
• at generates a speed change, not a directional change
• Note that these are simply the linear quantities
divided by the radius!
– Can we use this to find kinematic relationships?
x(t ) x0 v0t 1 at 2
2
rq t rq 0 rw0t ra t1
2
2
rq t r q 0 w0t 1 a t 2
2
q t q 0 w0t 1 a t 2
2
• This is the displacement equation for angular
motion!
• Can derive other kinematic equations in a
same way
• Notice that linear kinematic equation and
angular kinematic equation are almost the
same
– Replace linear quantities with angular
quantities to transform between them
q t q 0 w0t 1 a t 2
2
w t w0 a t
w 2 w02 2a q f q 0
Kinematic Quantity Linear Angular
Displacement x q
Velocity v w
Acceleration a a
Direction of angular motion
• Angular motion, like all motion, has a
direction
• Only stationary point is the center of the
circle
– Center defines axis of rotation
• Use right hand rule to define direction
– Curl fingers of right hand in direction of spin
– Thumb points in direction of motion