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

/

D

t = v / r

•

Angular acceleration corresponds to tangential

acceleration

–

a

=

Dw

/

D

t = a

t

/ r

•

a

t

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?

2

1

0 0

2

2

1

0 0

2

2

1

0 0

2

2

1

0 0

2

( )

x t x v t at

r t r r t r t

r t r t t

t t t

q q w a

q q w a

q q w a

•

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

2

1

0 0

2

0

2 2

0 0

2

f

t t t

t t

q q w a

w w a

w w aq q

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

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