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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