Angular Kinematics - tamuk physics

copygrouperMechanics

Nov 13, 2013 (3 years and 8 months ago)

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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