Transcript Angular Motion

Angular Motion
Chapter 10
Figure 10-1
Angular Position
Figure 10-2
Arc Length
Figure 10-3
Angular Displacement
Figure 10-4
Angular Speed and Velocity
Angular Speed is a Vector!


We use a “right hand rule” to
determine the vector direction of
a rotation. Using your right hand,
curl your fingers in the direction
of the rotation. Your thumb points
in the direction of the rotation.
Works for angular acceleration as well.
Figure 10-5
Angular Acceleration
Summary of angular motions.



t


t
Angular position, radians, measure counterclockwise.
Angular velocity, radians per second.
Angular acceleration, radians per
second squared.
Note that radians are a dimensionless quantity.
Radians = Degrees * p/180
Example: 180 degrees = 3.14 radians
Linear and Rotational Motion Compared

x

 x
v
t

 v
a
t


P  mv


F  ma
1 2
K  mv
2
Position
Velocity
Acceleration
Momentum
Force/Torque
Kinetic Energy





t

 

t


L  I


T  I
1 2
K  I
2

Figure 10-7
Angular and Linear Speed
Conceptual Checkpoint 10-1
How do the angular speeds compare?
V=r
How do the linear speeds compare?
Figure 10-8
Centripetal and Tangential Acceleration
IMPORTANT:
For uniform circular motion,
The centripetal acceleration is:
v2
ac 
r
For constant angular speed, at = 0. Then,
the acceleration is RADIAL, inwards.
Figure 10-9
Rolling Without Slipping
Figure 10-11
Velocities in Rolling Motion
Figure 10-10
Rotational and Translational Motions of a Wheel
Figure 10-12
Kinetic Energy of a Rotating Object
K
1 2
mv
2
But… v  r
So…
1 2
K  mv
2
1
2
 mr 
2
1
 mr 2  2
2
 
Define the moment of inertia, I…
I  mr 2
K ROT
1 2
 I
2
(it’s different for different shapes!)
Moment of Inertia
I   mi ri
i
Vi
2
K   Ki
i
Mi
Ri
Rigid body. Break up
into small pieces Mi.
What is the angular
speed of each piece?
1
  M iVi 2
i 2
1
  M i Ri2i2
2 i
1 2
   M i Ri2
2
i
1 2
  I
2


Rotational force: Torque
Torque is the “twisting force” that causes rotational motion. It
is equal to the magnitude of the component of an applied
force perpendicular to the arm transmitting the force.
F
A
R
The torque around point A is T = R x F
Example: torque’s in balance
2r
2m
4f
m