Rotational Kinematics
Chapter 8
8.1 Rotational Motion and Angular
Displacement
• Axis of Rotation: the line all points on a rotating
object rotate around. On a CD, it is an imaginary
line through the hole in the center of the CD.
• Angular Displacement: the angle ∆θ swept out by
a line passing through any point and intersecting
the axis of rotation perpendicularly.
• Positive: counterclockwise
• Negative: clockwise
• SI Unit: radian (rad)
Units, Units, and More Units
• Degree: 360° in a circle
• Revolution (rev): 1rev = 360°
• SI and useful: radian
arclength s


Since s and r are both
measurments of length, a
radian is considered to be
number without a unit
radius
r
Converting Units
• Remember: the arc length of an entire circle
is the circumference 2πr
• 1rev = 2πrad=360°
360
1rad 
 57.3
2
Read Conceptual Example 2 on pg. 225. Cool stuff really.
8.2 Angular Velocity and Angular
Acceleration
• SI Unit of angular velocity: rad/s
• SI Unit of angular acceleration: rad/s2
 -  0 


t - t0
t
Direction of angular velocity if
positive when counterclockwise and
negative when clockwise. For
angular acceleration, the sign
convention remains the same.
   0 


t  t0
t
Reminder: SAME CONCEPTS APPLIED TO
A DIFFERENT TYPE OF MOTION OR
SITUATION! THESE CONCEPTS SHOULD
BE FAMILIAR!
8.3 The Equations of Rotational
Kinematics
Applying the Concepts
The blades of an electric blender are
whirling with an angular velocity of
+375rad/s while the “puree” button is
pushed in. When the “blend” button
is pressed, the blades accelerate and
reach a greater angular velocity after
the blades have rotated through an
angular displacement of +44.0rad.
The angular acceleration has a
constant value of +1740rad/s2. Find
the final angular velocity of the
blades.
TRY THIS SELF ASSESSMENT TEST TOO!
8.4 Angular Variables and Tangential
Variables
• The tangential speed of any point rotating
about a fixed axis represents the velocity
vector which is tangent to that point on the
circle.
• The further from the axis of rotation for a
given rotational velocity, the higher the
tangential speed will be.
Relating Angular and Tangential
Variables Through Equations
v T  r
When using this equation, ω
MUST be in rad/sec because the
equation was derived using
these units!
a T  r
Again, α must be in units of
rad/sec2 !!
The main difference between using angular and tangential equations is that the
angular equations describe the entire object at one time but the tantential equations
only describe one particular point of a moving object at any given time. Depending
on the information given and needed, both will be useful.
Centripetal Acceleration and
Tangential Acceleration
• Remember from previous chapters that when
an object moves in a circle with a constant
velocity, the centripetal acceleration will be
pointed toward the center of the circle.
• Substituting (rω) into the centripetal
acceleration equation allows us to relate
centripetal acceleration to tangential
acceleration.
ac  r
2
Once again, use
rad/s for ω!
Putting it All Together
• Discus throwers often warm up by
standing with both feet flat on the
ground and throwing the discus with a
twisting motion of their bodies.
Starting from rest, a thrower
accelerates the discus to a final angular
velocity of +15.0rad/s in a time of
0.270s before releasing it. During the
acceleration, the discus moves on a
circular arc of radius 0.810m. Find the
magnitude (a) of the total acceleration
of the discus just before it is released.
8.6 Rolling Motion
• When an object is truly rolling, there is no
slipping at the point of contact where the
rolling object touches the ground.
• If a tire makes one complete revolution
(2πrad), the linear distance the center of the
tire will have moved will be equal to the
circumference of the wheel.
v  r
a  r
8.7 The Vector Nature of Angular
Variables
• So far, we haven’t discussed the vector direction
of angular quantities.
• When a rigid object rotates about a fixed axis, it is
the axis that identifies the motion and the
angular velocity points along the axis.
• Right Hand Rule: Grasp the axis of rotation with
your right hand, so that your fingers circle the
axis in the same sense as the rotation. Your
extended thumb points along the axis in the
direction of the angular velocity vector.
Vector Nature Continued
• Angular acceleration has the
same direction as the change in
angular velocity. It will always
point along the axis of rotation.
• When the angular velocity is
increasing, the acceleration
vector points in the same
direction as the angular velocity.
• When the angular velocity is
decreasing, the angular
acceleration points in the
direction opposite to the
angular velocity.
Final Notes
• In this chapter, we “put things together.” In a
sense, we have a more complete understanding
of the motion of objects moving in circles.
• Lots of new equations! UGGHH! Use an
equation sheet when solving problems for HW.
• Try both Chapter Assessments. The final one is
linked below.
• http://bcs.wiley.com/hebcs/Books?action=mininav&bcsId=3138&itemId=
0471663158&assetId=90800&resourceId=7780&
newwindow=true