Jan 7-11:
Mon: Rotational Kinematics (Ch. 8.1 - 8.3)
Tue:
Wed: Torque
(Ch. 9.1)
Center of Gravity
(Ch. 9.3)
Thurs: Hookes Law (springs)
(Ch. 10.1)
Elastic Potential Energy
(Ch. 10.3)
Fri: Pendulums
(Ch. 10.6)
Jan 14-18:
Review
Jan 21: MLK Day (no school)
Jan 22-25: Finals
Bulls-Eye Lab Practicum
Last three weeks of the Semester…
8.1 – 8.3
The angle through which the
object rotates is called the
angular displacement.
     o
Linear Motion
 x = displacement
 v = velocity
 a= acceleration
 t = time
Rotational Motion
 θ = displacement (theta)
 ω = velocity
(omega)
 α= acceleration (alpha)
 t = time
SI Units*:
Θ in rad
ω in rad/s
α in rad/s2
* ω and α might be in rev/s but use
dimensional analysis to make sure the
equations are “dimensionally balanced”!
Counter-clockwise is usually
taken to be positive.
Rotational Kinematics Equations
Linear Kinematics
x = x0 + v0 t + ½at2
Rotational Kinematics
θ = θ0 + ω 0 t + ½αt2
v = v0 + at
ω = ω0 + αt
x = ½(v0 + v)t
θ = ½ (ω0 + ω)t
v2 = v02 + 2ax
ω2 = ω02 + 2α θ
v = ½ (v0 + v)
ω = ½ (ω 0 + ω)
For CONSTANT linear
acceleration
For CONSTANT
rotational acceleration
Example #5 Blending with a Blender
The blades of an electric blender are
whirling with an angular velocity of 375
rad/s while the “puree” button is pushed
in. When the “blend” button is pressed
instead, the blades accelerate and reach a
greater angular velocity after the blades
have rotated an angular displacement of
44.0 rad. The angular acceleration is
constant at 1740 rad/s2. Find the final
angular velocity of the blades. (‘Blades’
refers to the TIP of the blade at all times.)
Catalogue variables:
Using ω2 = ω02 + 2α θ
v = 541.98 = 542 rad/s



If the tip of the blender-blade went around
one revolution, how many DEGREES did it
travel?
If the tip of the blender-blade went around
one revolution, how many RADIANS did it
travel?
If the tip of the blender-blade went around 8
revolutions, how many RADIANS did it travel?

Recall s = rθ
when θ is in radians

Example: Find the arclength between two
points on the circle if the radius is 4.23 x 107
m and the angle is 2 degrees.
Convert 2 degrees into radians.
Then s = 1.48 x 106 meters
Math-y Stuff, p. 3
In many of these problems, use proportionalities.
And use dimensional analysis to set up your problems!
And finally, think about them! You often ‘know’ how to solve these.
For instance, a circle has a radius of 4 meters. An object travels on
this circular path, and travels a distance of 11 meters. How many
radians did it travel?
The distance around the circle one time is 2π4 = 8π= 25.13 m
Proportion: 11/25.13 meters = θ/2 π = 2.75 radians
(=158 deg.)
Conceptual Example A Total Eclipse of the Sun
The diameter of the sun is about 400 times greater
than that of the moon. By coincidence, the sun is also
about 400 times farther from the earth than is the
moon.
For an observer on the earth, compare the angle subtended by
the moon to the angle subtended by the sun and explain why
this result leads to a total solar eclipse.
Using s = rθ, we can see that the angle is the same in both
cases, so the moon is able to completely block the sun!
DEFINITION OF AVERAGE ANGULAR VELOCITY
Average angular ve
locity 
Angular
displaceme
Elapsed time
 
 o
t  to


t
nt
Example Gymnast on a High Bar
A gymnast on a high bar swings through
two revolutions in a time of 1.90 s.
Find the average angular velocity
of the gymnast.
 2  rad 
    2 . 00 rev 
   12 . 6 rad
 1 rev 
 
 12 . 6 rad
1 . 90 s
  6 . 63 rad s
DEFINITION OF AVERAGE ANGULAR ACCELERATION
Changing angular velocity means that an angular
acceleration is occurring.
 
  o
t  to


t
Example A Jet Revving Its Engines
As seen from the front of the engine, the fan blades are
rotating with an angular speed of -110 rad/s. As the
plane takes off, the angular velocity of the blades reaches
-330 rad/s in a time of 14 s.
Find the angular acceleration, assuming it to be constant.
 
 330
rad s     110 rad s 
14 s
  16 rad s
2




P. 229 Check Understanding #6
P. 240 Focus #3-7
P. 241 Problems #20-25
Optional challenge #28.