Chapter 8
Rotational Motion
Objectives
• State the meaning of the symbols used in
the kinematics equations for uniformly
accelerated angular motion.
• describe uniformly accelerated angular
motion.
• Use the completed data table to solve
word problems related to angular
kinematics.
Elaboration
• Angular Acceleration Worksheet
8-2 Constant Angular Acceleration
The equations of motion for constant angular
acceleration are the same as those for linear
motion, with the substitution of the angular
quantities for the linear ones.
The Rotational Variables: Angular Acceleration
If the angular velocity of a rotating body is not constant, then the
body has an angular acceleration.
If w2 and w1 be its angular velocities at times t2 and t1, respectively,
then the average angular acceleration of the rotating body in the
interval from t1 to t2 is defined as
The instantaneous angular acceleration a, is the limit of this
quantity as Dt approaches zero.
These relations hold for every particle of that body. The unit of
angular acceleration is (rad/s2).
Angular Quantities as Vectors
# In rotation about a single fixed axis, the direction of angular
velocity and angular acceleration can be represented with a positive or
negative sign, but more generally they can be treated as vectors by
assigning a direction according to the right-hand rule convention.
(# A finite angular displacement is not a vector, since it does not
always follow rules of vectors, e.g. order in adding vectors, but an
infinitesimal angular displacement can be treated as a vector. )
Vector Nature of Angular
Quantities
 Angular displacement,
velocity and
acceleration are all
vector quantities
 Direction can be more
completely defined by
using the right hand
rule
 Grasp the axis of rotation
with your right hand
 Wrap your fingers in the
direction of rotation
 Your thumb points in the
direction of w
Velocity Directions
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
In (a), the disk
rotates clockwise,
the velocity is into
the page
In (b), the disk
rotates
counterclockwise,
the velocity is out
of the page
Acceleration Directions


If the angular acceleration and the
angular velocity are in the same
direction, the angular speed will
increase with time
If the angular acceleration and the
angular velocity are in opposite
directions, the angular speed will
decrease with time
What is the direction of
the angular velocity?
What is the direction of
the angular velocity?
What is the direction of
the angular velocity?
What is the direction of
the angular velocity?
What is the direction of
the angular velocity?
What is the direction of
the angular velocity?
Centripetal Acceleration

The magnitude of the centripetal
acceleration is given by

This direction is toward the center of
the circle
Centripetal Acceleration
and Angular Velocity


The angular velocity and the linear
velocity are related (v = wr)
The centripetal acceleration can
also be related to the angular
velocity
aC = w r
2
Forces Causing Centripetal
Acceleration

Newton’s Second Law says that
the centripetal acceleration is
accompanied by a force


FC = maC
FC stands for any force that keeps an
object following a circular path



Tension in a string
Gravity
Force of friction
8-3 Rolling Motion (Without Slipping)
In (a), a wheel is rolling without
slipping. The point P, touching
the ground, is instantaneously
at rest, and the center moves
with velocity v.
In (b) the same wheel is seen
from a reference frame where C
is at rest. Now point P is
moving with velocity –v.
The linear speed of the wheel is
related to its angular speed:
Rolling Motion
Many rotational motion situations involve rolling objects.
Rolling without slipping
involves both rotation and
translation.
w
Friction between the rolling
object and the surface it rolls
on is static, because the
rolling object’s contact point
this point on the wheel is
with the surface is always
instantaneously at rest if the
instantaneously at rest.
wheel does not slip (slide)
the illustration of w in this diagram is misleading;
the direction of w would actually be into the screen
Sample Problem 1 (similar
to problem 16 in book)

An automobile engine slows
down from 4000 rpm to 1200
rpm in 3.5 s. Calculate (a) its
angular acceleration, assumed
uniform, and (b) the total
number of revolutions the
engine makes in this time.
Answer to sample problem
fo = 4000 revolutions /minute f = 1200 revolutions / minute wo = 2f
wo = (4000 Revolutions/Minute)(2radians/revolution)(1 minute/60 sec) = 418.9
rad/s
w = (1200 Revolutions/Minute)(2radians/revolution)(1 minute/60 sec) = 125.7
rad/s
t = 3.5 s
Let's find the acceleration first:
wwo + at : wo = 418.9 rad/s; w = 125.7 rad/s; t = 3.5 s
a = -83.8 rad/s/s
And the displacement (Angular)
 = ½ (wo+w)t
Now let's plug in numbers:
 = (125.7 rad/s+418.9 rad/s)(3.5 s)/2 = 952.9 radians
But the problem wanted revolutions, so let's change the units:
 = (952.9 radians)(revolution/2radians) = 151.7 revolutions
Homework
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
Chapter 8 Problems
15, 17, 19, 21
Closure
Kahoot 8-2 & 8-3