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Rotation of a Rigid Object
About a Fixed Axis
10.1
Angular Displacement, Velocity,
and Acceleration
10.2
Rotational Kinematics: Rotational
Motion with Constant Angular
Acceleration
10.3
Angular and Linear Quantities
10.4
Rotational Energy
10.5
Calculation of Moments of
Inertia
10.6
Torque
10.7
Relationship Between Torque
and Angular Acceleration
10.8
Work, Power, and Energy in
Rotational Motion
Did you know that the CD inside this
player spins at different speeds, depend
ing on which song is playing? Why would
such a strange characteristic be incor
porated into the design of every CD
player?
(George Semple)
C h a p t e r O u t l i n e
292
P
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10.1
Angular Displacement, Velocity, and Acceleration
293
hen an extended object, such as a wheel, rotates about its axis, the motion
cannot be analyzed by treating the object as a particle because at any given
time different parts of the object have different linear velocities and linear
accelerations. For this reason, it is convenient to consider an extended object as a
large number of particles, each of which has its own linear velocity and linear
acceleration.
In dealing with a rotating object, analysis is greatly simplified by assuming that
the object is rigid. A
rigid object
is one that is nondeformable—that is, it is an
object in which the separations between all pairs of particles remain constant. All
real bodies are deformable to some extent; however, our rigidobject model is use
ful in many situations in which deformation is negligible.
In this chapter, we treat the rotation of a rigid object about a fixed axis, which
is commonly referred to as
pure rotational motion.
ANGULAR
DISPLACEMENT,
VELOCITY,
AND
ACCELERATION
Figure 10.1 illustrates a planar (ﬂat), rigid object of arbitrary shape confined to
the
xy
plane and rotating about a fixed axis through
O
. The axis is perpendicular
to the plane of the figure, and
O
is the origin of an
xy
coordinate system. Let us
look at the motion of only one of the millions of “particles” making up this object.
A particle at
P
is at a fixed distance
r
from the origin and rotates about it in a circle
of radius
r
. (In fact,
every
particle on the object undergoes circular motion about
O
.) It is convenient to represent the position of
P
with its polar coordinates (
r
, ),
where
r
is the distance from the origin to
P
and
is measured
counterclockwise
from
some preferred direction—in this case, the positive
x
axis. In this representation,
the only coordinate that changes in time is the angle
;
r
remains constant. (In
cartesian coordinates, both
x
and
y
vary in time.) As the particle moves along the
circle from the positive
x
axis (
0) to
P
, it moves through an arc of length
s
,
which is related to the angular position
through the relationship
(10.1a)
(10.1b)
It is important to note the units of
in Equation 10.1b. Because
is the ratio
of an arc length and the radius of the circle, it is a pure number. However, we com
monly give
the artificial unit
radian
(rad), where
s
r
s
r
10.1
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 Spring '10
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 Angular Momentum, Kinetic Energy, Moment Of Inertia, Rotation

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