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1011
REVIEW AND SUMMARY
This chapter was devoted to the study of the kinematics of rigid
bodies.
We first considered the
translation
of a rigid body [Sec. 15.2] and
observed that in such a motion,
all points of the body have the same
velocity and the same acceleration at any given instant.
We next considered the
rotation
of a rigid body about a fixed axis
[Sec. 15.3]. The position of the body is defined by the angle
u
that
the line
BP
, drawn from the axis of rotation to a point
P
of the body,
forms with a fixed plane (Fig. 15.39). We found that the magnitude
of the velocity of
P
is
v
5
ds
dt
5
r
u
.
sin
f
(15.4)
where
˙
u
is the time derivative of
u
. We then expressed the velocity
of
P
as
v
5
d
r
5
V
3
r
(15.5)
where the vector
V
5
v
k
5
u
˙
k
(15.6)
is directed along the fixed axis of rotation and represents the
angular
velocity
of the body.
Denoting by
A
the derivative
d
V
/
dt
of the angular velocity, we
expressed the acceleration of
P
as
a
5
A
3
r
1
V
3
(
V
3
r
)
(15.8)
Differentiating (15.6), and recalling that
k
is constant in magnitude
and direction, we found that
A
5
a
k
5
v
.
k
5
¨
u
k
(15.9)
The vector
A
represents the
angular acceleration
of the body and is
directed along the fixed axis of rotation.
Rigid body in translation
Rigid body in rotation
about a fixed axis
x
z
y
O
A
'
A
B
P
f
r
q
Fig. 15.39
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View Full DocumentNext we considered the motion of a representative slab located in a
plane perpendicular to the axis of rotation of the body (Fig. 15.40).
Since the angular velocity is perpendicular to the slab, the velocity of
a point
P
of the slab was expressed as
v
5
v
k
3
r
(15.10)
where
v
is contained in the plane of the slab. Substituting
V
5
v
k
and
A
5
a
k
into (15.8), we found that the acceleration of
P
could
be resolved into tangential and normal components (Fig. 15.41)
respectively equal to
a
t
5
a
k
3
r
a
t
5
r
a
a
n
5
2
v
2
r
a
n
5
r
v
2
(15.11
9
)
Recalling Eqs. (15.6) and (15.9), we obtained the following expres
sions for the
angular velocity
and the
angular acceleration
of the slab
[Sec. 15.4]:
v
5
d
u
dt
(15.12)
a
5
d
v
5
d
2
u
2
(15.13)
or
a
5
v
d
v
d
u
(15.14)
We noted that these expressions are similar to those obtained in
Chap. 11 for the rectilinear motion of a particle.
Two particular cases of rotation are frequently encountered:
uniform rotation
and
uniformly accelerated rotation.
Problems
involving either of these motions can be solved by using equations
similar to those used in Secs. 11.4 and 11.5 for the uniform rectilin
ear motion and the uniformly accelerated rectilinear motion of a
particle, but where
x
,
v
, and
a
are replaced by
u
,
v
, and
a
, respec
tively [Sample Prob. 15.1].
Rotation of a representative slab
Tangential and normal components
Angular velocity and angular
acceleration of rotating slab
x
y
O
r
P
w
k
v
=
w
k
×
r
Fig. 15.40
x
y
O
w
=
w
k
a
=
a
k
a
t
=
a
k
×
r
a
n
=
–
w
2
r
P
Fig. 15.41
1012
Kinematics of Rigid Bodies
1013
The
most general plane motion
of a rigid slab can be considered as
the
sum of a translation and a rotation
[Sec. 15.5]. For example, the
slab shown in Fig. 15.42 can be assumed to translate with point
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This note was uploaded on 12/05/2011 for the course MEEG 324 taught by Professor Ib during the Fall '10 term at The Petroleum Institute.
 Fall '10
 IB

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