905
REVIEW AND SUMMARY
In this chapter we analyzed the motion of
systems of particles,
i.e.,
the motion of a large number of particles considered together. In the
first part of the chapter we considered systems consisting of well-
defined particles, while in the second part we analyzed systems
which are continually gaining or losing particles, or doing both at the
same time.
We first defined the
effective force
of a particle
P
i
of a given system
as the product
m
i
a
i
of its mass
m
i
and its acceleration
a
i
with respect
to a newtonian frame of reference centered at
O
[Sec. 14.2]. We
then showed that
the system of the external forces acting on the
particles and the system of the effective forces of the particles are
equipollent;
i.e., both systems have the
same resultant
and the
same
moment resultant
about
O
:
O
n
i
5
1
F
i
5
O
n
i
5
1
m
i
a
i
(14.4)
O
n
i
5
1
(
r
i
3
F
i
)
5
O
n
i
5
1
(
r
i
3
m
i
a
i
)
(14.5)
Defining the
linear momentum
L
and the
angular momentum
H
O
about point O
of the system of particles [Sec. 14.3] as
L
5
O
n
i
5
1
m
i
v
i
H
O
5
O
n
i
5
1
(
r
i
3
m
i
v
i
)
(14.6, 14.7)
we showed that Eqs. (14.4) and (14.5) can be replaced by the
equations
o
F
5
L
.
o
M
O
5
H
.
O
(14.10, 14.11)
which express that
the resultant and the moment resultant about O
of the external forces are, respectively, equal to the rates of change
of the linear momentum and of the angular momentum about O of
the system of particles
.
In Sec. 14.4, we defined the mass center of a system of particles as
the point
G
whose position vector
r
satisfies the equation
m
r
5
O
n
i
5
1
m
i
r
i
(14.12)
Effective forces
Linear and angular momentum
of a system of particles
Motion of the mass center
of a system of particles

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906
Systems of Particles
where
m
represents the total mass
O
n
i
5
1
m
i
of the particles. Differ-
entiating both members of Eq. (14.12) twice with respect to
t
, we
obtained the relations
L
5
m
v
˙
L
5
m
a
(14.14, 14.15)
where
v
and
a
represent, respectively, the velocity and the accelera-
tion of the mass center
G
. Substituting for
L
.

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