6.2. NEWTON’S LAWS OF MOTION FOR SYSTEMS OF PARTICLES 87
r⃑
j
r⃑
i
y
z
O
O
x
r⃑
i
y
z
x
m
i
a⃑
i
P
i
P
j
f
⃑
ji
f
⃑
ij
F
⃑
i
F
⃑
j
P
i
=
Figure 6.2: Collinear vectors
*
r
j
*
r
i
and
*
f
ji
.
Adding all the internal forces of the system,
n
X
iD1
n
X
j D1
*
f
ij
D 0 (6.7)
and summing moments about O of these internal forces
n
X
iD1
n
X
j D1
*
r
i
*
f
ij
D 0: (6.8)
Equations (6.7) and (6.8) are simply the expressions of the fact that the resultant and moment
resultant of the internal forces of the system are zero.
By making use of Equation (6.7), Equation (6.3) reduces to
n
X
iD1
*
F
i
D
n
X
iD1
m
i
*
a
i
: (6.9)
and applying Equation (6.8), Equation (6.4) becomes
n
X
iD1
*
r
i
*
F
i
D
n
X
iD1
*
r
i
m
i
*
a
i
: (6.10)
Equations (6.9) and (6.10) state the fact that the system of external forces
*
F
i
and the system of the
effective forces m
i
*
a
i
possess the same resultant and the same moment resultant.
88 6. SYSTEMS OF PARTICLES
6.3 LINEAR AND ANGULAR MOMENTUM OF A SYSTEM
OF PARTICLES
Another approach for solution to the system of particles is to relate the effective forces of the
particles to their linear momenta. Writing the linear momentum of a system of particles as
*
L D
n
X
iD1
m
i
*
v
i
: (6.11)
In addition, one requires the angular momentum
*
H
O
about the origin O of the system of par-
ticles. By making use of the definition of angular momentum of a particle in Section 3.6, one
obtains
*
H
O
D
n
X
iD1
*
r
i
m
i
*
v
i
: (6.12)
Taking the time derivative of Equation (6.11),
*
L D
d
*
L
dt
D
n
X
iD1
m
i
d
*
v
i
dt
D
n
X
iD1
m
i
*
a
i
: (6.13)
Comparing the rhs of Equations (6.9) with that of the last equation, one concludes that the rate
of change of the linear momentum is equal to the effective forces of the system,
*
L D
n
X
iD1
*
F
i
D
n
X
iD1
m
i
*
a
i
: (6.14)
Taking the time derivative of Equation (6.12),
*
H
O
D
d
*
H
O
dt
D
d
P
n
iD1
*
r
i
m
i
*
v
i
dt
D
n
X
iD1
d
*
r
i
dt
m
i
*
v
i
C
*
r
i
m
i
d
*
v
i
dt
and since the first term inside the brackets on the rhs vanishes due to the fact that vector cross-
product of the vector itself is zero, the above equation becomes
*
H
O
D
n
X
iD1
*
r
i
m
i
d
*
v
i
dt
D
n
X
iD1
*
r
i
m
i
*
a
i
D
n
X
iD1
*
M
O
i
; (6.15)
where
*
M
O
i
is the moment of the particle P
i
about origin O.
us, Equations (6.14) and (6.15) state that the rates of change of the linear momentum and
of the angular momentum about fixed origin O of the external forces are respectively equal to the force
resultant and moment resultant about origin O of the system of particles.
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