6.5. ANGULAR MOMENTUM OF A SYSTEM OF PARTICLES ABOUT ITS MASS CENTER 91
But
P
n
iD1
m
i
*
v D m
*
v and
P
n
iD1
m
i
*
a D m
*
a ; therefore one obtains
n
X
iD1
m
i
*
v
0
i
D 0; (6.25a)
n
X
iD1
m
i
*
a
0
i
D 0: (6.25b)
6.5 ANGULAR MOMENTUM OF A SYSTEM OF PARTICLES
ABOUT ITS MASS CENTER
In Section 6.3, the linear and angular momentum of a system of particles has been expressed
w.r.t. the FFR. However, in some situations it is more convenient to consider the angular mo-
mentum w.r.t. the mass center G of the system of particles, as shown in Figure 6.4.
G
m
i
υ⃑
i
'
m
i
υ
⃑
i
O
r⃑
i
r⃑
i
'
_
r⃑
y
y'
z
x
z'
x'
P
i
Figure 6.4: Fixed frame of reference and mass center.
With reference to Figure 6.4 in which a frame of reference with G being the origin is
shown, the angular momentum
*
H
0
G
of the system of particles about the mass center of a system
of particles is defined by
*
H
0
G
D
n
X
iD1
*
r
0
i
m
i
*
v
0
i
: (6.26)
Taking the time derivative of
*
H
0
G
gives
*
H
0
G
D
d
*
H
0
G
dt
D
d
P
n
iD1
*
r
0
i
m
i
*
v
0
i
dt
D
n
X
iD1
d
*
r
0
i
dt
m
i
*
v
0
i
C
*
r
0
i
m
i
d
*
v
0
i
dt
!
92 6. SYSTEMS OF PARTICLES
in which the first term inside the brackets on the rhs vanishes due to the fact that vector cross-
product of the vector itself is zero. us, this equation reduces to
*
H
0
G
D
n
X
iD1
*
r
0
i
m
i
d
*
v
0
i
dt
D
n
X
iD1
*
r
0
i
m
i
*
a
0
i
: (6.27)
Now, taking the second time derivative of Equation (6.17), one obtains
*
a
i
D
*
a C
*
a
0
i
so that upon substituting this equation into Equation (6.27), one has
*
H
0
G
D
n
X
iD1
*
r
0
i
m
i
*
a
i
*
a
D
n
X
iD1
*
r
0
i
m
i
*
a
i
n
X
iD1
*
r
0
i
m
i
!
*
a :
By making use of Equation (6.19) the second term on the rhs is zero, and therefore this
equation reduces to
*
H
0
G
D
n
X
iD1
*
r
0
i
m
i
*
a
i
D
n
X
iD1
*
r
0
i
*
F
i
D
n
X
iD1
*
M
G
i
; (6.28)
where
*
F
i
D m
i
*
a
i
has been applied and
*
M
G
i
is the moment of the particle P
i
about origin
G. us, Equation (6.28) simply states that the sum of all the moments about G of all particles in
the system is equal to the rate of change of the angular momentum about G of the system of particles.
It is interesting to note that by taking the first time derivative of Equation (6.17) and
substituting the resulting equation into Equation (6.26), one can show that
*
H
0
G
D
n
X
iD1
*
r
0
i
m
i
*
v
0
i
D
n
X
iD1
*
r
0
i
m
i
*
v
i
D
*
H
G
: (6.29)
Equation (
6.29) simply states that the sum of all the angular momenta about G based on the velocities
of the particles with reference to G is identical to that of all angular momenta about G based on the
velocities of the particles with reference to the origin O of the FFR.
Furthermore, with reference to Figure 6.4, the sum of angular momenta of all particles
about O is
*
H
O
D
n
X
iD1
*
r
i
m
i
*
v
i
D
n
X
iD1
*
r C
*
r
0
i
m
i
*
v
i
;
by Equation (6.17).
Expanding,
*
H
O
D
n
X
iD1
*
r m
i
*
v
i
C
n
X
iD1
*
r
0
i
m
i
*
v
i
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