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Moment of Momentum (Euler
`
s 2
nd
Law)
Consider system of
N
particles as in Euler
`
s First Law
C
:
mass center
O
:
fixed point in inertial frame
P
:
arbitrary point (may be moving)
P
r
i
m
i
F
i
f
ij
f
ji
m
j
F
j
O
x
r
ji
r
j
i
a
j
a

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`
s 2
nd
Law for
m
i
:
1
N
ii
j
i
i
j
m
=
+=
∑
Ff
a
Take moment about
P
(cross with
r
i
):
1
N
i
i
j
i
j
m
=
×
+
×
=
×
∑
rF
r
f
r
a
(using distributive rule)
Add:
r
i
!
F
i
=
M
P
{
+
r
i
!
f
ij
j
=
1
N
"
i
=
1
N
"
=
0
1 2
4
3
4
=
r
i
!
m
i
a
i
i
=
1
N
"
i
=
1
N
"
0
(co linearity)
r
1
!
f
12
+
r
2
!
f
21
(typical pair, recall
f
ii
=
0)
r
1
!
f
12
+
r
2
!
(
"
f
12
) (equal & opposite - 3rd Law)
(
r
1
!
r
2
)
!
f
12
(distributive rule)
11
NN
Pi
i
i
m
==
=
×
∑∑
Mr
a

Define:
1
moment of momentum about
N
Pi
i
i
i
mP
=
=
×
∑
Hr
v
()
PC
i
i
i
m
=+
×
∑
rv
r
=
r
PC
!
m
i
v
i
i
=
1
N
!
L
12
4 3
4
+
!
m
i
v
i
i
=
1
N
!
H
C
1 2
44
3
So:
P
C
×
HH
r
L
P
r
i
m
i
C
O
x
y
PC
r
i
r
i
i
i
i
i
mm
=
×
+
×
∑∑
v
r
C
v
i
v
i
r

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