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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Newton
s 2
nd
Law for
m
i
:
1
N
i
ij
i
i
j
m
=
+
=
∑
F
f
a
Take moment about
P
(cross with
r
i
):
1
N
i
i
i
ij
i
i
i
j
m
=
×
+
×
=
×
∑
r
F
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)
1
1
N
N
P
i
i
i
i
i
m
=
=
=
×
∑
∑
M
r
a