Angular Momenta and Torques
Shina Tan
Angular momentum and torque about any fixed point
By a fixed point we mean a point that is at rest in the given coordinate system.
Definition 1–
The angular momentum of a system of particles about any fixed reference
point
o
is defined as
L
(
o
)
(
t
) =
X
α
[
r
α
(
t
)

o
]
×
m
α
˙
r
α
(
t
)
,
(1)
where [
r
α
(
t
)

o
] is the position vector of the
α
th particle with respect to the reference
point.
Definition 2–
The torque about any fixed reference point
o
due to external forces acting
on the system is defined as
N
(
o
)
(
t
) =
X
α
r
α
(
t
)

o
×
F
(
e
)
α
(
t
)
,
(2)
where
F
(
e
)
α
(
t
) is the force acting on the
α
th particle by objects external to the system.
Corollary 1–
If the system’s total linear momentum is zero, its angular momentum does
not depend on the choice of the reference point.
Corollary 2–
If the vector sum of all external forces acting on the system is zero, the
torque does not depend on the choice of the reference point.
Theorem 1–
The rate of change of a system’s angular momentum about any fixed refer
ence point is equal to the total external torque exerted on the system about the same reference
point:
d
dt
L
(
o
)
(
t
) =
N
(
o
)
(
t
)
.
(3)
To prove this, we note that
d
dt
L
(
o
)
(
t
) =
X
α
˙
r
α
(
t
)
×
m
α
˙
r
α
(
t
) +
X
α
[
r
α
(
t
)

o
]
×
m
α
¨
r
α
(
t
)
.
Because the vector product of any two parallel vectors is zero,
d
dt
L
(
o
)
(
t
) =
X
α
[
r
α
(
t
)

o
]
×
m
α
¨
r
α
(
t
)
=
X
α
[
r
α
(
t
)

o
]
×
F
(
e
)
α
(
t
) +
X
β
f
αβ
(
t
)
=
N
(
o
)
(
t
) +
X
αβ
[
r
α
(
t
)

o
]
×
f
αβ
(
t
)
1
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Since
α
and
β
and dummy indices, their names can be interchanged, so the last term is
equal to
∑
βα
[
r
β
(
t
)

o
]
×
f
βα
(
t
). According to Newton’s Third Law,
f
βα
(
t
) =

f
αβ
(
t
). So
the last term is also equal to
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 Spring '11
 Tan
 mechanics, Angular Momentum, Momentum, Newton’s Third Law

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