Linear Momentum
Shina Tan
Definition–
The total linear momentum of any system is defined to be the vector sum
of the linear momenta of all its constituent parts:
1
2
P
=
X
α
m
α
˙
r
α
(
t
)
.
(1)
Theorem A: Momentum and the Center of Mass Motion
Because
P
(
t
) =
X
α
m
α
˙
r
α
(
t
) =
d
dt
X
α
m
α
r
α
(
t
) =
d
dt
h
M
1
M
X
α
m
α
r
α
(
t
)
i
=
d
dt
[
M
R
(
t
)] =
M
˙
R
(
t
)
,
we find
P
(
t
) =
M
˙
R
(
t
)
.
(2)
Theorem A–
The total linear momentum of any system is equal to its total mass times
the velocity of the center of mass.
In particular, if a system’s center of mass is not moving, its total linear momentum is
zero, even though its individual parts may be moving.
Theorem B: Rate of Change of the Momentum
The
α
th particle within the system experiences two forces in general: a force
F
(
e
)
α
(
t
) whose
origin is external to the system, and the sum of forces acting on the particle by all other
particles within the system:
m
α
¨
r
α
(
t
) =
F
(
e
)
α
(
t
) +
X
β
f
αβ
(
t
)
.
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 Spring '11
 Tan
 mechanics, Derivative, Center Of Mass, Force, Mass, Momentum, total linear momentum

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