Environmental Engineering Reference
In-Depth Information
Let us consider an extended system, which we take to be composed of
N
particles
whose positions relative to an origin are given by the position vectors
r
j
,where
j
1
,
2
,
3
,...,N
is an index labelling the particles. The
j
th
=
particle has mass
m
j
d
r
j
d
t
. We shall define the total angular momentum of the system
L
about the origin to be the vector sum of the particle angular momenta:
and velocity
v
j
=
N
L
=
r
j
×
(m
j
v
j
).
(4.6)
j
=
1
Let's now compute the rate of change of
L
:
N
N
d
L
d
t
=
×
+
×
v
j
(m
j
v
j
)
r
j
(m
j
a
j
),
(4.7)
j
=
1
j
=
1
d
v
j
d
t
is the acceleration of particle
j
. The first term on the right-hand
side is identically zero so we have
where
a
j
=
N
d
L
d
t
=
r
j
×
(m
j
a
j
).
(4.8)
j
=
1
It is tempting to do as we did for the single particle and use Newton's Second Law
to introduce the force on the
j
th
particle,
F
j
=
m
j
a
j
. Doing this yields
N
d
L
d
t
=
=
1
τ
j
=
τ
.
(4.9)
j
However, such a substitution is valid only if we are working in an inertial frame of
reference and we would like to be more general than that. Suppose instead that the
origin of the co-ordinate system (i.e. the point about which we compute the angular
momentum and torque) is accelerating relative to some inertial frame, which we
generically refer to as the lab frame. Provided that this accelerating co-ordinate
system is not rotating
1
we can write
a
j
=
A
+
a
j
,
(4.10)
where
A
is the acceleration of the origin (of the accelerating co-ordinate system)
and
a
j
is the acceleration of the
j
th
particle, both determined in the lab frame.
Substitution for
a
j
in Eq. (4.8) gives
N
d
L
d
t
m
j
r
j
(
a
j
−
=
A
)
(4.11)
j
=
1
1
Do not agonize over this caveat at this stage. We shall discuss rotating frames of reference in some
detail later on.
