Environmental Engineering Reference
In-Depth Information
and then
N
N
d
L
d
t
m
j
a
j
−
×
=
r
j
×
m
j
r
j
A
.
(4.12)
j
=
1
j
=
1
Now we can go ahead and use Newton's Second Law to simplify the first term:
it is the torque calculated about the accelerating origin in the lab. The second
term is generally not zero but notice that
j
=
1
m
j
r
j
is the position vector of the
centre-of-mass. Therefore, we can eliminate it if we choose the origin to coincide
with the centre of mass of the system of particles. In which case we have
N
d
L
c
d
t
=
1
τ
cj
=
τ
c
(4.13)
j
=
and the subscript
c
reminds us that we are to compute the angular momentum and
torque about the centre of mass.
Despite the apparently wide range of applicability of Eq. (4.13) and Eq. (4.9),
these equations involve sums over the mutual interactions between constituent
particles in a system to determine the net torque
and these sums rapidly become
impossible to handle with increasing numbers of particles. To go further with
systems composed of many particles we need to make the distinction between
internal and external forces:
τ
N
F
(e)
j
F
j
=
+
F
jk
,
(4.14)
k
=
1
where again
F
jk
is the force exerted on particle
j
due to particle
k
and
F
(e
j
is the
net force on particle
j
coming from some source outside of the system. Therefore
we now have
F
(e)
j
,
N
N
N
d
L
d
t
=
1
τ
=
r
j
×
+
F
jk
(4.15)
=
=
=
j
j
1
k
1
which, upon expanding the bracket, gives
N
N
N
r
j
×
F
jk
.
d
L
d
t
=
F
(e)
j
r
j
×
+
(4.16)
j
=
1
j
=
1
k
=
1
=
j
=
1
r
j
×
F
(e)
j
(e)
τ
is
the
net
external
torque
on
the
system
and
j
=
1
k
=
1
r
j
F
jk
is the sum of all internal torques, i.e. torques exerted by
particles within the system on other particles within the system. In our analogous
discussion of the linear motion of extended bodies in Section 2.2.1, Newton's
Third Law came to our rescue and told us that the sum of the internal
forces
is
the null vector. We cannot use the same argument to show that the sum of the
internal
torques
vanishes. To illustrate the issue, consider the situation depicted in
Figure 4.2. Two particles have position vectors
r
1
and
r
2
and are exerting forces
×
