Environmental Engineering Reference
In-Depth Information
to a force equal in magnitude but opposite in direction to that acting on
A
.Ifsuch
a mutual interaction cannot be identified, the only conclusion that can be reached
(other than we did not look hard enough) is that the force is fictitious; a result of
starting from a non-inertial frame.
The Third Law also allows us to generalize from the mechanics of particles
to the mechanics of extended bodies. To do this we will consider a body as being
composed of
N
classical particles. These particles may interact with each other
as well as with other particles that are not part of the body. We consider two
particles within the body
i
and
j
(see Figure 2.7). The mutual interaction between
theses particles consists of two forces:
F
ij
acting on particle
i
and
F
ji
acting on
particle
j
. The Third Law states that these forces must be equal in magnitude but
opposite in direction. The net external force acting on particle
i
is
F
(e)
i
and for
particle
j
it is
F
(e
j
. Since the remote particles responsible for these forces, and
the nature of the interactions, are unspecified, we cannot deduce any relationship
between
F
(e)
i
and
F
(e)
j
in the general case.
F
j
(
e
)
F
ji
F
ij
F
i
(
e
)
Limit of Body
Figure 2.7
Internal and external forces on particles
i
and
j
in an extended body.
Using this separation into internal and external forces, the net force on particle
i
may be written
N
F
(e)
i
F
i
=
+
F
ij
,
(2.20)
j
=
1
where the sum over
j
does not include a contribution from
j
i
(i.e. particles do
not act upon themselves). We now consider the total force acting on the body. This
is the sum
=
N
F
(e)
i
F
=
F
i
=
+
F
ij
.
(2.21)
i
=
1
i
i
j
The second term is a summation of all the forces that arise from mutual interactions
between particles within the body. We can group terms corresponding to the same
