Environmental Engineering Reference
In-Depth Information
F
d
r
Figure 3.1
The forces involved when pushing a block against a wall.
B
∆
r
i
∆
W
i
=
F
i
·
∆
r
i
A
Figure 3.2
The path between
A
and
B
is made up of a series of discrete steps.
considering a given smooth path to be approximated by a series of
N
small but
finite displacements
(
r
i
)
each corresponding to an interval of time
t
. Figure 3.2
illustrates this for a particle making the journey from point
A
to point
B
.You
might like to consider the vertices on the path as successive positions of the particle
when illuminated by a stroboscope where the interval between the flashes is
t
.
The displacement vector that takes us from
A
to
B
can then be written as
N
r
AB
=
r
B
−
r
A
=
r
i
,
(3.2)
i
=
1
where
r
A
and
r
B
are the position vectors of
A
and
B
, respectively. For each of the
steps we can calculate the work done:
W
i
=
F
i
·
r
i
,
(3.3)
where
F
i
is the force vector acting on the particle at the
i
th
step. We can then
approximate the total work done in going from
A
to
B
as
N
W
AB
≈
W
i
.
(3.4)
i
=
1
This approximation becomes more accurate as
t
gets smaller (i.e.
N
gets larger)
and the displacement in each step gets smaller. In the limit that
N
→∞
the
displacement
r
i
→
d
r
and it becomes parallel to a tangent vector to the curve.