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.
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