Environmental Engineering Reference
In-Depth Information
B
d
r
d
W
=
F
· d
r
A
Figure 3.3 The path taken by a particle as it moves from
A
to
B
. The work done is the
integral of
F
·
d
r
along the path. The direction of d
r
is indicated.
r
i
t
Simultaneously the ratio
v
(t)
, the velocity at time
t
. This limit is shown in
Figure 3.3. Now the work done in going from
A
to
B
is expressed exactly as the
integral
→
B
W
AB
=
F
(
r
)
·
d
r
,
(3.5)
A
where the force
F
(
r
)
is written explicitly as a function of position.
Example 3.1.1
Calculate the work done by gravity on a projectile of mass m that
is fired to an altitude h.
Solution 3.1.1
We use a co-ordinate system where
i
is horizontal and
j
is vertical.
Then (see Figure 3.4)
d
r
=
d
x
i
+
d
y
j
.
150
dx
dy
dr
100
h
50
0
0
200
400
600
800
x
Figure 3.4
Parabolic path of a projectile launched from the origin.
