Environmental Engineering Reference
In-Depth Information
The force of gravity is represented (for low altitudes) as
F
=−
mg
j
.So
d
W
=−
mg
j
·
(
d
x
i
+
d
y
j
)
=−
mg
d
y,
i.e. only the vertical component of the path contributes to the work. We integrate
the above expression to obtain
W
=
d
W
=−
mgh.
So although the path is parabolic we need only know the maximum height reached
in order to determine the work done.
We are now ready to establish why work is such a useful concept. To do this we
shall use Newton's Second Law in order to evaluate the work done on a particle.
Consider
B
B
d
(m
v
)
d
t
W
AB
=
F
(
r
)
·
d
r
=
·
d
r
.
(3.6)
A
A
We can use d
r
=
v
d
t
, allowing us to write
B
B
d
(m
v
)
d
t
m
d
(
v
)
d
t
W
AB
=
·
=
·
v
d
t
v
d
t.
(3.7)
A
A
This integral is easily evaluated once we recognise that
d
(v
2
)
d
t
d
(
v
·
v
)
d
v
d
t
,
=
=
2
v
·
d
t
where we have used the product rule for differentiation of the scalar product. We
are therefore able to rewrite Eq. (3.7) as
d
2
mv
2
d
t
B
1
2
mv
B
−
1
2
mv
A
=
W
AB
=
d
t
=
T
B
−
T
A
,
(3.8)
A
where
v
A
and
v
B
are the speeds of the particle at positions
A
and
B
respectively.
The quantity
1
2
mv
2
T
=
(3.9)
is the kinetic energy and the effect of the force is to alter the kinetic energy by
doing work. Eq. (3.8) is called the Work-Energy Theorem.
Example 3.1.2
Let us consider again the example of the projectile. We can now
use Eq. (3.8) to calculate the speed of the projectile at an altitude y having been
launched from y
=
0
.
