Environmental Engineering Reference
In-Depth Information
where
F(x)
is the magnitude of
F
(x)
. Putting
x
=
x
1
/N
allows us to write, for
fixed
x
1
,
F(x)
x
F(x
1
)
x
1
=
=
k.
(2.3)
Since Eq. (2.3) must hold for any
N
the function
g(x)
F(x)/x
must take on
the same value at an infinite number of different points, i.e.
g(x
1
)
=
=
g(x
1
/
2
)
=
1
,
2and3etc.Assumingthat
F(x)
is smooth allows
us then to conclude that
F(x)/x
g(x
1
/
3
)
= ···
for
N
=
k
for all values of
x
. Thus we deduce that the
force produced by a stretched spring is proportional to its extension
x
,aresult
known as Hooke's Law:
=
F
=−
kx.
(2.4)
The choice of sign fixes the direction of the force and
k
is some constant charac-
teristic of the spring, usually called the “spring constant”.
It may seem rather restrictive that we should be using mechanical springs to
define the magnitude of a force. But remember, once we have defined our standard
force meter we can in principle use it to measure the magnitude of any other force.
Also, and as we shall see in Section 3.2.3, very many systems actually behave just
like springs, in that for small deviations from equilibrium they experience restoring
forces that satisfy Hooke's Law.
Now we have established a force meter we can begin to look at other forces.
The condition of static equilibrium allows us to put forces that arise from differ-
ent sources on an equal footing. An example of this is illustrated schematically
in Figure 2.5 in which a mass is in static equilibrium. There are two forces act-
ing on the mass: the elastic force of the stretched spring is given by Hooke's
Law, and is upwards with a magnitude
kx
; and the gravitational force or weight.
Experiments like this on different masses show that the gravitational force is
proportional to the amount of matter in the block, i.e. for identically consti-
tuted blocks, a doubling of the volume doubles the gravitational force (etc.) and
so
F
m
,where
m
∝
characterizes the amount of matter within the body and
kx
x
Mass
m
′
g
′
(
m
′)
Figure 2.5 Static equilibrium of a mass under the influence of the force of gravity and that
of a stretched spring.
