Environmental Engineering Reference
In-Depth Information
In other words, a particle in static equilibrium behaves like an isolated particle. Thus
observers in all inertial frames agree upon the static nature of our force experiment.
For the moment, let us not delve too deeply into the complexities that arise
when forces act upon extended bodies (ensembles of particles). We will work on
this in detail in the next section when we show that the centre of mass of a system
of particles behaves very much like a classical particle. For a particle in static
equilibrium the vector sum of all forces acting on it is the null vector. For example,
if forces
F
1
,
F
2
and
F
3
act on a particle, the condition for static equilibrium is
F
1
+
F
2
+
F
3
=
0
.
(2.1)
This condition may be interpreted geometrically (Figure 2.3) as the three vectors
forming a triangle when placed head-to-tail. For larger numbers of forces,
F
i
,
i
1
,
2
,
3
...N,
static equilibrium occurs when
i
=
1
F
i
=
=
0
.
F
1
F
2
F
3
Figure 2.3 Forces in static equilibrium.
To measure the magnitude of forces we need a force meter. Let's figure out how
we might make one. Imagine a situation of static equilibrium whereby one extended
spring is balanced by
N
others. Figure 2.4 shows the setup for
N
=
3. For ideal
springs, the extension of the single spring will be
N
times that of the springs on
the other side, i.e.
x
1
=
Nx
N
. Using this result we can obtain an expression for the
force exerted by a spring as a function of distance. Since we have static equilibrium
and all of the springs are collinear we can write
F(x
1
)
=
NF(x
N
)
=
NF(x
1
/N),
(2.2)
x
N
x
1
Mass
Figure 2.4 Static equilibrium with several identical springs. The extensions of the springs
x
1
and
x
N
are measured relative to the length of an unstretched spring.
