Graphics Reference
In-Depth Information
Furthermore, the above formulae can be extended to incorporate any number
of ratio divisions. For example,
A
can be divided into the ratio
r
:
s
:
t
by the
following:
r
r
+
s
+
t
A,
s
r
+
s
+
t
A
t
r
+
s
+
t
A
and
similarly
r
r
+
s
+
t
+
r
+
s
+
t
+
t
s
r
+
s
+
t
=1
These expressions are very important as they show the emergence of barycen-
tric coordinates. For the moment, though, just remember their structure and
we will investigate some ideas associated with balancing weights.
11.3 Mass Points
We begin by calculating the centre of mass - the centroid - of two masses.
Consider the scenario shown in Figure 11.3 where two masses
m
A
and
m
B
are
placed at the ends of a massless rod.
If
m
A
=
m
B
a state of equilibrium is achieved by placing the fulcrum mid-
way between the masses. If the fulcrum is moved towards
m
A
,mass
m
B
will
have a turning advantage and the rod rotates clockwise.
To calculate a state of equilibrium for a general system of masses, consider
the geometry illustrated in Figure 11.4, where two masses
m
A
and
m
B
,are
positioned
x
A
and
x
B
at
A
and
B
respectively. When the system is in balance
we can replace the two masses by a single mass
m
A
+
m
B
at the centroid
defined by
x
.
A balance condition arises when the LHS turning moment equals the RHS
turning moment. The turning moment being the product of a mass by its
offset from the fulcrum.
Equating turning moments, equilibrium is reached when
m
B
(
x
B
−
x
)=
m
A
(
x
−
x
A
)
m
B
m
A
Fig. 11.3.
Two masses fixed at the ends of a massless rod.
A
B
(
m
A
+
m
B
)
m
A
x
A
m
B
x
B
x
x
−
x
A
x
B
−
x
Fig. 11.4.
The geometry used for equating turning moments.
