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C
m C
m A
m A + m c
m B
m B + m c
a
a
m A + m c
B
m B + m C
A
P
m c
m A + m c b
m c
m B + m c a
C
A
m A
m B
B
m B
m A + m B
m A + m B
m A
m A + m B c
c
Fig. 11.10. How the masses determine the positions of A ,B and C .
To summarize, given three masses m A ,m B and m C located at A , B and
C , the centroid
P is given by
m A
m A + m B + m C A +
m B
m A + m B + m C B +
m C
m A + m B + m C C
P =
(11.9)
If we accept that m A ,m B and m C can have any value, including zero, then
the barycentric coordinates of P will be affected by these values. For example,
if m B = m C = 0 and m A =1,then P will be located at A with barycentric
coordinates (1, 0, 0). Similarly, if m A = m C =0and m B =1,then P will be
located at B with barycentric coordinates (0, 1, 0). And if m A = m B =0and
m C =1,then P will be located at C with barycentric coordinates (0, 0, 1).
Now let's examine a 3D example as illustrated in Figure 11.11. The fig-
ure shows three masses 4, 8 and 12 and their equivalent mass 24 located at
( x, y, z ).
The magnitude and coordinates of three masses are shown in the follow-
ing table, together with the barycentric coordinate t i . The column headed t i
Y
4
12
24
8
y
z
x
X
Z
Fig. 11.11. Three masses can be represented by a single mass located at the system's
centroid.
 
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