Biomedical Engineering Reference
In-Depth Information
Fig. 3.8 Illustration of the
same extensive body of
Fig. 3.6 , but with a rotation
of x and y axis, in a way that
the x -axis is now vertical
and parallel to the
elementary weights and to
the resultant weight applied
at the center of gravity, C.
G., whose coordinates we
want to determine. The
downward arrows represent
some elementary weights
x
m 1
m n
C.G.
m 2
0
0
y n
y 2
y 1
y
m 1 y 1 þ
m 2 y 2 þþ
m n y n
y C : G : ¼
(3.9)
:
M
An analogous analysis performed to locate the coordinates x C.G. and y C.G. of
the center of gravity could be done for a third coordinate, z , orthogonal to the
coordinates x and y . This would complete the determination of the coordinates of
the center of mass of an extensive body, with volume. Such a coordinate should be
given by ( 3.10 ):
m 1 z 1 þ
m 2 z 2 þþ
m n z n
z C : G : ¼
(3.10)
:
M
The expressions ( 3.6 ), ( 3.9 ), and ( 3.10 ) for the coordinates of the center of
gravity correspond to the center of mass of the body. This identity is due to the
fact that in the process of deduction of the coordinates of the center of gravity, the
acceleration, g , due to the gravity was eliminated in the mathematical expressions.
As already written, this can only be done if the value of g can be considered
constant, which is true for small bodies compared to the size of earth. When this
is not the case, the center of gravity does not correspond to the center of mass.
An important observation should be made about the origin of the system of
coordinates. To obtain the coordinates of the center of mass, we have placed the
origin at the stipulated axis of rotation, and we have based our analysis on the torque
of the weight force. The mathematical equations obtained involve, however, only
the elementary masses and their respective coordinates, limiting them to the spatial
distribution of the mass of a body. In this case, it is possible to choose any origin for
the system of coordinates, arbitrarily. Then, it is recommended, for convenience,
that the choice should be done to simplify the calculations. One such example
would be to choose an origin such that the significant part of the lever arms of the
elementary weights should be zero for one of the coordinates. Observe, moreover,
that the coordinates of the elements of mass can be positive or negative.
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