Biomedical Engineering Reference
In-Depth Information
y
m
1
C.G.
m
2
m
n
0
axis of rotation
0
x
2
x
n
x
1
x
Fig. 3.7
Illustration of an extensive body composed of several elementary masses. In this case,
the
y
-axis is vertical and parallel to the force (elementary weights) vectors and to the resultant
weight acting at the center of gravity, C.G., whose coordinates are to be determined. The
vertical
downward arrows
represent the elementary weights
Substituting the weights by the product of the masses and the acceleration of
gravity,
g
, we obtain (
3.5
):
Mgx
C
:
G
:
¼
m
1
gx
1
þ
m
2
gx
2
þþ
m
n
gx
n
:
(3.5)
The coordinate
x
C.G.
of the center of gravity is obtained, isolating it from
(
3.5
). Observe further that the acceleration of gravity can be canceled on both
sides of (
3.5
), if we consider it constant, at the location of the body. Then, we
can write
X
n
m
i
x
i
M
:
m
1
x
1
þ
m
2
x
2
þþ
m
n
x
n
i¼
1
x
C
:
G
:
¼
¼
(3.6)
M
The determination of the coordinate
y
C.G.
of the center of gravity can be done
similarly. For this, we have to rotate the system of coordinates without changing the
disposition of the body about the
x
and
y
axes. Figure
3.8
illustrates the new situation
where the
x
-axis is vertical, with the same direction of the line of action of the weight.
Again, the resultant torque can be calculated from the sum of each elementary torque
or from the torque of the total weight applied at the center of gravity. In the case of
Fig.
3.8
, the lever arms of elementary weights about the origin of the system of
coordinates correspond to the coordinate
y
of each elementary mass.
Analogously to what has been done to determine the coordinate
x
C.G.
of the
center of gravity, we can write (
3.7
), (
3.8
), and (
3.9
):
Wy
C
:
G
:
¼
W
1
y
1
þ
W
2
y
2
þþ
W
n
y
n
,
(3.7)
Mgy
C
:
G
:
¼
m
1
gy
1
þ
m
2
gy
2
þþ
m
n
gy
n
,
(3.8)
