Center of Gravity - Biomechanics of the Human Body - page 45

m 1

C.G.

m 2

m n

Fig. 3.6 Illustration of an extensive body composed of several elementary masses. The net weight
of this body acts at the general center of gravity, C.G., represented in the figure. The vertical
downward arrows represent the elementary weight forces. Only three of them have been drawn

and, hence, the total weight of this body, W , will correspond to the sum of weights
of each of these pieces. The general center of gravity, C.G., is represented in
the figure.
The mathematical relations describing the total mass and weight are given by
( 3.1 ), ( 3.2 ), and ( 3.3 ):

M ¼ m 1 þ m 2 þþ m n , (3.1)

W ¼ W 1 þ W 2 þþ W n , (3.2)

Mg ¼ m 1 g þ m 2 g þþ m n g , (3.3)

where g is the acceleration due to the gravity.
Now let us determine the coordinates of the center of gravity, analytically.
For this, let us analyze the previous flat figure, defining a point to be the origin of
the system of coordinates ( x , y ). Beyond this, let us suppose that this body is
suspended at the origin (0, 0) of the system and that the y -axis is vertical as
shown in Fig. 3.7 .
As in the case of analysis by the practical method, the suspended body is subject
to the torque, due to the weight, about the origin of system of coordinates which is
also the axis of rotation in this case. The resultant or net torque can be calculated
from the sum of each elementary torque, due to the action of weight of each
elementary mass or from the torque of resultant total weight acting at the center
of gravity where all the mass of the body is considered to be concentrated, whose
coordinates we want to determine. In the case of Fig. 3.7 , the lever arms of the
elementary weights about the origin of the system of coordinates correspond to the
x coordinates of each elementary mass.
The mathematical relation between the torque due to the total weight W of the
body acting at the center of gravity and the sum of torques due to the elementary
weights is given by ( 3.4 ):

Wx C : G : ¼ W 1 x 1 þ W 2 x 2 þþ W n x n : (3.4)