Machinery's Handbook, 31st Edition
234 Center of Gravity Center of Gravity of Figures of Any Outline.— If the figure is symmetrical about a center line, as in Fig. 1, the center of gravity will be located on that line. To find the exact location on that line, the simplest method is by taking moments with reference to any convenient axis at right angles to this center line. Divide the area into geometrical figures the centers of gravity of which can be easily found. In the example shown, divide the figure into three rectangles KLMN, EFGH and OPRS. Call the areas of these rectangles A , B and C , respectively, and find the center of gravity of each. Then select any convenient axis, such as X-X, at right angles to the center line Y-Y, and determine distances a , b and c . The distance y of the center of gravity of the complete figure from the axis X-X is then found from the equation: y A B C Aa Bb Cc = + + + +
Y
Y
c 1
C S E F R P O
C
B
B
x b 1
c
c
K H
G
N
y
b
b
y
A
A
a
a
L
M
a 1
X
X
X
X
Y
Y
Fig. 1. Fig. 2. Example 1: Assume that the area A is 24 square inches, B , 14 square inches, and C , 16 square inches, and that a = 3 inches, b = 7.5 inches, and c = 12 inches. Then: . . y 24 14 16 24314751612 54 369 6 83 inches # # # = + + + + = = If the figure whose center of gravity is to be found is not symmetrical about any axis, as in Fig. 2, then moments must be taken with relation to two axes X-X and Y-Y, centers of grav- ity of which can be easily found, the same as before. The center of gravity is determined by the equations: x A B C Aa Bb Cc y A B C Aa Bb Cc 1 1 1 = + + + + = + + + + Example 2: In Fig. 2, let A = 14 cm 2 , B = 18 cm 2 , and C = 20 cm 2 . Let a = 3 cm, b = 7 cm, and c = 11.5 cm. Let a 1 = 6.5 cm, b 1 = 8.5 cm, and c 1 = 7 cm. Then:
+
+
. 14 18 20 14651885207 . # # # + +
52 384 52 398
=
= =
. 738
cm
x
+
+
. 14 18 20 14318720115 # # # = = In other words, the center of gravity is located at a distance of 7.65 cm from the axis X–X and 7.38 cm from the axis Y–Y. . 765 y cm = + +
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