46 APPLIED MECHANICS 
The centroid of a straight line is at its middle point. 
The centroid of a triangle is at the intersection of its medians. 
The centroid of a parallelogram is at the intersection of its diagonals. 
_If a plane figure is symmetrical about a straight line, the centroid of 
the figure is in that straight line. 
65. Examples on the Determination of Centroids.—(1) ABC is 
a triangle AB=BC=23% inches, AC=33 inches. D is a point within 
the triangle ABC 24 inches from A and 1} inches from B. Small bodies 
are placed at the points A, B, C, and D, their masses being proportional 
to the numbers 3, 3°5, 2°5, and 5 respectively. It is required to find the 
centre of gravity of the four bodies. 
The graphic method of working this example is fully explained in. 
Art. 63, and is illustrated by Fig. 43. The dimensions in Fig. 43 are, 
however, not the same as given above. (If G be the required centre of 
gravity, then AG = 1-94 inches, and BG = 1:21 inches.) 
(2) A piece of wire of uniform thickness is bent to the form ABCD 
a = © o 2 cep 3 
edbaes ta ine : 
w& 
Fia. 44. 
(Fig. 44). The parts AB, CD, and DA are straight, and the part BC is 
an are of a circle whose centre is A. AB=24 inches, CD=1 inch, 
DA = 2 inches, and the are BC subtends an angle of 60° at A. It is 
required to find the centre of gravity of the frame ABCD. 
The are BC is divided into four equal parts, and the centre of gravity 
of each of these is assumed to be at its middle point. This assumption 
only involves a small error, because the ares are small compared with the 
radius of the circle. It may also be assumed that the weights of these 
small arcs of wire are proportional to the length of their chords. The 
weights of the straight sides are proportional to their lengths, and their 
centres of gravity are at their middle points. The weight of each part 
into which the frame is divided may be supposed to act at its centre of 
gravity, and the problem becomes similar to the preceding one. G is the. 
required centre of gravity. ‘é 
Scales to be used.—For the frame, full size. For the forces, 1 inch 
equal to the weight of 2 inches of wire. 
