Surfaces are a two-dimensional whereas the solids are three dimensional bodies. For two –dimensional bodies, are to be determined but for three dimensional bodies volume is to be determined. This chapter deals with determination of area of two-dimensional bodies and volume of three dimensional bodies. Also the first moment of area and second moments f areas of plane sections such as rectangle, triangle, circle, T-section and I-section are discussed. The concept of mass content of intertia, principle moment of intertia and principle axis of intertia is also given.
Centre of Gravity
Centre of gravity of a body is the point through which the whole weight of the body acts. A body is having only one centre of gravity for all positions of the body. It is represented by C.G. or simply G.
The point at which the total area of a plane figure (like rectangle, square, triangle, quadrilateral, circle etc.) is assumed to be concentrated, is known as the centroid of that area. The centroid is also represented by C.G. or simply G. The centroid and centre of gravity are at the same point.
Centroid of Centre of Gravity of simple Plane Figures.
(i) The centre of gravity (C.G.) of a uniform rod lies at its middle point.
(ii) The centre of gravity of a triangle lies at the point where the three medians* of the triangle meet.
(iii) The centre of gravity of a rectangle or of a parallelogram is at the point, where its diagonal meet each other. It is also the point of intersection of the lines joining the middle points of the opposites sides.
(iv) The centre of gravity of a circle is at its centre.
Centroid (or Centre of Gravity) of Areas of Plane Figures by the Method of Moments.
Fig. 4.1 shows a plane figure of total area A whose centre of gravity is to be determined. Let this area A is composed of a number of small area a1, a2, a3, a4, …. etc.
Centroid or (Centre of Gravity) of a Line.
The centre of gravity of a line which may be straight or curve, is obtained by dividing the given line, into a large number of small lengths as shown in Fig. 4.1 (a)
i. The axis, about which moments of areas are taken, is known as axis of reference. In the above article, axis OX and OY are called axis of reference.
ii. The axis of reference, of plane figures, is generally taken as the lowest line of the figure for determining , and left line of the figure for calculating ¯x
iii. If the given section is symmetrical about X-X axis or Y-Y axis, then the C.G. of the section will lie on the axis is symmetry.