Consider a thick lamina of area A as shown in Fig. 4.18
Let x = Distance of the C.G. of area A from the axis OY.
y = Distance of the C.G. of area A from the axis OX.
Then moment of area about the OY
= Area × perpendicular distance of C.G. of area from axis OY
=Ax … (4.3 D)
Equation (4.3 D) is known as first moment of area about the axis OY. This first moment of area is used to determine the centroid of the area.
If the moment of area given by equation (4.3 D) is again multiplied by the perpendicular distance between the C.G. of the area and axis OY (i.e., distance x), then the quantity (Ax).x=Ax2 is known as moment of the moment of area or second moment of area or area moment of intertia about the axis OY. This second moment of area is used in the study of mechanics of fluids and mechanics of solids.
Similarly, the moment of area (or first moment of area) about the axis OX=Ay.
And second moment of area (or area moment of intertia) about the axis OX=(Ay).y=Ay2.
If, instead of area, the mass (m) of the body is taken into consideration then the second moment is known as second moment of mass. This second moment of mass is also known as mass moment of intertia.
Hence moment of intertia when mass is taken into consideration about the axis OY = mx2 and about the axis OX = my2.
Hence the product of the area (or mass) and the square of the distance of the centre of gravity of the area (or mass) from an axis is known as moment of intertia of the area (or mass) about that axis. Moment of intertia is represented by I. Hence moment of intertia about the axis OX is represented by IXX whereas about the axis OY by IYY.
The product of the area (or mass) and the square of the distance of the centre of gravity of the area (or mass) from an axis perpendicular to the plane of the area is known as polar moment of intertia and is represented by J.
Consider a plane area which is split up into small area a1, a2, a3 … etc. Let the C.G. of the small circle areas from a given axis be at a distance of r1, r2, r3, …. etc as shown in Fig. 4.19.
4.5.1. Radius of Gyration.
Radius of gyration of a body (or a given lamina) about an axis is a distance such that its square multiplied by the area gives moment of intertia of the area about the given axis.
For the Fig. 4.19, the moment of intertia about the given axis is given by equation (4.4) as