Consider a body mass M as shown in Fig. 4.42.
Let x = Distance of the centre of gravity of mass M from axis OY
y = Distances of the C.G. of mass M from axis OX
Then moment of the mass about the axis OY = M.x
The above equations is known as first moment of mass about the axis OY.
If the moment of mass given by the above equation is again multiplied by the perpendicular distance between the C.G. of the mass and axis OY, then the quantity (M.x).x =M.x2 is known as second moment of mass about the axis OY. This second moment of the mass (i.e., quantity M . x2) is known as mass moment of inertia about the axis OY.
Similarly, the second moment of mass or mass moment of inertia about the axis OX
= (M. y) . y = M . y2
Hence the product of the mass and the square of the distance of the centre of gravity of the mass from an axis is known as the mess moment Bf inertia about that axis. Mass moment of inertia is represented by Im. Hence mass moment of inertia about the axis OX is represented by (Im)xx whereas about the axis OY by (Im)yy.
Consider a body which is split up into small masses m1, m2, m3 etc. Let the C.G. of the small areas from a given axis be at a distance of r1, r2, r3 … etc. as shown in Fig. 4.43. Then mass moment of inertia of the body about the given axis is given by
1m= m1r12 + m2r22 + m3r32 + …