Find out the moment of inertia of an annular area:

Find out the M. I. of an annular area among two circles where difference of radius is quite small, around any centroidal axis.

Solution

The centroid G of such a thin circular strip of small thickness t is at its geometrical centre as illustrated in Figure

Figure 4.36

We have, dA = ds × t

where  ds = Arc length = a × d θ

Letting the plane of the area in XOY plane (O and G coincide),

= 2 π a³ × t

= 2 π at × a ²

= (Area of annular strip) × a ²

=  A′ a ²

As the annular strip of area A′ is also axis symmetric, then I _x  = I _y ,

Hence, we have I _x  =  I _y  =    I _z   /2 = A′ a²/2