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 a^{3} × 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′ a^{2}/2