Find out the moment of inertia of a semicircle:
Find out the M. I. of a semicircle ABM around its periphery AB and also around centroidal axes x and y as illustrated in Figure.
Solution
The position of centroid C of the semicircle is already calculated earlier. It is
x¯ = 4 a /3 π ≈ 0.424 a
I_{(AB)} for semicircle = 1/2 [I_{ AB} for full circle] = π a ^{4} / 8
I_{GY} + A (x¯)^{2 } = I ( AB) for semicircle = (½) I _{(AB)} for full circle
∴ I _{y} = π a /8 - (π a^{2} /2)(0.424)^{2} a^{2}
I _{y } = a^{4 } (0.3927 - 0.2824)
I y = 0.1103 a^{ 4}