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 a² /2)(0.424)² a²

I _y  = a⁴  (0.3927 - 0.2824)

I y  = 0.1103 a ⁴