Find out the centroid of the shaded :
Find out the centroid of the shaded area shown in Figure
Solution
Net area of shaded portion of Figure
= The area of full circle of radius r (A_{1}) - the area of cut out circle of radius r/2 ( A )
= π r ^{2 }- π r ^{2} / 4 = 3 π r ^{2} / 4
Area A_{2} is to be considers as negative area.
Let moment of areas around G (the needed C. G.); ∑ A_{i} x_{i} = 0 since the lever-arm of the Resultant area A regarding G is zero,
A_{1} x_{1} + (- A _{2} ) x_{2} = A × (0) = 0
∴ π r ^{2} (- x) + ( - π r ^{2} /4) × (- (r/2)+ x¯ )} = 0
(an anticlockwise moment) (an anticlockwise moment)
∴ - (3 π r /4 ) x¯ + π r^{3}/(4 × 2) = 0
∴ x¯ = + r/6
Positive sign of x show that with respect to the origin O of reference axes x and y, x¯ = OG is along with positive direction of x axis. As the centres O and A of the two area (A_{1}) and (- A_{2}) are taken along x axis; G lies on AO.