Find out the centroid of the shaded area, Mechanical Engineering

Find out the centroid of the shaded :

Find out the centroid of the shaded area shown in Figure

689_Find out the centroid of the shaded.jpeg

Solution

Net area of shaded portion of Figure

= The area of full circle of radius r (A1) -   the area of cut out circle of radius r/2 ( A )

= π r 2 - π r 2 / 4 = 3 π r 2 / 4

 Area A2 is to be considers as negative area.

Let moment of areas around G (the needed C. G.); ∑ Ai xi = 0 since the lever-arm of the Resultant area A regarding G is zero,

A1  x1 + (- A 2 ) x2  =  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 r3/(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 (A1) and (- A2) are taken along x axis; G lies on AO.

Posted Date: 1/29/2013 12:45:06 AM | Location : United States







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