To determine Find the centroid of the plane area. Explanation Calculation: Divide the plane area as shown in Figure 1. Sketch the cross-section of plane area as shown in Figure 1. Refer to Figure 1. Calculate the radius ( r ) of the semi-circular segment as shown below. r 2 = 60 2 + 60 2 r 2 = 2 × 60 2 r = 60 2 mm Calculate the angle ( α ) as shown below. tan α = 60 60 α = 45 ° α = π 4 Consider the section (1) as triangle, section (2) as semicircle, and section (3) as triangle. For section (1), The base of the triangle is b = 120 mm and height is h = 90 mm . For section (2), The radius of the semicircle is r = 60 2 mm . For section (2), The base of the triangle is b = 120 mm and height is h = 60 mm . Calculate the location of the centroid for section (2) ( y ¯ ) as shown below. y ¯ = 2 r sin α 3 α Substitute 60 2 mm for r and π 4 for α . y ¯ = 2 × 60 2 sin π 4 3 × π 4 = 50.93 mm Calculate the area using the relation as shown below. For semicircular segment. A = π r 2 2 For triangle. A = 1 2 b h Due to the symmetry of the plane area about y axis, the centroid of the plane area about x axis is X ¯ = 0 . Hence, the centroid of the plane area along x axis is X ¯ = 0 _ . Calculate the sum of the plane area using the relation as shown below. Reduce the area and centroid of the triangle in section (3) from the summation of area and centroid of the section (1) and (2). Sum = ( 1 ) + ( 2 ) − ( 3 )

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ISBN: 8220102809888

Author: BEER

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Chapter 5.1, Problem 5.15P

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Find the centroid of the plane area.

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