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. r 2 = 60 2 + 60 2 r 2 = 2 × 60 2 r = 60 2 mm Calculate the angle ( α ) . 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 ¯ ) using the relation. 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. For semicircular segment. A = π r 2 2 (1) For triangle. A = 1 2 b h (2) Due to the symmetry of the plane area about y axis, the centroid of the plane area about x axis is X ¯ = 0 . Therefore, the centroid of the plane area along x axis is X ¯ = 0 _ . Calculate the sum of the plane area using the relation. 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 )

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 5.1, Problem 5.15P

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The centroid of the plane area.

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