Explanation Given information: The specific weight of the plate, γ p l a t e = 180 lb / ft 3 . The thickness of the plate, t = 0.25 f t . Calculation: Show the area of the differential element as in Figure 1. Show the reaction at the plate as in Figure 2. Use Figure 1, Express the parabolic equation, y . y = x − 8 x 1 / 2 + 16 Compute the area of the differential element. d A = y d x Substitute x − 8 x 1 / 2 + 16 for y . d A = ( x − 8 x 1 / 2 + 16 ) d x Compute the centroid of the differential element along the x- axis using the formula. x ˜ = x Compute the centroid of the differential element along the y- axis using the formula. y ˜ = x − 8 x 1 / 2 + 16 2 Determine the area of a plate by applying the limits from 0 to 16 ft. A = ∫ 0 16 ft d A = ∫ 0 16 ( x − 8 x 1 / 2 + 16 ) d x (1) Determine the location of the centre of gravity of the plate along the x -axis ( x ¯ ) using the relation. x ¯ = ∫ A x ˜ d A A (2) Determine the location of the centre of gravity of the plate along the y -axis ( y ¯ ) using the relation. y ¯ = ∫ A y ˜ d A A (3) Use Figure 2, Determine the total weight on the plate. W = γ p l a t e ( A ) t (4) Here, the area of the plate is A and thickness of the plate is t . Determine the total moment along x direction to calculate T A . ∑ M x = 0 x ¯ ( W ) − T A ( 16 ) = 0 (5) Determine the total moment along y direction to calculate T C . ∑ M y = 0 T C ( 16 ) − W ( y ¯ ) = 0 (6) Determine the total force in the z -direction to find T B . ∑ F z = 0 T B + T A + T C − W = 0 (7) Conclusion: Integrate Equation (1). A = ∫ 0 16 ( x − 8 x 1 / 2 + 16 ) d x = [ 1 2 x 2 − 16 3 x 3 / 2 + 16 x ] 0 16 ft = 42.67 ft 2 Apply the limits from 0 to 16 ft and substitute 42.67 ft 2 for A , x for x ˜ , and ( x − 8 x 1 / 2 + 16 ) d x for dA in Equation (2). x ¯ = ∫ 0 16 x ( x − 8 x 1 / 2 + 16 ) d x 42.67 = [ 1 3 x 3 − 16 5 x 5 / 2 + 8 x 2 ] 0 16 ft 4

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ISBN: 8220102955295

Author: HIBBELER

Publisher: PEARSON

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Chapter 6.1, Problem 19P

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Find the location of center of gravity (x¯,y¯) of a plate

Find the tensions (TA,TB, and TC) in each of the cords used to support the plate.

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Find the location of center of gravity ( x ¯ , y ¯ ) of a plate Find the tensions ( T A , T B , and T C ) in each of the cords used to support the plate.

Solution Summary: The author shows the location of center of gravity (stackrel x,y) of a plate, and the tensions in each of the cords used to support

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Find the location of center of gravity (x¯,y¯) of a plate

Find the tensions (TA,TB, and TC) in each of the cords used to support the plate.

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Chapter 6 Solutions

Find the location of center of gravity ( x ¯ , y ¯ ) of a plate Find the tensions ( T A , T B , and T C ) in each of the cords used to support the plate.

Solution Summary: The author shows the location of center of gravity (stackrel x,y) of a plate, and the tensions in each of the cords used to support

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Find the location of center of gravity (x¯,y¯) of a plate Find the tensions (TA,TB, and TC) in each of the cords used to support the plate.

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Chapter 6 Solutions

Find the location of center of gravity (x¯,y¯) of a plate

Find the tensions (TA,TB, and TC) in each of the cords used to support the plate.