Explanation Given information: The length of the shaded area is b . The height of the shaded area is h . Calculation: Show the area of the differential element as in Figure 1. Refer Figure 1, Express the parabolic values, x 1 and x 2 . x 1 = a h y x 2 = ( a − b h ) y + b Compute the area of the element. d A = ( x 2 − x 1 ) d y Here, the area of the element is dA . Substitute ( a − b h ) y + b for x 2 and a h y for x 1 . d A = ( ( a − b h ) y + b − ( a h y ) ) d y = ( ( a h y ) − ( b h y ) + b − ( a h y ) ) d y = ( b − b h y ) d y Calculate the centroid of the differential element along the x axis using the formula: x ˜ = x 1 + x 2 − x 1 2 = 1 2 ( x 1 + x 2 ) (1) Substitute ( a − b h ) y + b for x 2 and a h y for x 1 in Equation (1). x ˜ = 1 2 ( a h y + ( a − b h ) y + b ) = a h y − b 2 h y + b 2 Calculate the location of the centre of gravity of the homogeneous rod along the x axis ( x ¯ ) using the relation: x ¯ = ∫ A x ˜ d A ∫ A d A (2) Conclusion: Apply the limits from 0 to h and substitute a h y − b 2 h y + b 2 for x ˜ and ( b − b h y ) d y for dA in Equation (2). x ¯ = ∫ 0 h ( a h y − b 2 h y + b 2 ) ( b − b h y ) d y ∫ 0 h ( b − b h y ) d y (3) Calculate the numerator part from Equation (3)

Statics and Mechanics of Materials - Modified Access

5th Edition

ISBN: 9780134392363

Author: HIBBELER

Publisher: PEARSON

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Center of Gravity and Centroid

The term center of gravity indicates the position or location of a point about which the whole weight of a body or system is supposed to be concentrated, which means the whole weight of the body or system works through this point. It is closely related t…

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Chapter 6.1, Problem 24P

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Find the centroid x¯ of the shaded area.

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Statics and Mechanics of Materials - Modified Access

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Find the centroid x ¯ of the shaded area.

Solution Summary: The author calculates the centroid of the differential element along the x axis using the formula: cstackrel

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Find the centroid x¯ of the shaded area.

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Chapter 6 Solutions