Explanation Given information: The length of the shaded area is 100 mm. The height of the shaded area is 100 mm. Calculation: Show the area of the differential element as in Figure 1. Refer Figure 1, Express the parabolic value, y 1 to calculate for x 2 . y = 1 100 x 2 2 x 2 = 10 y 1 / 2 Compute the area of the element. d A = ( x 2 − x 1 ) d y Here, the area of element is dA . Substitute 10 y 1 / 2 for x 2 and y for x 1 . d A = ( 10 y 1 / 2 − y ) d y Compute the centroid of the differential element along the x axis using the formula. y ˜ = y Determine the location of the center of gravity of the homogeneous rod along the x axis ( x ¯ ) using the relation. y ¯ = ∫ A y ˜ d A ∫ A d A (1) Conclusion: Apply the limits from 0 to 100 mm and substitute y for y ˜ and ( 10 y 1 / 2 − y ) d y for dA in Equation (1). x ¯ = ∫ 0 100 mm ( y ) ( 10 y 1 / 2 − y ) d y ∫ 0 100 mm ( 10 y 1 / 2 − y ) d y (2) Calculate the numerator part using Equation (2), ∫ 0 100 mm ( y ) ( 10 y 1 / 2 − y ) d y = y ( 20 3 y 3 / 2 − y 2 2 ) − ∫ 0 100 mm 1 ⋅ ( 20 3 y 3 / 2 − y 2 2 ) d y = y ( 20 3 </

Statics and Mechanics of Materials (5th Edition)

5th Edition

ISBN: 9780134382593

Author: Russell C. 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 23P

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Find the centroid y¯ of the shaded area.

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Chapter 6.1, Problem 22P

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Chapter 6 Solutions

Statics and Mechanics of Materials (5th Edition)

Ch. 6.1 - In each case, use the element shown and specify...Ch. 6.1 - Prob. 1FP Ch. 6.1 - Determine the centroid (x,y) of the area. Prob....Ch. 6.1 - Determine the centroid y of the area. Prob. F63 Ch. 6.1 - Locate the center of gravity x of the straight rod...Ch. 6.1 - Prob. 5FP Ch. 6.1 - Locate the centroid z of the homogeneous solid...Ch. 6.1 - Locate the centroid x of the area. Prob. 61 Ch. 6.1 - Locate the centroid of the area. Prob. 62 Ch. 6.1 - Locate the centroid x of the area. Probs. 63/4

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Find the centroid y ¯ of the shaded area.

Solution Summary: The author calculates the area of the differential element as shown in Figure 1.

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