Explanation Given information: The element has a thickness of dz and a height of z . Calculation: Show the thin disc element shaded as in Figure 1. Express the parabolic value. z = 4 y 2 / 3 (1) Compute the volume of the element. d V = π y 2 d z (2) Rearrange Equation (I) to calculate y . z = 4 y 2 / 3 y = 1 8 z 3 / 2 Substitute 1 8 z 3 / 2 for y in Equation (2). d V = π ( 1 8 z 3 / 2 ) 2 d z = π 64 z 3 d z Compute the centroid of the differential element along the z axis using the formula, z ˜ = z Determine the location of the centre of gravity of the homogeneous rod along the z axis ( z ¯ ) using the relation, z ¯ = ∫ V z ˜ d V ∫ V d V (3) Conclusion: Apply the limits from 0 to 16 in., substitute z for z ˜ and π 64 z 3 d z for dV in Equation (3). z ¯ = ∫ 0 16 in . z ( π 64 z 3 d z ) ∫ 0 16 in . π 64 z 3 d z (4) Calculate the numerator part from Equation (4). ∫ 0 16 in . z ( π 64 z 3 d z ) = π 64 ⋅ ∫ 0 16 in

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 33P

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Find the center of gravity z¯ of the solid.

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Chapter 6.1, Problem 32P

Chapter 6.1, Problem 34P

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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 center of gravity z ¯ of the solid.

Solution Summary: The author calculates the center of gravity of the homogeneous rod along the z axis using the formula.

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Find the center of gravity z¯ of the solid.

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Chapter 6 Solutions