Explanation Given: The centroid of the differential element along the x , y , and z axis are x ˜ , y ˜ , and z ˜ , respectively. The mass of each segment is represented by m . The density of material 1, ρ 1 = 2.70 Mg m 3 . The density of material 2, ρ 2 = 5.70 Mg m 3 . The density of material 3, ρ 3 = 7.80 Mg m 3 . Convert the unit of density for material 1 from Mg m 3 to kg mm 3 . ρ 1 = 2.70 Mg m 3 = 2.70 Mg m 3 × ( 10 3 mm ) 3 m 3 × 10 3 kg Mg = 2.70 × 10 − 6 kg mm 3 . Convert the unit of density for material 2 from Mg m 3 to kg mm 3 . ρ 2 = 5.70 Mg m 3 = 5.70 Mg m 3 × ( 10 3 mm ) 3 m 3 × 10 3 kg Mg = 5.70 × 10 − 6 kg mm 3 . Convert the unit of density for material 3 from Mg m 3 to kg mm 3 . ρ 3 = 7.80 Mg m 3 = 7.80 Mg m 3 × ( 10 3 mm ) 3 m 3 × 10 3 kg Mg = 7.80 × 10 − 6 kg mm 3 . Show the block is divided into three segments as shown in Figure (1). Determine the location of the center of gravity of the steel and aluminum plate along the x axis ( x ¯ ) using the relation, x ¯ = ∑ x ˜ m ∑ m (I) Here, the total mass is ∑ m . Determine the location of the center of gravity of the steel and aluminum plate along the y axis ( y ¯ ) using the relation, y ¯ = ∑ y ˜ m ∑ m (II) Determine the location of the center of gravity of the steel and aluminum plate along the z axis ( z ¯ ) using the relation, z ¯ = ∑ z ˜ m ∑ m (III) Conclusion: Refer to Figure (1) and use the spreadsheet to calculate the value of x ˜ m for the three segments as shown in Table (1). Materials m ( kg ) = ρ A d x ˜ ( mm ) x ˜ m ( kg ⋅ mm ) 1 ( ρ 1 ) 1 2 ( l ) ( b ) h = [ ( 2.70 × 10 − 6 ) 1 2 ( 30 ) ( 60 ) ( 20 ) ] = 0.0486 10 0.486 2 ( ρ 2 ) ( l ) ( b ) h = [ ( 5.70 × 10 − 6 ) ( 40 ) ( 60 ) ( 20 ) ] = 0.2736 10 2.736 3 ( ρ 3 ) ( l ) ( b ) h = [ ( 7.80 × 10 − 6 ) ( 60 ) ( 60 ) ( 20 ) ] = 0

Engineering Mechanics: Statics and Study Pack (13th Edition)

13th Edition

ISBN: 9780133027990

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Chapter 9.2, Problem 79P

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The location (x¯,y¯,z¯) of the center of mass of the block.

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Chapter 9.2, Problem 78P

Chapter 9.2, Problem 80P

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Chapter 9 Solutions

Engineering Mechanics: Statics and Study Pack (13th Edition)

Ch. 9.1 - Determine the centroid (x,y) of the shaded area....Ch. 9.1 - Determine the centroid (x,y) of the shaded area....Ch. 9.1 - Determine the centroid of the shaded area. Prob....Ch. 9.1 - Locate the center of mass x of the straight rod if...Ch. 9.1 - Locate the centroid of the homogeneous solid...Ch. 9.1 - Locate the centroid z of the homogeneous solid...Ch. 9.1 - Locate the center of mass of the homogeneous rod...Ch. 9.1 - Prob. 2P Ch. 9.1 - Prob. 3P Ch. 9.1 - Prob. 4P

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The location ( x ¯ , y ¯ , z ¯ ) of the center of mass of the block.

The location (x¯,y¯,z¯) of the center of mass of the block.

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Chapter 9 Solutions