Explanation Given: The total height of the beam's cross-sectional area is 4 in . The total width of the beam's cross-sectional area is 8 in . To calculate the distance y ¯ to the centroid of the beam's cross-sectional area, subdivide the cross-sectional area into 3 segments as follows: Part 1: Rectangle Part 2: Rectangle Part 3: Rectangle Show the subdivided beam's cross-sectional area along with their respective centroids as in Figure 1: To calculate the moment of inertia about the centroid x ′ axis, subdivide the cross-sectional area into 3 segments as follows: Part 1: Rectangle Part 2: Rectangle Part 3: Rectangle Show the subdivided beam's cross-sectional area along with their respective centroids as in Figure 2: From Figure (1), Write the height and the width of the rectangle in part 1: b 1 = 1 in . h 1 = 6 in . Here, the height of rectangle in part 1 is h 1 and the width of rectangle in part 1 is b 1 . Write the height and the width of the rectangle in part 2. b 2 = 4 in . h 2 = 1 in . Here, the height of rectangle in part 2 is h 2 and the width of the rectangle in part 2 is b 2 . Write the height and width of the rectangle in part 3: b 3 = 4 in . h 3 = 1 in . Here, the height of the rectangle in part 3 is h 3 and the width of the rectangle in part 3 is b 3 . Centroid calculations: Calculate the distance between centroid and y axis about the x axis for rectangle in part 1: y ¯ 1 = b 1 2 (I) Substitute 1 in . for b 1 in Equation (I). y ¯ 1 = 1 2 = 0.5 in . Calculate the distance between centroid and y axis about the axis for rectangle in part 2: y ¯ 2 = b 2 2 (II) Substitute 4 in . for b 2 in Equation (II). y ¯ 2 = 4 2 = 2 in . Calculate the distance between centroid and y axis about the x axis for rectangle in part 3: y ¯ 3 = b 3 2 (III) Substitute 4 in . for b 3 in Equation (III). y ¯ 3 = 4 2 = 2 in . Conclusion: Calculate the location y ¯ . y ¯ = ∑ y ¯ A ∑ A = y ¯ 1 A 1 + y ¯ 2 A 2 + y ¯ 3 A 3 A 1 + A 2 + A 3 = y ¯ 1 [ ( b 1 ) ( h 1 ) ] + y ¯ 2 [ ( b 2 ) ( h 2 ) ] + y ¯ 3 [ ( b 3 ) ( h 3 ) ] ( b 1 ) ( h 1 ) + ( b 2 ) ( h 2 ) + ( b 3 ) ( h 3 ) (IV) Here, the area is A , the area of rectangle 1 is A 1 , the area of rectangle 2 is A 2 , and the area of rectangle 3 is A 3 . Substitute 0 .5 in . for y ¯ 1 , 2 in . for y ¯ 2 and y ¯ 3 respectively, 1 in . for b 1 , 6 in . for h 1 , 4 in . for b 2 , 4 in . for b 3 , 1 in . for h 2 , and 1 in . for h 3 in Equation (4). y ¯ = ( 0.5 ) [ ( 1 ) ( 6 ) ] + ( 2 ) [ ( 4 ) ( 1 ) ] + ( 2 ) [ ( 4 ) ( 1 ) ] ( 1 ) ( 6 ) + ( 4 ) ( 1 ) + ( 4 ) ( 1 ) = 3 + 8 + 8 6 + 4 + 4 = 19 in . 3 14 in . 2 = 1.357 in . ≈ 1.36 in . Hence, the distance y ¯ to the centroid of the beam's cross-sectional area is 1 .36 in . _ . From Figure (2), Write the height and width of the rectangle in part 1 about the x ′ axis

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

See similar textbooks

Concept explainers

Moment of Inertia

Inertia can be termed as a resistance offered by a body to change its state. A rigid body initially at rest will apply resistance when acted by an external force that tends to change its state of rest or velocity, or a body initially at motion will provi…

Videos

Question

Chapter 10.4, Problem 40P

To determine

The distance y¯ and the moment of inertia about the centroid x′ axis.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.4, Problem 39P

Chapter 10.4, Problem 41P

Students have asked these similar questions

Please sovle this for me and please don't use ai

Chapter 10 Solutions

Engineering Mechanics: Statics & Dynamics (14th Edition)

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

The distance y ¯ and the moment of inertia about the centroid x ′ axis.

Solution Summary: The author explains how to calculate the moment of inertia about the centroid xprime axis by subdividing the cross-sectional area into 3 segments.

Concept explainers

Videos

The distance y¯ and the moment of inertia about the centroid x′ axis.

Want to see the full answer?

Chapter 10 Solutions