Explanation Given information: The height of the shaded area is 1 m . The width of the shaded area is 2 m . Calculation: Show the area of the differential element parallel to the y axis as in Figure 1. From the Figure 1, Compute the area of the differential element parallel to the y axis as shown with shaded area from the Figure 1. d A = y d x Here, the centroid of the differential element along y axis is y . Compute the centroid of the differential element along x and y axis using the formula. y ˜ = y 2 Here, the centroid of the differential element along y axis is y ˜ . Solve the given equation for the shaded area to obtain the centroid of the differential element along y axis. x 2 + 4 y 2 = 4 y 2 = 4 − x 2 4 y 2 = 1 4 − x 2 4 y = 1 4 − x 2 4 y = 1 2 4 − x 2 Compute the moment of inertia of the differential element parallel to the y axis about the x axis. d I x = d I ¯ x ′ + d A y ˜ 2 (1) Here, the first integral of the moment of inertia of the area about the centroid axis is I ¯ x ′ and the centroid of the differential element along y axis is y ˜ . Substitute 1 12 ( d x ) y 3 for d I ¯ x ′ , y d x for d A , and y / 2 for y ˜ in Equation (1). d I x = 1 12 ( d x ) y 3 + y d x ( y / 2 ) 2 = [ y 3 12 + y ( y 2 / 4 ) ] d x = [ y 3 12 + y 3 4 ] d x = 4 y 3 12 d x = y 3 3 d x (2) Substitute 1 2 4 − x 2 for y in Equation (2). d I x = [ 1 2 4 − x 2 ] 3 3 d x = 1 24 ( 4 − x 2 ) 3 d x Conclusion: Express the moment of inertia for the shaded area about the x axis

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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…

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Chapter 6.4, Problem 68P

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Chapter 6 Solutions

EBK STATICS AND MECHANICS OF MATERIALS

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 moment of inertia about the x axis.

Solution Summary: The author calculates the moment of inertia of the differential element parallel to the x axis.

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