Explanation Given information: The height of the section is a . The width of the section is a . The curve equation is y = k x Calculation: The curve equation as shown below: y = k x (1) The horizontal distance of the given area is a . Write the x coordinate as follows: x = a The vertical distance of the given area is - a (below the axis). Write the y coordinate as follows: y = − a Substitute a for x and − a for y in equation (1). − a = k a Rearrange the above equation for k . k = − a 2 Substitute − a 2 for k in equation (1). y = − a 2 x Express the differential version of the parallel-axis theorem as below: d I x y = d I x ' y ' + x ¯ e l y ¯ e l d A (2) Here, x ¯ e l and y ¯ e l is centroid of differential element. The element is symmetrical with respect to x ' and y ' axes. d I x ' y ' = 0 Express the centroid of the element about x axis x ¯ e l as shown below: x ¯ e l = x Express the centroid of the element about y axis y ¯ e l as below: y ¯ e l = y 2 Express the area ( d A ) of the rectangular strip in Figure 1 as below: d A = y d x Substitute y d x for d A , x for x ¯ e l , y 2 for y ¯ e l , 0 for d I x ' y ' in equation (2)

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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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 9.3, Problem 9.70P

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Find the product of inertia (Ixy) of the given area with respect to x and y axes using direct integration.

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Chapter 9.3, Problem 9.69P

Chapter 9.3, Problem 9.71P

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Vector Mechanics for Engineers: Statics and Dynamics

Ch. 9.1 - 9.1 through 9.4 Determine by direct integration...Ch. 9.1 - 9.1 through 9.4 Determine by direct integration...Ch. 9.1 - Prob. 9.3P Ch. 9.1 - Prob. 9.4P Ch. 9.1 - 9.5 through 9.8 Determine by direct integration...Ch. 9.1 - 9.5 through 9.8 Determine by direct integration...Ch. 9.1 - Prob. 9.7P Ch. 9.1 - Prob. 9.8P Ch. 9.1 - 9.9 through 9.11 Determine by direct integration...Ch. 9.1 - Prob. 9.10P

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Find the product of inertia ( I x y ) of the given area with respect to x and y axes using direct integration.

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Find the product of inertia (Ixy) of the given area with respect to x and y axes using direct integration.

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Chapter 9 Solutions