Explanation Calculation: Sketch the quarter ellipse as shown in Figure 1. Refer to problem 9.79. Refer to Figure 9.12 “Moments of inertia of common geometric Shapes”. Find the moment of inertia ( I x ) for quarter ellipse as shown below: ( I x ) = π 16 ( a b 3 ) (1) Here, a is moments and products of area for a quarter of a circle of radius. Substitute 2 a for a and a for b in Equation (1). ( I x ) = π 16 ( 2 a ) ( a ) 3 = π 16 2 a 4 = π 8 a 4 Find the moment of inertia ( I y ) for quarter ellipse as shown below: ( I y ) = π 16 ( a 3 b ) (2) Substitute 2 a for a and a for b in Equation (2). ( I y ) = π 16 ( 2 a ) 3 ( a ) = π 16 8 a 4 = π 2 a 4 Refer to problem 9.67. Write the curve Equation as shown below: x 2 4 a 2 + y 2 a 2 = 1 (3) Modify Equation (3). x 2 4 a 2 + y 2 a 2 = 1 y 2 a 2 = 1 − x 2 4 a 2 y a = 1 − x 2 4 a 2 y = a 1 − x 2 4 a 2 y = 1 2 4 a 2 − x 2 Select a vertical strip as differential element of area. Applying the parallel axis theorem. d I x y = d I x ′ y ′ + x ¯ E L y ¯ E L d A Here, x ¯ E L and y ¯ E L are the coordinates of the centroid of the element of area d A . When the x axis, the y axis or both axis of symmetry. d I ¯ x ′ y ′ = 0 Express the variables in terms of x and y . x ¯ E L = x Find the coordinate of centroid element y ¯ E L as shown below: y ¯ E L = 1 2 y (4) Substitute 1 2 4 a 2 − x 2 for y in Equation (4). y ¯ E L = 1 2 4 a 2 − x 2 Consider the element strip as follows: d A = y d x = 1 2 4 a 2 − x 2 d x Integrating d I x apply the limits as shown below: I x y = ∫ d I x y = ∫ 0 2 a x ( 1 4 4 a 2 − x 2 ) ( 1 2 4 a 2 − x 2 ) d x = 1 8 ∫ 0 2 a ( 4 a 2 x − x 3 ) d x = 1 8 [ 4 a 2 x 2 2 − x 4 4 ] 0 2 a = 1 8 [ 2 a 2 ( 2 a ) 2 − ( 2 a ) 4 4 ] = 1 8 [ 8 a 4 − 4 a 4 ] = 1 8 [ 4 a 4 ] I x y = 1 2 a 4 Find the orientation of the principal axes at the origin using the equation as follows: Refer to Equation 9

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 9.85P

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Find the orientation of the principal axes at the origin and the corresponding value of moment of inertia.

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Chapter 9.3, Problem 9.84P

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Determine for the quarter ellipse of Prob. 9.67 the moments of inertia and the product of inertia with respect to new axes obtained by rotating the x and y axes about O (a) through 45° counterclockwise, (b) through 30° clockwise.(Reference to Problem 9.67):Determine by direct integration the product of inertia of the given area with respect to the x and y axes.

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Determine the moments of inertia and the product of inertia of the L3 x 2 x 1/4-14 angle cross section of Prob. 9.74 with respect to new centroidal axes obtained by rotating the x and y axes 30° clockwise.(Reference to Problem 9.75):Using the parallel-axis theorem, determine the product of inertia of the area shown with respect to the centroidal x and y axes.

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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 orientation of the principal axes at the origin and the corresponding value of moment of inertia.

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