
Find the moment and product of inertia of the area with respect to x and y axes about through

Answer to Problem 9.192RP
The moment of inertia of the area with respect to x about through
The moment of inertia of the area with respect to y about through
The product of inertia of the area with respect to x and y axes about through
Explanation of Solution
Sketch the cross section as shown in Figure 1.
Refer to Figure 9.13.
The moment of inertia
The moment of inertia
Refer to Problem 9.191.
Sketch the cross section as shown in Figure 2.
Express the product of inertia as shown below:
Here,
Applying parallel axis theorem,
When the x and y axis is symmetry.
Refer to Figure 1.
Find the area of semicircle section 1 as shown below:
Here,
Substitute
Find the area of rectangle section 2 as shown below:
Here,
Substitute
Find the centroid for section 1 about x axis
Find the centroid for section 1 about x axis
Find the centroid for section 1 about y axis
Find the centroid for section 2 about x axis
Find the product of inertia of the area with respect to x and y axes by using parallel axis theorem as shown below:
Substitute
The Mohr circle is defined by the diameter XY, where
Find the average moment of inertia
Here,
Substitute
Find the radius (R) using the relation as shown below:
Here, R is radius and
Substitute
Sketch the Mohr circle as shown in Figure 3.
Refer to Figure 2.
Substitute
Find the angle
Find the moment of inertia of the area with respect to x about through
Here,
Substitute
Thus, the moment of inertia of the area with respect to x about through
Find the moment of inertia of the area with respect to y about through
Substitute
Thus, the moment of inertia of the area with respect to y about through
Find the product of inertia of the area with respect to y about through
Substitute
Thus, the product of inertia of the area with respect to x and y axes about through
(b)
Find the orientation of the principal axes through the centroid and corresponding values.
(b)

Answer to Problem 9.192RP
The orientation of the principal axes at the origin is
The maximum moment of inertia is
The minimum moment of inertia is
Explanation of Solution
Calculation:
Find the orientation of the principal axes through at origin as shown below.
Refer part a.
Thus, the orientation of the principal axes at the origin is
Sketch the orientation axis as shown in Figure 4.
Find the maximum moment
Substitute
Thus, the maximum moment of inertia is
Find the minimum moment
Substitute
Thus, the minimum moment of inertia is
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Chapter 9 Solutions
EBK VECTOR MECHANICS FOR ENGINEERS: STA
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