Concept explainers
9.75 through 9.78 Using the parallel-axis theorem, determine the product of inertia of the area shown with respect to the centroidal x and y axes.
Fig. P9.75
9.82 Determine the moments of inertia and the product of inertia of the area of Prob. 9.75 with respect to new centroidal axes obtained by rotating the x and y axes 45° clockwise.
Find the moment of inertia and product of inertia with respect new centroid axes obtained x and y axes
Answer to Problem 9.82P
The moment of inertia with respect new centroid axes obtained x axes
The moment of inertia with respect new centroid axes obtained y axes
The product of inertia with respect new centroid axes obtained x axes
Explanation of Solution
Calculation:
Refer to problem 9.75.
The product of inertia of the area with respect to x and y axes by using direct parallel axis theorem is
Sketch the rectangular section as shown in Figure 1.
Find the moment of inertia
Here,
Substitute
Find the moment of inertia
Here,
Substitute
Find the total moment of inertia
Substitute
Find the moment of inertia
Substitute
Find the moment of inertia
Substitute
Find the total moment of inertia
Substitute
Find the value of
Find the value of
Find the moment of inertia with respect new centroid axes obtained x axes
Refer to Equation 9.18 in section 9.3B in the textbook.
Substitute
Thus, the moment of inertia with respect new centroid axes obtained x axes
Find the moment of inertia with respect new centroid axes obtained y axes
Refer to Equation 9.19 in section 9.3B in the textbook.
Substitute
Thus, the moment of inertia with respect new centroid axes obtained y axes
Find the product of inertia with respect new centroid axes obtained x and y axes
Substitute
Thus, the product of inertia with respect new centroid axes obtained x axes
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Chapter 9 Solutions
Vector Mechanics for Engineers: Statics
- For the area indicated, determine the orientation of the principal axes at the origin and the corresponding values of the moments of inertia.The L152 × 102 × 12.7-mm angle cross section of Prob. 9.78(Reference to Problem 9.78):Using the parallel-axis theorem, determine the product of inertia of the area shown with respect to the centroidal x and y axes.arrow_forwardUsing the polar moment of inertia of the isosceles triangle of Prob. 9.28, show that the centroidal polar moment of inertia of a circular area of radius r is π4/2. ( Hint: As a circular area is divided into an increasing number of equal circular sectors, what is the approximate shape of each circular sector?)(Reference to Problem 9.28):Determine the polar moment of inertia and the polar radius of gyration of the isosceles triangle shown with respect to point O.arrow_forwardTwo L76 × 76 × 6.4-mm angles are welded to a C250 × 22.8 channel. Determine the moments of inertia of the combined section with respect to centroidal axes that are respectively parallel and perpendicular to the web of the channel.arrow_forward
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- For the figure shown knowing that a = 1.00 cm and b = 4.00 cm, determine:(a) The centroid of the figure shown.b) The moments of inertia Ix and Iy and the product of inertia Ixy about the x and y axes.c) The angle m corresponding to the location of the principal axes.d) For the u-v axes shown and obtained by rotating an angle = 34.0° clockwise, the corresponding momentsand product of inertia using Mohr's circle.arrow_forwardDetermine by direct integration the mass moment of inertia with respect to the x axis of the tetrahedron shown, assuming that it has a uniform density and a mass m.Fig. P9.123arrow_forwarddetermine the centroidal polar moment of inertia of a rectangle 100mm wide by 200 mm higharrow_forward
- Determine the moments of inertia and the product of inertia of the area of Prob. 9.73 with respect to new centroidal axes obtained by rotating the x and y axes 60° counterclockwise.(Reference to Problem 9.73):Using the parallel-axis theorem, determine the product of inertia of the area shown with respect to the centroidal x and y axes.arrow_forwardDetermine 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.arrow_forwardTwo L4 x 4 x 1/2-in angles are welded to a steel plate as shown. Determine the moments of inertia of the combined section with respect to centroidal axes respectively parallel and perpendicular to the plate.arrow_forward
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