Concept explainers
9.47 and 9.48 Determine the polar moment of inertia of the area shown with respect to (a) point O, (b) the centroid of the area.
Fig. P9.48
(a)
Find the polar moment of inertia of the area with respect to point O.
Answer to Problem 9.48P
The polar moment of inertia of the area with respect to point O is
Explanation of Solution
Calculation:
Sketch the cross section as shown in Figure 1.
Refer to Figure1.
It is divided into 4 parts as shown above.
Find the area of section 1 ellipsoid using the relation:
Substitute
Find the area of section 2 ellipsoid using the relation:
Here,
Substitute
Find the area of section 3 ellipsoid using the relation:
Substitute
Find the area of section 4 ellipsoid using the relation:
Substitute
Find the total are of section (A) as shown below:
Substitute
Find the centroid
Find the centroid
Find the centroid
Find the centroid
Find the centroid
Substitute
Find the polar moment of inertia
Substitute
Find the polar moment of inertia
Substitute
Find the polar moment of inertia
Substitute
Find the polar moment of inertia
Substitute
Find the total moment of inertia
Substitute
Thus, the polar moment of inertia of the area with respect to point O is
(b)
Find the centroid of area.
Answer to Problem 9.48P
The centroid of area is
Explanation of Solution
Calculation:
Find the centroid of area using the relation:
Substitute
Thus, the centroid of area is
Want to see more full solutions like this?
Chapter 9 Solutions
VEC MECH 180-DAT EBOOK ACCESS(STAT+DYNA)
- 1.3 cm 1.0 cm -0.5 cm 3.8 cm 0.5 cm AI B 3.6 cm PROBLEM 9.44 Determine the moments of inertia I, and I, of the area shown with respect to centroidal axes respectively parallel and perpendicular to side AB.arrow_forward1. Determine the moment of inertia of the above section (Fig. 1) about Ox axis. 2. Determine the moment of inertia of the above section (Fig. 2) about the vertical axis passing through the centroid.arrow_forwardA channel and a plate are welded together as shown to form a section that is symmetrical with respect to the y axis. Determine the moments of inertia of the combined section with respect to its centroidal x and y axes.arrow_forward
- For the area indicated, determine the orientation of the principal axes at the origin and the corresponding values of the moments of inertia.Area of Prob. 9.75(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_forward9.31 and 9.32 Determine the moment of inertia and the radius of gyration of the shaded area with respect to the x axis. 2 in. 2 in. 3 in. -3 in.+ 1 in. 3 in. 2 in. 2 in. 3 in. 1 in. 1 in.- -1 in. Fig. P9.32.arrow_forwardDetermine the polar moment of inertia of the area shown with respect to (a) point O, (b) the centroid of the area.arrow_forward
- Two L4 × 4 × 1/2-in. angles are welded to a steel plate as shown.Determine the moments of inertia of the combined section with respect to the centroidal axes that are respectively parallel and perpendicular to the plate.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_forwardTwo L6 × 4 × 1/2-in. angles are welded together to form the section shown. Determine the moments of inertia and the radii of gyration of the combined section with respect to the centroidal x and y axes.arrow_forward
- Q.1) Determine the moment of inertia of the area below. Use integration. y y = mx b h Xarrow_forwardProblem 09.086 - Orientation of the principal axes and the corresponding moments of inertia For the area indicated, determine the orientation of the principal axes at the origin and the corresponding values of the moments of inertia when b= 76 mm and h = 56 mm. b The value of mis The value of 0m2 is The value of Imax is The value of I min is 180 270 1.429 7292 × h x 106 mm4. 106 mm4arrow_forward(a) Determine by direct integration the polar moment of inertia of the annular area shown. (b) Using the result of part a, determine the moment of inertia of the given area with respect to the x axis.Fig. P9.18arrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY