Concept explainers
For the component described in the problem indicated,determine (a) the principal mass moments of inertia at the origin (b) the principal axes of inertia at the origin. Sketch the body and show the orientation of the principal axes of inertia relative to the x, y, and z axes.
Prob. B.58
(a)
The principal mass moments of inertia at origin.
Answer to Problem B.73P
The principal mass moments of inertia at origin are
Explanation of Solution
Given information:
The thickness of steel sheet is
The following figure illustrates different centroidal axis.
Figure-(1).
Write the expression for the mass of section 1.
Here, the mass of the square section 1 is
Write the expression for the mass of section 2.
Here, the mass of the square section 2 is
Write the expression for hypotenuse side
Here, the hypotenuse side in right angled triangle
Write the expression for hypotenuse side
Here, the hypotenuse side in right angled triangle
Write the expression of position vector of the line joining
Here, the vector along the line joining the points
Write the expression for unit vector along
Here, the unit vector along
Write the expression of mass moment of inertia of section 1 about
Here, the mass moment of inertia of section 1 about
Write the expression of mass moment of inertia of section 2 about
Here, the mass moment of inertia of section 2 about
Write the expression of total mass moment of inertia about
Here, the total mass moment of inertia about
Write the expression of mass moment of inertia of section 1 about
Here, the mass moment of inertia of section 1 about
Write the expression of mass moment of inertia of section 2 about
Here, the mass moment of inertia of section 2 about
Write the expression of total mass moment of inertia about
Write the expression of mass moment of inertia of section 1 about
Here, the mass moment of inertia of section 1 about
Write the expression of mass moment of inertia of section 2 about
Here, the mass moment of inertia of section 2 about
Write the expression of total mass moment of inertia about
Here, the total mass moment of inertia about
From, the symmetry in above figure about
Write the expression for product of mass moment of inertia in
Here, the product mass moment of inertia is
Write the expression for product mass moment of inertia in
Here, the product mass moment of inertia is
Write the expression for product mass moment of inertia in
Here, the product mass moment of inertia in
Write the expression of mass moments of inertia at origin.
Here, the principle mass moment of inertia at origin is
Calculation:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Conclusion:
The principal mass moments of inertia at origin are
(b)
The principal axes of inertia at the origin.
Answer to Problem B.73P
The principal axes of inertia at the origin at
The principal axes of inertia at the origin at
The principal axes of inertia at the origin at
Explanation of Solution
Write the expression for direction cosine in terms of
Write the expression for direction cosine in terms of
Write the expression for the direction cosines at x, y and z axis.
Write the expression for the principal axes of the inertia at x-axis.
Write the expression for the principal axes of the inertia at y-axis.
Write the expression for the principal axes of the inertia at z-axis.
Write the expression for direction cosine in terms of
Write the expression for direction cosine in terms of
Write the expression for the direction cosines at x, y and z axis.
Write the expression for the principal axes of the inertia at x-axis.
Write the expression for the principal axes of the inertia at y-axis.
Write the expression for the principal axes of the inertia at z-axis.
Write the expression for direction cosine in terms of
Write the expression for direction cosine in terms of
Write the expression for the direction cosines at x, y and z axis.
Write the expression for the principal axes of the inertia at x-axis.
Write the expression for the principal axes of the inertia at y-axis.
Write the expression for the principal axes of the inertia at z-axis.
Calculation:
Substitute
Substitute
Add Equation (XXXIX) and Equation (XL).
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Add Equation (XLIII) and Equation (XLIV).
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Add Equation (XLVII) and Equation (XLVIII).
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Conclusion:
The principal axes of inertia at the origin at
The principal axes of inertia at the origin at
The principal axes of inertia at the origin at
Want to see more full solutions like this?
Chapter B Solutions
Vector Mechanics For Engineers
- The machine element shown is fabricated from steel, where h = 64 mm. The density of Steel is y 40 mm 20 mm- 80 mm 40 mm 20 mm 40 mm Determine the mass moment of inertia of the assembly with respect to the z axis. The mass moment of inertia of the assembly with respect to the z axis is x 10-3 kg-m².arrow_forward100 mm Problem (3) A 3-mm thick piece of aluminum sheet metal is cut and bent into the machine component shown. The density of aluminum is 2770 kg/m³. Determine the mass moment of inertia of the component with respect to the y-axis. 180 mm 160 mm 240 mm 160 mmarrow_forwardh Determine the moment of inertia and radius of gyration of the composite shape with respect to the x- and y-axes knowing that b=4 cm, h = 5 cm, T₁ = 2 cm, and r₁ = 1.333 cm. For the x-axis L₂ = b For the y axis I₂ = k₂ = ky =arrow_forward
- By the method of this article, determine the moments of inertia about the x-and y-axes of the trapezoidal area. 1.5b 1.5b 1.3b b Answers: Ix= i ba baarrow_forwardThe thin plate shown is composed of a square plate of mass 80 kg and a quarter-circular plate of mass 50 kg. What is the mass moment of inertia of the shape about the y-axis?arrow_forwardDetermine the moment of inertia about the AB axis. Fig (3)arrow_forward
- The two small spheres of mass m each are connected by the light rigid rod which lies in the plane. Determine the mass moments of inertia of the assembly about the x-, y-, and z- axes. Use the values m = 2.9 kg and L = 545 mm. m Answers: Ixx = lyy Izz 11 m kg.m² kg-m² kg.m²arrow_forwardFor the component described in the problem indicated, determine (a) the principal mass moments of inertia at the origin, (b) the principal axes of inertia at the origin. Sketch the body and show the orientation of the principal axes of inertia relative to the x, y, and z axes.Prob. 9.141Reference to Problem 9.141:arrow_forward1. (10 points) A thin plate of mass m was cut in the shape of a parallelogram as shown. Determine the mass moment of inertia of the plate with respect to the y -axis. Please simplify your answer as much as possible. A B'arrow_forward
- Determine by direct integration the mass moment of inertia with respect to the z axis of the right circular cylinder shown, assuming that it has a uniform density and a mass m.arrow_forward2. A rectangular prism (brick) has dimensions a (in x) b (in y) and c (in z) directions. Taking the origin at the center determine the mass moment of inertia about the y axis in terms of total mass m.arrow_forwardTwo identical solid spheres are attached on the 2cm diameter solid rod with the density of 7750 'kg/m³. Calculate moment inertia about P- P axis of a lamina. The measurements are in cm. 3 kg 5 kg 10 cm 50 cm 40 cmarrow_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