Concept explainers
The mass moment of inertia of the machine component with respect to the axis through the origin characterized by the unit
Answer to Problem B.61P
The mass moment of inertia of the machine component with respect to the axis through the origin by the unit vector is
Explanation of Solution
Given information:
The diameter of the formed steel wire is
The below figure represents the schematic diagram of the wire.
Figure-(1)
Concept used:
Expression of mass of the steel wire.
Here, density of the steel wire is
Expression of density of the steel wire.
Here, the specific weight of the steel wire is
Substitute
For section (1).
Substitute
Expression of volume of the steel wire after formation for section (1).
Here, the diameter of the formed wire is
For section (2).
Substitute
Expression of volume of the steel wire after formation for section (2).
For section (3).
Substitute
Expression of volume of the steel wire after formation for section (3).
For section (4).
Substitute
Expression of volume of the steel wire after formation for section (4).
Expression of Moment of inertia about
Here, the moment of inertia of section (1) about the
Expression of moment of inertia of section (1) about the
Expression of moment of inertia of section (2) about the
Expression of moment of inertia of section (3) about the
Expression of moment of inertia of section (4) about the
Expression of Moment of inertia about
Here, the moment of inertia of section (1) about the
Expression of moment of inertia of section (1) about the
Expression of moment of inertia of section (2) about the
Expression of moment of inertia of section (3) about the
Expression of moment of inertia of section (4) about the
Expression of Moment of inertia about
Here, the moment of inertia of section (1) about the
Expression of moment of inertia of section (1) about the
Expression of moment of inertia of section (2) about the
Expression of moment of inertia of section (3) about the
Expression of moment of inertia of section (4) about the
Mass products of inertia
Here. Mass products of inertia is
Mass products of inertia
Here. Mass products of inertia is
Mass products of inertia
Here. Mass products of inertia is
Calculation:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Thus, the mass moment of inertia of the wire with respect to
Substitute
Substitute
Substitute
Substitute
Substitute
Thus, the mass moment of inertia of the wire with respect to
Substitute
Substitute
Substitute
Substitute
Substitute
Product of inertia
Substitute
Substitute
Substitute
Mass moment of inertia of the machine component with respect to the axis through the origin by the unit vector is calculated as follows:
Conclusion:
The mass moment of inertia of the machine component with respect to the axis through the origin by the unit vector is
Want to see more full solutions like this?
Chapter B Solutions
Vector Mechanics For Engineers
- For the shape below, the moment :of inertia about X-axis is 25 mm 250 mm 25 mm 25 mm |x= 1791666667 mm^4 O |x= 1792226667 mm^4 O Ix= 222135416.7 mm^4 O 200 mmarrow_forwardFind the center of mass and the moment of inertia about the y-axis of a thin plate bounded by the x-axis, the lines x tl, and the parabola y = x if 6 (x, y) 7y + 1.arrow_forwardProblem 7: Determine the moment of inertia for the shaded blue area in the figure below about the x- axis. 1 ft- y = x- X1 (x, y) -y=x² dy (x₁.y) 1 ftarrow_forward
- Determine the moment of inertia about the y-axis of the shaded area of the figure shown:*arrow_forwardBy the method of this article, determine the rectangular and polar radii of gyration of the shaded area about the axes shown. Assume r = 0.36a. a Answers: ky = a ky = i a kz = i aarrow_forwardA rigid body is rotating around O in the vertical plane. It consists of two slender rods, each of mass m and length L. At the instant shown, the rigid body has angular velocity w and angular acceleration a. Determine the following: L 0 8 Motion L ,لياarrow_forward
- Consider Figure Q3, Determine the mass moment of inertia of the composite body about the x and y axes. Mass of rod = 2 kg/m Mass of disc = 15 kg/m2arrow_forwardIn the figure the L- shaped machine part is composed of two homogeneous bars . Bar 1 is tungsten alloy with mass density 14,000 kg/m3 , and bar 2 is steel with mass density 7800 kg/m3 . Determine its moment of inertia about the x axis.arrow_forwardFind the moment of inertia and radius of gyration of the section of this bar about an axis parallel to x-axis going through the center of gravity of the bar. The bar is symmetrical about the axis parallel to y-axis and going through the center of gravity of the bar and about the axis parallel to z-axis and going through the center of gravity of the bar. The dimensions of the section are: l=55 mm, h=22 mm The triangle: hT=12 mm, lT=19 mm and the 2 circles: diameter=8 mm, hC=6 mm, dC=8 mm. A is the origin of the referential axis. Provide an organized table and explain all your steps to find the moment of inertia and radius of gyration about an axis parallel to x-axis and going through the center of gravity of the bar. Does the radius of gyration make sense? Enter the y position of the center of gravity of the bar in mm with one decimal.arrow_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