
Concept explainers
Find the polar moment of inertia and polar radius of gyration of the shaded area shown with respect to point P.

Answer to Problem 9.23P
The polar moment of inertia of the shaded area shown with respect to point P is
The polar radius of gyration of the shaded area shown with respect to point P is
Explanation of Solution
Given information:
The equation of the curve is
The equation of the curve is
Calculation:
Sketch the vertical strip shaded portion as shown in Figure 1.
Write the curve
Write the curve
Refer to Figure 1.
At
For
Find the constant
Substitute
Substitute 0 for x and a for y in Equation (2).
Refer to Figure 1.
At
For
Find the constant
Substitute
Substitute
Substitute
Determine the area of the strip element
Substitute
Find the shaded area (A) using the relation:
Substitute
Determine the moment of inertia
Refer Equation 9.2 in section 9.1B determining the moment of inertia of an area by integration:
Substitute
Integrate Equation (6) with respect to x.
Determine the moment of inertia
Integrate Equation (7) with respect to x.
Find the polar moment of inertia
Here,
Substitute
Thus, the polar moment of inertia of the shaded area shown with respect to point P is
Find the polar radius of gyration of the shaded area shown with respect to point P.
Here,
Substitute
Thus, the polar radius of gyration of the shaded area shown with respect to point P is
Want to see more full solutions like this?
Chapter 9 Solutions
Loose Leaf for Vector Mechanics for Engineers: Statics and Dynamics
- A hollow metal sphere of outer radius or = 2 cm is heated internally with a variable output electricheater. The sphere loses heat from its surface by convection and radiation. The heat transfercoefficient is 22 W/ m2 - o C and surface emissivity is 0.92. The ambient fluid temperature is 20 o C andthe surroundings temperature is 14 oC. Construct a graph of the surface temperature corresponding toheating rates ranging from zero to 100 watts. Assume steady state. Use a simplified model forradiation exchange based on a small gray surface enclosed by a much larger surface at 14 o C.arrow_forward2. A program to make the part depicted in Figure 26.A has been created, presented in figure 26.B, but some information still needs to be filled in. Compute the tool locations, depths, and other missing information to present a completed program. (Hint: You may have to look up geometry for the center drill and standard 0.5000 in twist drill to know the required depth to drill). Dashed line indicates - corner of original stock Intended toolpath-tangent - arc entry and exit sized to programmer's judgment 026022 (Slot and Drill Part) (Setup Instructions. (UNITS: Inches (WORKPIECE MAT'L: SAE 1020 STEEL (Workpiece: 3.25 x 2.00 x0.75 in. Plate (PRZ Location G54: ( XY 0.0 Upper Left of Fixture ( TOP OF PART 2-0 (Tool List: ) ( T04 T02 0.500 IN 4 FLUTE FLAT END MILL) #4 CENTER DRILL ' T02 0.500 TWIST DRILL N010 GOO G90 G17 G20 G49 G40 G80 G54 N020 M06 T02 (0.5 IN 4-FLUTE END MILL) R0.750 N030 S760 M03 G00 x N040 043 H02 2 Y (P1) (RAPID DOWN -TLO) P4 NO50 MOB (COOLANT ON) N060 G01 X R1.000 N070…arrow_forward6–95. The reaction of the ballast on the railway tie can be assumed uniformly distributed over its length as shown. If the wood has an allowable bending stress of σallow=1.5 ksi, determine the required minimum thickness t of the rectangular cross section of the tie to the nearest 18 in. Please include all steps. Also if you can, please explain how you found Mmax using an equation rather than using just the moment diagram. Thank you!arrow_forward
- 6–53. If the moment acting on the cross section is M=600 N⋅m, determine the resultant force the bending stress produces on the top board. Please explain each step. Please explain how you got the numbers and where you plugged them in to solve the problem. Thank you!arrow_forwardSolving coplanar forcesarrow_forwardComplete the following problems. Show your work/calculations, save as.pdf and upload to the assignment in Blackboard. 1. What are the x and y dimensions for the center position of holes 1,2, and 3 in the part shown in Figure 26.2 (below)? 6.0000 7118 Zero reference point 1.0005 1.0000 1.252 Bore C' bore 1.250 6.0000 .7118 0.2180 deep (3 holes) 2.6563 1.9445 3.000 diam. slot 0.3000 deep. 0.3000 wide 2.6563 1.9445arrow_forward
- Complete the following problems. Show your work/calculations, save as.pdf and upload to the assignment in Blackboard. missing information to present a completed program. (Hint: You may have to look up geometry for the center drill and standard 0.5000 in twist drill to know the required depth to drill). 1. What are the x and y dimensions for the center position of holes 1,2, and 3 in the part shown in Figure 26.2 (below)? 6.0000 Zero reference point 7118 1.0005 1.0000 1.252 Bore 6.0000 .7118 Cbore 0.2180 deep (3 holes) 2.6563 1.9445 Figure 26.2 026022 (8lot and Drill Part) (Setup Instructions--- (UNITS: Inches (WORKPIECE NAT'L SAE 1020 STEEL (Workpiece: 3.25 x 2.00 x0.75 in. Plate (PRZ Location 054: ' XY 0.0 - Upper Left of Fixture TOP OF PART 2-0 (Tool List ( T02 0.500 IN 4 FLUTE FLAT END MILL #4 CENTER DRILL Dashed line indicates- corner of original stock ( T04 T02 3.000 diam. slot 0.3000 deep. 0.3000 wide Intended toolpath-tangent- arc entry and exit sized to programmer's judgment…arrow_forwardA program to make the part depicted in Figure 26.A has been created, presented in figure 26.B, but some information still needs to be filled in. Compute the tool locations, depths, and other missing information to present a completed program. (Hint: You may have to look up geometry for the center drill and standard 0.5000 in twist drill to know the required depth to drill).arrow_forwardWe consider a laminar flow induced by an impulsively started infinite flat plate. The y-axis is normal to the plate. The x- and z-axes form a plane parallel to the plate. The plate is defined by y = 0. For time t <0, the plate and the flow are at rest. For t≥0, the velocity of the plate is parallel to the 2-coordinate; its value is constant and equal to uw. At infinity, the flow is at rest. The flow induced by the motion of the plate is independent of z. (a) From the continuity equation, show that v=0 everywhere in the flow and the resulting momentum equation is მu Ət Note that this equation has the form of a diffusion equation (the same form as the heat equation). (b) We introduce the new variables T, Y and U such that T=kt, Y=k/2y, U = u where k is an arbitrary constant. In the new system of variables, the solution is U(Y,T). The solution U(Y,T) is expressed by a function of Y and T and the solution u(y, t) is expressed by a function of y and t. Show that the functions are identical.…arrow_forward
- Part A: Suppose you wanted to drill a 1.5 in diameter hole through a piece of 1020 cold-rolled steel that is 2 in thick, using an HSS twist drill. What values if feed and cutting speed will you specify, along with an appropriate allowance? Part B: How much time will be required to drill the hole in the previous problem using the HSS drill?arrow_forward1.1 m 1.3 m B 60-mm diameter Brass 40-mm diameter Aluminum PROBLEM 2.52 - A rod consisting of two cylindrical portions AB and BC is restrained at both ends. Portion AB is made of brass (E₁ = 105 GPa, α = 20.9×10°/°C) and portion BC is made of aluminum (Ę₁ =72 GPa, α = 23.9×10/°C). Knowing that the rod is initially unstressed, determine (a) the normal stresses induced in portions AB and BC by a temperature rise of 42°C, (b) the corresponding deflection of point B.arrow_forward30 mm D = 40 MPa -30 mm B C 80 MPa PROBLEM 2.69 A 30-mm square was scribed on the side of a large steel pressure vessel. After pressurization, the biaxial stress condition at the square is as shown. For E = 200 GPa and v=0.30, determine the change in length of (a) side AB, (b) side BC, (c) diagnonal AC.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





