International Edition---engineering Mechanics: Statics, 4th Edition
International Edition---engineering Mechanics: Statics, 4th Edition
4th Edition
ISBN: 9781305501607
Author: Andrew Pytel And Jaan Kiusalaas
Publisher: CENGAGE L
bartleby

Concept explainers

bartleby

Videos

Question
Book Icon
Chapter 9, Problem 9.51P
To determine

(a)

The principal moments of inertia and the principal directions at the centroid C for the semicircular region.

Expert Solution
Check Mark

Answer to Problem 9.51P

The principal moments of inertia:

  I1=81.43×106 mm4

  I2=22.77×106 mm4

Principal directions are along x and y axes

Explanation of Solution

Given information:

The semicircular region:

International Edition---engineering Mechanics: Statics, 4th Edition, Chapter 9, Problem 9.51P , additional homework tip  1

Calculations:

International Edition---engineering Mechanics: Statics, 4th Edition, Chapter 9, Problem 9.51P , additional homework tip  2

Because to symmetry, the x- and y- axes are the principal axes at C.

Hence,

  I1=Iy=π8R4=π8(120)4I1=81.43×106 mm4I2=Ix=0.1098R4=0.1098(120)4I2=22.77×106 mm4

Conclusion:

The principal moments of inertia at the centroid C for the semicircular region shown are I1=81.43×106 mm4 and I2=22.77×106 mm4. And the principal directions are along x and y axes.

To determine

(b)

The moments and the products of inertia about the u-v-axes for the semicircular region shown.

Expert Solution
Check Mark

Answer to Problem 9.51P

Moments of inertia:

  Iu=33.2×106mm4

  Iv=71.0×106mm4

Products of inertia:

  Iuv=22.5×106mm4

Explanation of Solution

Given information:

For the semicircular region shown:

  I1=Iy=81.43×106 mm4

  I2=Ix=22.77×106 mm4

Calculations:

  12(Ix+Iy)=12(22.77+81.43)×106=52.10×106 mm412(IxIy)=12(22.7781.43)×106=29.33×106 mm4Moments of inertia about the u-v-axes, using the relations:Iu=12(Ix+Iy)+12(IxIy)cos2θIxysin2θIu=[52.1029.33 cos( 50 o)0]×106Iu=33.2×106mm4Iv=12(Ix+Iy)12(IxIy)cos2θ+Ixysin2θIv=[52.10+29.33 cos( 50 o)+0]×106Iv=71.0×106mm4Hence, Products of inertia about the u-v-axes:Iuv=12(IxIy)sin2θ+Ixycos2θIuv=[29.33 sin( 50 o)+0]×106Iuv=22.5×106mm4

Conclusion:

For the semicircular region, the moments of inertia about the u-v axes are Iu=33.2×106mm4 and Iv=71.0×106mm4. And the products of inertia about the u-v axes is Iuv=22.5×106mm4.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
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…
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).
We 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.…

Chapter 9 Solutions

International Edition---engineering Mechanics: Statics, 4th Edition

Ch. 9 - Compute the moment of inertia about the x-axis for...Ch. 9 - By integration, find the moment of inertia about...Ch. 9 - Figure (a) shows the cross section of a column...Ch. 9 - Compute the dimensions of the rectangle shown in...Ch. 9 - Compute Ix and Iy for the W867 shape dimensioned...Ch. 9 - Figure (a) shows the cross-sectional dimensions...Ch. 9 - A W867 section is joined to a C1020 section to...Ch. 9 - Compute Ix and Iy for the region shown.Ch. 9 - Prob. 9.19PCh. 9 - Calculate Ix for the shaded region, knowing that...Ch. 9 - Compute Iy for the region shown, given that...Ch. 9 - Prob. 9.22PCh. 9 - Prob. 9.23PCh. 9 - Determine Ix for the triangular region shown.Ch. 9 - Determine the distance h for which the moment of...Ch. 9 - A circular region of radius R/2 is cut out from...Ch. 9 - Prob. 9.27PCh. 9 - Determine the ratio a/b for which Ix=Iy for the...Ch. 9 - As a round log passes through a sawmill, two slabs...Ch. 9 - Prob. 9.30PCh. 9 - By numerical integration, compute the moments of...Ch. 9 - Use numerical integration to compute the moments...Ch. 9 - The plane region A is submerged in a fluid of...Ch. 9 - Use integration to verify the formula given in...Ch. 9 - For the quarter circle in Table 9.2, verify the...Ch. 9 - Determine the product of inertia with respect to...Ch. 9 - The product of inertia of triangle (a) with...Ch. 9 - Prob. 9.38PCh. 9 - For the region shown, Ixy=320103mm4 and Iuv=0....Ch. 9 - Prob. 9.40PCh. 9 - Calculate the product of inertia with respect to...Ch. 9 - Prob. 9.42PCh. 9 - Prob. 9.43PCh. 9 - The figure shows the cross section of a standard...Ch. 9 - Prob. 9.45PCh. 9 - Prob. 9.46PCh. 9 - Prob. 9.47PCh. 9 - Use numerical integration to compute the product...Ch. 9 - Determine the dimension b of the square cutout so...Ch. 9 - For the rectangular region, determine (a) the...Ch. 9 - Prob. 9.51PCh. 9 - Prob. 9.52PCh. 9 - Prob. 9.53PCh. 9 - Prob. 9.54PCh. 9 - Prob. 9.55PCh. 9 - The u- and v-axes are the principal axes of the...Ch. 9 - The x- and y-axes are the principal axes for the...Ch. 9 - Prob. 9.58PCh. 9 - The inertial properties of the region shown with...Ch. 9 - Determine Iu for the inverted T-section shown....Ch. 9 - Using Ix and Iu from Table 9.2, determine the...Ch. 9 - Show that every axis passing through the centroid...Ch. 9 - Prob. 9.63PCh. 9 - The L806010-mm structural angle has the following...Ch. 9 - Compute the principal centroidal moments of...Ch. 9 - Prob. 9.66PCh. 9 - Determine the principal axes and the principal...Ch. 9 - Compute the principal centroidal moments of...Ch. 9 - Find the moments and the product of inertia of the...Ch. 9 - Determine the moments and product of inertia of...Ch. 9 - Find the principal moments of inertia and the...Ch. 9 - Determine the moments and product of inertia of...Ch. 9 - Prob. 9.73PCh. 9 - Prob. 9.74PCh. 9 - The u- and v-axes are the principal axes of the...Ch. 9 - The x- and y-axes are the principal axes for the...Ch. 9 - Prob. 9.77PCh. 9 - The L806010-mm structural angle has the following...Ch. 9 - Prob. 9.79RPCh. 9 - Prob. 9.80RPCh. 9 - By integration, show that the product of inertia...Ch. 9 - Compute Ix and Iy for the shaded region.Ch. 9 - Using integration, evaluate the moments of inertia...Ch. 9 - The inertial properties at point 0 for a plane...Ch. 9 - Compute Ix and Iy for the shaded region.Ch. 9 - The flanged bolt coupling is fabricated by...Ch. 9 - Prob. 9.87RPCh. 9 - Compute Ix,Iy, and Ixy for the shaded region.Ch. 9 - Determine Ix and Ixy for the shaded region shown.Ch. 9 - Calculate Ix,Iy, and Ixy for the shaded region...Ch. 9 - For the shaded region shown, determine (a) Ix and...Ch. 9 - Use integration to find Ix,Iy, and Ixy for the...Ch. 9 - Determine the principal moments of inertia and the...Ch. 9 - The properties of the unequal angle section are...
Knowledge Booster
Background pattern image
Mechanical Engineering
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
International Edition---engineering Mechanics: St...
Mechanical Engineering
ISBN:9781305501607
Author:Andrew Pytel And Jaan Kiusalaas
Publisher:CENGAGE L
moment of inertia; Author: NCERT OFFICIAL;https://www.youtube.com/watch?v=A4KhJYrt4-s;License: Standard YouTube License, CC-BY