Vector Mechanics for Engineers: Statics, 11th Edition
11th Edition
ISBN: 9780077687304
Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek
Publisher: McGraw-Hill Education
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Chapter 9.4, Problem 9.103P
(a)
To determine
Find the product of inertia
(b)
To determine
Find the orientation of the principal axes.
(C)
To determine
Find the maximum moment of inertia.
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Find ȳ and the moment of inertia about the X-axis, Y-axis, and X'-axis of the cross-sectional area, given: L1 = 11 in, L2 = 1.4 in, L3 = 7.5 in, L4 = 1 in.
The moments and product of inertia of an L4 x 3 x 1/4-in. angle cross section with respect to two rectangular axes x and y through C are, respectively, Ix = 1.33 in4. Iy = 2.75 in4, and Ixy <0, with the minimum value of the moment of inertia of the area with respect to any axis through C being Imim = 0.692 in 4 Using Mohr’s circle, determine (a) the product of inertia Ixy of the area, (b) the orientation of the principal axes, (c) the value of Imax.
Determine the moment reaction at the fixed support A. The forces P = 218 N, F =436 N, and Q = 218 N are parallel to the y, x and z axis
respectively. The moment MCB = 436 N.m is around the line CB.
P
B
2 m
1 m
2.5 m
McB
2 m
D
F
O A. MA = -872.0k N.m
O B. MA = 218i + -654.0j + -327.0k N.m
O C. MA = -436.0j N.m
O D. MA = -654.0j + -545.0k N.m
O E. MA = 545.0i + -545.0j + -545.0k N.m
Chapter 9 Solutions
Vector Mechanics for Engineers: Statics, 11th Edition
Ch. 9.1 - 9.1 through 9.4 Determine by direct integration...Ch. 9.1 - 9.1 through 9.4 Determine by direct integration...Ch. 9.1 - 9.1 through 9.4 Determine by direct integration...Ch. 9.1 - 9.1 through 9.4 Determine by direct integration...Ch. 9.1 - 9.5 through 9.8 Determine by direct integration...Ch. 9.1 - 9.5 through 9.8 Determine by direct integration...Ch. 9.1 - Prob. 9.7PCh. 9.1 - Prob. 9.8PCh. 9.1 - 9.9 through 9.11 Determine by direct integration...Ch. 9.1 - Prob. 9.10P
Ch. 9.1 - 9.9 through 9.11 Determine by direct integration...Ch. 9.1 - 9.12 through 9.14 Determine by direct integration...Ch. 9.1 - Prob. 9.13PCh. 9.1 - 9.12 through 9.14 Determine by direct integration...Ch. 9.1 - 9.15 through 9.16 Determine the moment of inertia...Ch. 9.1 - 9.15 through 9.16 Determine the moment of inertia...Ch. 9.1 - 9.17 through 9.18 Determine the moment of inertia...Ch. 9.1 - Prob. 9.18PCh. 9.1 - Prob. 9.19PCh. 9.1 - Prob. 9.20PCh. 9.1 - Prob. 9.21PCh. 9.1 - 9.21 and 9.22 Determine the polar moment of...Ch. 9.1 - 9.23 and 9.24 Determine the polar moment of...Ch. 9.1 - 9.23 and 9.24 Determine the polar moment of...Ch. 9.1 - (a) Determine by direct integration the polar...Ch. 9.1 - (a) Show that the polar radius of gyration kQ of...Ch. 9.1 - Determine the polar moment of inertia and the...Ch. 9.1 - Determine the polar moment of inertia and the...Ch. 9.1 - Using the polar moment of inertia of the isosceles...Ch. 9.1 - Prove that the centroidal polar moment of inertia...Ch. 9.2 - 9.31 and 9.32 Determine the moment of inertia and...Ch. 9.2 - 9.31 and 9.32 Determine the moment of inertia and...Ch. 9.2 - 9.33 and 9.34 Determine the moment of inertia and...Ch. 9.2 - 9.33 and 9.34 Determine the moment of inertia and...Ch. 9.2 - 9.35 and 9.36 Determine the moments of inertia of...Ch. 9.2 - Prob. 9.36PCh. 9.2 - 9.37 The centroidal polar moment of inertia of...Ch. 9.2 - 9.38 Determine the centroidal polar moment of...Ch. 9.2 - 9.39 Determine the shaded area and its moment of...Ch. 9.2 - 9.40 Knowing that the shaded area is equal to 6000...Ch. 9.2 - 9.41 through 9.44 Determine the moments of inertia...Ch. 9.2 - 9.41 through 9.44 Determine the moments of inertia...Ch. 9.2 - 9.41 through 9.44 Determine the moments of inertia...Ch. 9.2 - Prob. 9.44PCh. 9.2 - 9.45 and 9.46 Determine the polar moment of...Ch. 9.2 - 9.45 and 9.46 Determine the polar moment of...Ch. 9.2 - Prob. 9.47PCh. 9.2 - Prob. 9.48PCh. 9.2 - 9.49 Two channels and two plates are used to form...Ch. 9.2 - 9.50 Two . angles are welded together to form the...Ch. 9.2 - Four L3 3 14 - in. angles are welded to a rolled...Ch. 9.2 - Two 20-mm steel plates are welded to a rolled S...Ch. 9.2 - A channel and a plate are welded together as shown...Ch. 9.2 - The strength of the rolled W section shown is...Ch. 9.2 - Two L76 76 6.4-mm angles are welded to a C250 ...Ch. 9.2 - Two steel plates are welded to a rolled W section...Ch. 9.2 - 9.57 and 9.58 The panel shown forms the end of a...Ch. 9.2 - 9.57 and 9.58 The panel shown forms the end of a...Ch. 9.2 - Prob. 9.59PCh. 9.2 - Prob. 9.60PCh. 9.2 - A vertical trapezoidal gate that is used as an...Ch. 9.2 - The cover for a 0.5-m-diameter access hole in a...Ch. 9.2 - Determine the x coordinate of the centroid of the...Ch. 9.2 - Determine the x coordinate of the centroid of the...Ch. 9.2 - Show that the system of hydrostatic forces acting...Ch. 9.2 - Show that the resultant of the hydrostatic forces...Ch. 9.3 - 9.67 through 9.70 Determine by direct integration...Ch. 9.3 - Prob. 9.68PCh. 9.3 - 9.67 through 9.70 Determine by direct integration...Ch. 9.3 - 9.67 through 9.70 Determine by direct integration...Ch. 9.3 - 9.71 through 9.74 Using the parallel-axis theorem,...Ch. 9.3 - 9.71 through 9.74 Using the parallel-axis theorem,...Ch. 9.3 - 9.71 through 9.74 Using the parallel-axis theorem,...Ch. 9.3 - Prob. 9.74PCh. 9.3 - 9.75 through 9.78 Using the parallel-axis theorem,...Ch. 9.3 - 9.75 through 9.78 Using the parallel-axis theorem,...Ch. 9.3 - 9.75 through 9.78 Using the parallel-axis theorem,...Ch. 9.3 - Prob. 9.78PCh. 9.3 - Determine for the quarter ellipse of Prob. 9.67...Ch. 9.3 - Determine the moments of inertia and the product...Ch. 9.3 - Determine the moments of inertia and the product...Ch. 9.3 - 9.75 through 9.78 Using the parallel-axis theorem,...Ch. 9.3 - Determine the moments of inertia and the product...Ch. 9.3 - Determine the moments of inertia and the product...Ch. 9.3 - Prob. 9.85PCh. 9.3 - 9.86 through 9.88 For the area indicated,...Ch. 9.3 - 9.86 through 9.88 For the area indicated,...Ch. 9.3 - 9.86 through 9.88 For the area indicated,...Ch. 9.3 - 9.89 and 9.90 For the angle cross section...Ch. 9.3 - 9.89 and 9.90 For the angle cross section...Ch. 9.4 - Using Mohrs circle, determine for the quarter...Ch. 9.4 - Using Mohrs circle, determine the moments of...Ch. 9.4 - Using Mohrs circle, determine the moments of...Ch. 9.4 - Using Mohrs circle, determine the moments of...Ch. 9.4 - Using Mohrs circle, determine the moments of...Ch. 9.4 - Using Mohrs circle, determine the moments of...Ch. 9.4 - For the quarter ellipse of Prob. 9.67, use Mohrs...Ch. 9.4 - Prob. 9.98PCh. 9.4 - 9.98 though 9.102 Using Mohrs circle, determine...Ch. 9.4 - 9.98 though 9.102 Using Mohrs circle, determine...Ch. 9.4 - 9.98 through 9.102 Using Mohrs circle, determine...Ch. 9.4 - 9.98 through 9.102 Using Mohrs circle, determine...Ch. 9.4 - Prob. 9.103PCh. 9.4 - 9.104 and 9.105 Using Mohrs circle, determine the...Ch. 9.4 - 9.104 and 9.105 Using Mohrs circle, determine the...Ch. 9.4 - Prob. 9.106PCh. 9.4 - it is known that for a given area Iy = 48 106 mm4...Ch. 9.4 - Prob. 9.108PCh. 9.4 - Using Mohrs circle, prove that the expression...Ch. 9.4 - Using the invariance property established in the...Ch. 9.5 - A thin plate with a mass m is cut in the shape of...Ch. 9.5 - A ring with a mass m is cut from a thin uniform...Ch. 9.5 - Prob. 9.113PCh. 9.5 - The parabolic spandrel shown was cut from a thin,...Ch. 9.5 - Prob. 9.115PCh. 9.5 - Fig. P9.115 and P9.116 9.116 A piece of thin,...Ch. 9.5 - Prob. 9.117PCh. 9.5 - Prob. 9.118PCh. 9.5 - Prob. 9.119PCh. 9.5 - The area shown is revolved about the x axis to...Ch. 9.5 - Prob. 9.121PCh. 9.5 - 9.122 Determine by direct integration the mass...Ch. 9.5 - Prob. 9.123PCh. 9.5 - Determine by direct integration the mass moment of...Ch. 9.5 - Prob. 9.125PCh. 9.5 - A thin steel wire is bent into the shape shown....Ch. 9.5 - Shown is the cross section of an idler roller....Ch. 9.5 - Shown is the cross section of a molded flat-belt...Ch. 9.5 - Prob. 9.129PCh. 9.5 - Prob. 9.130PCh. 9.5 - Prob. 9.131PCh. 9.5 - Prob. 9.132PCh. 9.5 - After a period of use, one of the blades of a...Ch. 9.5 - Determine the mass moment of inertia of the 0.9-lb...Ch. 9.5 - 9.135 and 9.136 A 2-mm thick piece of sheet steel...Ch. 9.5 - 9.135 and 9.136 A 2 -mm thick piece of sheet steel...Ch. 9.5 - Prob. 9.137PCh. 9.5 - A section of sheet steel 0.03 in. thick is cut and...Ch. 9.5 - Prob. 9.139PCh. 9.5 - Prob. 9.140PCh. 9.5 - The machine element shown is fabricated from...Ch. 9.5 - Determine the mass moments of inertia and the...Ch. 9.5 - Determine the mass moment of inertia of the steel...Ch. 9.5 - Fig. P9.143 and P9.144 9.144 Determine the mass...Ch. 9.5 - Determine the mass moment of inertia of the steel...Ch. 9.5 - Aluminum wire with a weight per unit length of...Ch. 9.5 - The figure shown is formed of 18-in.-diameter...Ch. 9.5 - A homogeneous wire with a mass per unit length of...Ch. 9.6 - Determine the mass products of inertia Ixy, Iyz,...Ch. 9.6 - Determine the mass products of inertia Ixy, Iyz,...Ch. 9.6 - Prob. 9.151PCh. 9.6 - Determine the mass products of inertia Ixy, Iyz,...Ch. 9.6 - Prob. 9.153PCh. 9.6 - Prob. 9.154PCh. 9.6 - 9.153 through 9.156 A section of sheet steel 2 mm...Ch. 9.6 - 9.153 through 9.156 A section of sheet steel 2 mm...Ch. 9.6 - The figure shown is formed of 1.5-mm-diameter...Ch. 9.6 - Prob. 9.158PCh. 9.6 - 9.159 and 9.160 Brass wire with a weight per unit...Ch. 9.6 - Fig. P9.160 9.159 and 9.160 Brass wire with a...Ch. 9.6 - Complete the derivation of Eqs. (9.47) that...Ch. 9.6 - Prob. 9.162PCh. 9.6 - Prob. 9.163PCh. 9.6 - Prob. 9.164PCh. 9.6 - Shown is the machine element of Prob. 9.141....Ch. 9.6 - Determine the mass moment of inertia of the steel...Ch. 9.6 - The thin, bent plate shown is of uniform density...Ch. 9.6 - A piece of sheet steel with thickness t and...Ch. 9.6 - Determine the mass moment of inertia of the...Ch. 9.6 - 9.170 through 9.172 For the wire figure of the...Ch. 9.6 - Prob. 9.171PCh. 9.6 - 9.172 Prob. 9.146 9.146 Aluminum wire with a...Ch. 9.6 - For the homogeneous circular cylinder shown with...Ch. 9.6 - For the rectangular prism shown, determine the...Ch. 9.6 - Prob. 9.175PCh. 9.6 - Prob. 9.176PCh. 9.6 - Consider a cube with mass m and side a. (a) Show...Ch. 9.6 - Prob. 9.178PCh. 9.6 - Prob. 9.179PCh. 9.6 - 9.180 through 9.184 For the component described in...Ch. 9.6 - 9.180 through 9.184 For the component described in...Ch. 9.6 - Prob. 9.182PCh. 9.6 - 9.180 through 9.184 For the component described in...Ch. 9.6 - 9.180 through 9.184 For the component described in...Ch. 9 - Determine by direct integration the moments of...Ch. 9 - Determine the moment of inertia and the radius of...Ch. 9 - Determine the moment of inertia and the radius of...Ch. 9 - Determine the moments of inertia Ix and Iy of the...Ch. 9 - Determine the polar moment of inertia of the area...Ch. 9 - Two L4 4 12-in. angles are welded to a steel...Ch. 9 - Using the parallel-axis theorem, determine the...Ch. 9 - Prob. 9.192RPCh. 9 - Fig. P9.193 and P9.194 9.193 A thin plate with a...Ch. 9 - Fig. P9.193 and P9.194 9.194 A thin plate with...Ch. 9 - A 2-mm-thick piece of sheet steel is cut and bent...Ch. 9 - Determine the mass moment of inertia of the steel...
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- Find ȳ and the moment of inertia about the X-axis, Y-axis, and X'-axis of the cross-sectional area, given: L1 = 10 in, L2 = 1 in, L3 = 9 in, L4 = 0.8 in.arrow_forwardFind the moment of inertia about centroidal X-axis and centroidal Y-axis of the given geometry Figure: 2a and 2barrow_forwardFind the moment of inertia of the shaded area about the x-axis and y-axis, given: y2 = x2 and L = 4 mmarrow_forward
- 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_forwardThe shaded area shown is bounded by x,y axes and the curve y2 = 3.24 − 0.5x m2 , where x is in m. Suppose that a = 6.48 m and h = 1.8 m . A) Determine the moment of inertia for the shaded area about the y axis. Iy = ?arrow_forwardFour L3 × 3 × -in. angles are welded to a rolled W section as shown. Determine the moments of inertia and the radii of gyration of the combined section with respect to the centroidal x and yaxes if a = 5 in. y с W 8 X 31 L3×3× // x The moment of inertia with respect to the x axis is The radius of gyration with respect to the x axis is The moment of inertia with respect to the y axis is The radius of gyration with respect to the y axis is in 4. in. in4 in.arrow_forward
- Determine the moment of inertia (in mm4) of the given figure with respect to y axis if y=96 mm and z=54 mm. Round off only on the final answer expressed in whole number.arrow_forwardFind the moment of inertia for the cross-sectional shape about the x and y axes, given the function: y = 2·x2 and L = 1.6 m.arrow_forwardDetermine the following statement that is true regarding Mohr's Circle. Reference points on Mohr's Circle from two perpendicular axes on the cross-section can lie at angles smaller than 180 degrees relative to each other. Mohr's Circle can't be used to identify angles between principal and non-principal axes. Mohr's Circle contains all possible moment of inertia and product of inertia values for a given fixed area about all rotated axes around the same origin. A reference point on Mohr's Circle that corresponds to one of the principal axes can lie at coordinates of (492, 18.7). The center of Mohr's Circle can only be calculated using principal moments of inertia.arrow_forward
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moment of inertia; Author: NCERT OFFICIAL;https://www.youtube.com/watch?v=A4KhJYrt4-s;License: Standard YouTube License, CC-BY