Engineering Mechanics: Statics and Dynamics

R. C. Hibbeler

Chapter 10 Moments of Inertia - all with Video Answers

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Chapter Questions

02:56

Problem 1

Determine the moment of inertia about the $x$ axis.

Ryan Pollard

Numerade Educator

01:49

Problem 2

Determine the moment of inertia about the $y$ axis

Ryan Pollard

Numerade Educator

02:10

Problem 3

Determine the moment of inertia for the shaded area about the $x$ axis.

Ryan Pollard

Numerade Educator

02:05

Problem 4

Determine the moment of inertia for the shaded area about the $y$ axis.

Ryan Pollard

Numerade Educator

01:37

Problem 5

Determine the moment of inertia for the shaded area about the $x$ axis.

Ryan Pollard

Numerade Educator

01:20

Problem 6

Determine the moment of inertia for the shaded area about the $y$ axis.

Ryan Pollard

Numerade Educator

06:26

Problem 7

Determine the moment of inertia for the shaded area about the $x$ axis.

João Gabriel Alencar Caribé

Numerade Educator

07:33

Problem 8

Determine the moment of inertia for the shaded area about the $y$ axis.

João Gabriel Alencar Caribé

Numerade Educator

10:12

Determine the moment of inertia of the area about the $x$ axis. Solve the problem in two ways, using rectangular differential elements:
(a) having a thickness $d x$ and
(b) having a thickness of $d y$

Khoobchandra Agrawal

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03:29

Problem 10

Determine the moment of inertia of the area about the $x$ axis.

Khoobchandra Agrawal

Numerade Educator

03:31

Problem 11

Determine the moment of inertia for the shaded area about the $x$ axis.

Khoobchandra Agrawal

Numerade Educator

03:45

Problem 12

Determine the moment of inertia for the shaded area about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

04:51

Problem 13

Determine the moment of inertia about the $x$ axis.

Khoobchandra Agrawal

Numerade Educator

02:56

Problem 14

Determine the moment of inertia about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

02:47

Problem 15

Determine the moment of inertia for the shaded area about the $x$ axis.

Khoobchandra Agrawal

Numerade Educator

03:30

Problem 16

Determine the moment of inertia for the shaded area about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

03:40

Problem 17

Determine the moment of inertia for the shaded area about the $x$ axis.

Khoobchandra Agrawal

Numerade Educator

03:10

Problem 18

Determine the moment of inertia for the shaded area about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

03:16

Problem 19

Determine the moment of inertia for the shaded area about the $x$ axis.

Khoobchandra Agrawal

Numerade Educator

03:20

Problem 20

Determine the moment of inertia for the shaded area about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

03:26

Problem 21

Determine the moment of inertia for the shaded area about the $x$ axis.

Khoobchandra Agrawal

Numerade Educator

03:16

Problem 22

Determine the moment of inertia for the shaded area about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

03:37

Problem 23

Determine the moment of inertia for the shaded area about the $x$ axis.

Susan Hallstrom

Numerade Educator

03:19

Problem 24

Determine the moment of inertia for the shaded area about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

03:56

Problem 25

Determine the moment of inertia of the composite area about the $x$ axis.

Khoobchandra Agrawal

Numerade Educator

03:16

Problem 26

Determine the moment of inertia of the composite area about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

03:46

Problem 27

The polar moment of inertia for the area is $J_{C}=642\left(10^{6}\right) \mathrm{mm}^{4},$ about the $z^{\prime}$ axis passing through the centroid $C .$ The moment of inertia about the $y^{\prime}$ axis is $264\left(10^{6}\right) \mathrm{mm}^{4},$ and the moment of inertia about the $x$ axis is $938\left(10^{6}\right) \mathrm{mm}^{4} .$ Determine the area $A$

Khoobchandra Agrawal

Numerade Educator

07:24

Problem 28

Determine the location $\bar{y}$ of the centroid of the channel's cross-sectional area and then calculate the moment of inertia of the area about this axis.

Khoobchandra Agrawal

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09:39

Problem 29

Determine $\bar{y},$ which locates the centroidal axis $x^{\prime}$ for the cross-sectional area of the T-beam, and then find the moments of inertia $I_{x^{\prime}}$ and $I_{y^{\prime}}$

Khoobchandra Agrawal

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05:02

Problem 30

Determine the moment of inertia for the beam's cross-sectional area about the $x$ axis.

Khoobchandra Agrawal

Numerade Educator

05:49

Problem 31

Determine the moment of inertia for the beam's cross-sectional area about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

06:28

Problem 32

Determine the moment of inertia $I_{x}$ of the shaded area about the $x$ axis.

Khoobchandra Agrawal

Numerade Educator

06:21

Problem 33

Determine the moment of inertia $I_{x}$ of the shaded area about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

02:36

Problem 34

Determine the moment of inertia of the beam's cross-sectional area about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

04:24

Problem 35

Determine $\bar{y},$ which locates the centroidal axis $x^{\prime}$ for the cross-sectional area of the T-beam, and then find the moment of inertia about the $x^{\prime}$ axis.

Khoobchandra Agrawal

Numerade Educator

04:01

Problem 36

Determine the moment of inertia about the $x$ axis.

Khoobchandra Agrawal

Numerade Educator

01:37

Problem 37

Determine the moment of inertia about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

06:59

Problem 38

Determine the moment of inertia of the shaded area about the $x$ axis.

Khoobchandra Agrawal

Numerade Educator

06:00

Problem 39

Determine the moment of inertia of the shaded area about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

07:02

Problem 40

Determine the distance $\bar{y}$ to the centroid of the beam's cross-sectional area; then find the moment of inertia about the centroidal $x^{\prime}$ axis.

Khoobchandra Agrawal

Numerade Educator

06:50

Problem 41

Determine the moment of inertia for the beam's cross-sectional area about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

02:21

Problem 42

Determine the moment of inertia of the beam's cross-sectional area about the $x$ axis.

Khoobchandra Agrawal

Numerade Educator

02:36

Problem 43

Determine the moment of inertia of the beam's cross-sectional area about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

06:41

Problem 44

Determine the distance $\bar{y}$ to the centroid $C$ of the beam's cross-sectional area and then compute the moment of inertia $\bar{I}_{x^{\prime}}$ about the $x^{\prime}$ axis.

Narayan Hari

Numerade Educator

05:08

Problem 45

Determine the distance $\bar{x}$ to the centroid $C$ of the beam's cross-sectional area and then compute the moment of inertia $\bar{I}_{y^{\prime}}$ about the $y^{\prime}$ axis.

Khoobchandra Agrawal

Numerade Educator

08:23

Problem 46

Determine the moment of inertia for the shaded area about the $x$ axis.

Khoobchandra Agrawal

Numerade Educator

08:40

Problem 47

Determine the moment of inertia for the shaded area about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

01:08

Problem 48

Determine the moment of inertia of the parallelogram about the $x^{\prime}$ axis, which passes through the centroid $C$ of the area.

Khoobchandra Agrawal

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02:26

Problem 49

Determine the moment of inertia of the parallelogram about the $y^{\prime}$ axis, which passes through the centroid $C$ of the area.

Khoobchandra Agrawal

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11:11

Problem 50

Locate the centroid $\bar{y}$ of the cross section and determine the moment of inertia of the section about the
$x^{\prime}$ axis.

Khoobchandra Agrawal

Numerade Educator

03:57

Problem 51

Determine the moment of inertia for the beam's cross-sectional area about the $x^{\prime}$ axis passing through the centroid $C$ of the cross section.

Khoobchandra Agrawal

Numerade Educator

03:14

Problem 52

Determine the moment of inertia of the area about the $x$ axis.

Khoobchandra Agrawal

Numerade Educator

02:22

Problem 53

Determine the moment of inertia of the area about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

02:01

Problem 54

Determine the product of inertia of the thin strip of area with respect to the $x$ and $y$ axes. The strip is oriented at an angle $\theta$ from the $x$ axis. Assume that $t \ll l$

Khoobchandra Agrawal

Numerade Educator

02:48

Problem 55

Determine the product of inertia of the shaded area with respect to the $x$ and $y$ axes.

Khoobchandra Agrawal

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03:04

Problem 56

Determine the product of inertia for the shaded portion of the parabola with respect to the $x$ and $y$ axes.

Khoobchandra Agrawal

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05:46

Problem 57

Determine the product of inertia of the shaded area with respect to the $x$ and $y$ axes, and then use the parallel-axis theorem to find the product of inertia of the area with respect to the centroidal $x^{\prime}$ and $y^{\prime}$ axes.

Khoobchandra Agrawal

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02:37

Problem 58

Determine the product of inertia for the parabolic area with respect to the $x$ and $y$ axes.

Khoobchandra Agrawal

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03:10

Problem 59

Determine the product of inertia of the shaded area with respect to the $x$ and $y$ axes.

Khoobchandra Agrawal

Numerade Educator

02:27

Problem 60

Determine the product of inertia of the shaded area with respect to the $x$ and $y$ axes.

Khoobchandra Agrawal

Numerade Educator

02:59

Problem 61

Determine the product of inertia of the shaded area with respect to the $x$ and $y$ axes.

Khoobchandra Agrawal

Numerade Educator

01:38

Problem 62

Determine the product of inertia for the beam's cross-sectional area with respect to the $x$ and $y$ axes.

Khoobchandra Agrawal

Numerade Educator

04:46

Problem 63

Determine the moments of inertia of the shaded area with respect to the $u$ and $v$ axes.

Khoobchandra Agrawal

Numerade Educator

03:11

Problem 64

Determine the product of inertia for the beam's cross-sectional area with respect to the $u$ and $v$ axes.

Khoobchandra Agrawal

Numerade Educator

01:55

Problem 65

Determine the product of inertia for the shaded area with respect to the $x$ and $y$ axes.

Khoobchandra Agrawal

Numerade Educator

02:27

Problem 66

Determine the product of inertia of the crosssectional area with respect to the $x$ and $y$ axes.

Khoobchandra Agrawal

Numerade Educator

03:03

Problem 67

Determine the location $(\bar{x}, \bar{y})$ to the centroid $C$ of the angle's cross-sectional area, and then compute the product of inertia with respect to the $x^{\prime}$ and $y^{\prime}$ axes.

Khoobchandra Agrawal

Numerade Educator

07:46

Problem 68

Determine the distance $\bar{y}$ to the centroid of the area and then calculate the moments of inertia $I_{u}$ and $I_{v}$ of the channel's cross-sectional area. The $u$ and $v$ axes have their origin at the centroid $C .$ For the calculation, assume all corners to be square.

Khoobchandra Agrawal

Numerade Educator

05:50

Problem 69

Determine the moments of inertia $I_{u}, I_{v}$ and the product of inertia $I_{u v}$ for the beam's cross-sectional area. Take $\theta=45^{\circ}$

Khoobchandra Agrawal

Numerade Educator

05:33

Problem 70

Determine the moments of inertia $I_{u}, I_{v}$ and the product of inertia $I_{u v}$ for the rectangular area. The $u$ and $v$ axes pass through the centroid $C$.

Khoobchandra Agrawal

Numerade Educator

07:56

Problem 71

Solve Prob. 10-70 using Mohr's circle. Hint: To solve, find the coordinates of the point $P\left(I_{u}, I_{u v}\right)$ on the circle, measured counterclockwise from the radial line $O A$ (See Fig. $10-19 .)$ The point $Q\left(I_{v},-I_{u v}\right)$ is on the opposite side of the circle.

Khoobchandra Agrawal

Numerade Educator

06:30

Problem 72

Determine the directions of the principal axes having an origin at point $O,$ and the principal moments of inertia for the triangular area about the axes.

Khoobchandra Agrawal

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05:21

Problem 73

Solve Prob. $10-72$ using Mohr's circle.

Khoobchandra Agrawal

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09:38

Problem 74

Determine the orientation of the principal axes having an origin at point $C,$ and the principal moments of inertia of the cross section about these axes.

Khoobchandra Agrawal

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07:22

Problem 75

Solve Prob. $10-74$ using Mohr's circle.

Khoobchandra Agrawal

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06:35

Problem 76

Determine the orientation of the principal axes having an origin at point $O,$ and the principal moments of inertia for the rectangular area about these axes.

Khoobchandra Agrawal

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07:02

Problem 77

Solve Prob. $10-76$ using Mohr's circle.

Khoobchandra Agrawal

Numerade Educator

03:27

Problem 78

The area of the cross section of an airplane wing has the following properties about the $x$ and $y$ axes passing through the centroid $C: \bar{I}_{x}=450$ in $^{4}, \quad \bar{I}_{y}=1730$ in $^{4}$ $\bar{I}_{x y}=138$ in $^{4} .$ Determine the orientation of the principal axes and the principal moments of inertia.

Khoobchandra Agrawal

Numerade Educator

04:50

Problem 79

Solve Prob. $10-78$ using Mohr's circle.

Khoobchandra Agrawal

Numerade Educator

05:47

Problem 80

Determine the moments and product of inertia for the shaded area with respect to the $u$ and $v$ axes.

Khoobchandra Agrawal

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06:24

Problem 81

Solve Prob. $10-80$ using Mohr's circle

Khoobchandra Agrawal

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06:41

Problem 82

Determine the directions of the principal axes with origin located at point $O,$ and the principal moments of inertia for the area about these axes.

Khoobchandra Agrawal

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09:34

Problem 83

Solve Prob. $10-82$ using Mohr's circle.

Khoobchandra Agrawal

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01:53

Problem 84

Determine the moment of inertia of the thin ring about the $z$ axis. The ring has a mass $m.$

Khoobchandra Agrawal

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03:44

Problem 85

Determine the moment of inertia of the ellipsoid with respect to the $x$ axis and express the result in terms of the mass $m$ of the ellipsoid. The material has a constant density $\rho$

Khoobchandra Agrawal

Numerade Educator

04:34

Problem 86

Determine the radius of gyration $k_{x}$ of the paraboloid. The density of the material is $\rho=5 \mathrm{Mg} / \mathrm{m}^{3}$

Khoobchandra Agrawal

Numerade Educator

03:24

Problem 87

The paraboloid is formed by revolving the shaded area around the $x$ axis. Determine the moment of inertia about the $x$ axis and express the result in terms of the total mass $m$ of the paraboloid. The material has a constant density $\rho$

Khoobchandra Agrawal

Numerade Educator

07:36

Problem 88

Determine the moment of inertia of the homogenous triangular prism with respect to the $y$ axis. Express the result in terms of the mass $m$ of the prism. Hint: For integration, use thin plate elements parallel to the $x-y$ plane having a thickness of $d z$

Khoobchandra Agrawal

Numerade Educator

04:37

Problem 89

Determine the moment of inertia of the semiellipsoid with respect to the $x$ axis and express the result in terms of the mass $m$ of the semiellipsoid. The material has a constant density $\rho$

Khoobchandra Agrawal

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05:58

Problem 90

Determine the radius of gyration $k_{x}$ of the solid formed by revolving the shaded area about $x$ axis. The density of the material is $\rho$

Khoobchandra Agrawal

Numerade Educator

03:48

Problem 91

The concrete shape is formed by rotating the shaded area about the $y$ axis. Determine the moment of inertia $I_{y} .$ The specific weight of concrete is $\gamma=150 \mathrm{lb} / \mathrm{ft}^{3}$

Khoobchandra Agrawal

Numerade Educator

04:11

Problem 92

Determine the moment of inertia $I_{x}$ of the sphere and express the result in terms of the total mass $m$ of the sphere. The sphere has a constant density $\rho$

Khoobchandra Agrawal

Numerade Educator

05:00

Problem 93

The right circular cone is formed by revolving the shaded area around the $x$ axis. Determine the moment of
inertia $I_{x}$ and express the result in terms of the total mass $m$ of the cone. The cone has a constant density $\rho$.

Khoobchandra Agrawal

Numerade Educator

04:08

Problem 94

Determine the mass moment of inertia $I_{y}$ of the solid formed by revolving the shaded area around the $y$ axis. The total mass of the solid is $1500 \mathrm{kg}$.

Khoobchandra Agrawal

Numerade Educator

02:36

Problem 95

The slender rods have a mass of $4 \mathrm{kg} / \mathrm{m} .$ Determine the moment of inertia of the assembly about an axis perpendicular to the page and passing through point $A$

Khoobchandra Agrawal

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05:18

Problem 96

The pendulum consists of a 8 -kg circular disk $A,$ a $2-\mathrm{kg}$ circular disk $B,$ and a $4-\mathrm{kg}$ slender rod. Determine the radius of gyration of the pendulum about an axis perpendicular to the page and passing through point $O$

Supratim Pal

Numerade Educator

00:54

Problem 97

Determine the moment of inertia $I_{z}$ of the frustum of the cone which has a conical depression. The material has a density of $200 \mathrm{kg} / \mathrm{m}^{3}$

Khoobchandra Agrawal

Numerade Educator

02:26

Problem 98

The pendulum consists of the 3 -kg slender rod and the $5-\mathrm{kg}$ thin plate. Determine the location $\bar{y}$ of the center of mass $G$ of the pendulum; then find the mass moment of inertia of the pendulum about an axis perpendicular to the page and passing through $G$

Khoobchandra Agrawal

Numerade Educator

05:11

Problem 99

Determine the mass moment of inertia of the thin plate about an axis perpendicular to the page and passing through point $O .$ The material has a mass per unit area of $20 \mathrm{kg} / \mathrm{m}^{2}$

Khoobchandra Agrawal

Numerade Educator

03:14

Problem 100

The pendulum consists of a plate having a weight of 12 lb and a slender rod having a weight of 4 lb. Determine the radius of gyration of the pendulum about an axis perpendicular to the page and passing through point $O$

Khoobchandra Agrawal

Numerade Educator

05:53

Problem 101

If the large ring, small ring and each of the spokes weigh $100 \mathrm{lb}, 15 \mathrm{lb},$ and $20 \mathrm{lb}$, respectively, determine the mass moment of inertia of the wheel about an axis perpendicular to the page and passing through point $A$.

Khoobchandra Agrawal

Numerade Educator

07:41

Problem 102

Determine the mass moment of inertia of the assembly about the $z$ axis. The density of the material is $7.85 \mathrm{Mg} / \mathrm{m}^{3}$.

Khoobchandra Agrawal

Numerade Educator

01:00

Problem 103

Each of the three slender rods has a mass $m$ Determine the moment of inertia of the assembly about an axis that is perpendicular to the page and passes through the center point $O$

Khoobchandra Agrawal

Numerade Educator

03:55

Problem 104

The thin plate has a mass per unit area of $10 \mathrm{kg} / \mathrm{m}^{2}$ Determine its mass moment of inertia about the $y$ axis.

Khoobchandra Agrawal

Numerade Educator

04:01

Problem 105

The thin plate has a mass per unit area of $10 \mathrm{kg} / \mathrm{m}^{2}$ Determine its mass moment of inertia about the $z$ axis.

Khoobchandra Agrawal

Numerade Educator

02:22

Problem 106

Determine the moment of inertia of the assembly about an axis that is perpendicular to the page and passes through the center of mass $G .$ The material has a specific weight of $\gamma=90 \mathrm{lb} / \mathrm{ft}^{3}$

Khoobchandra Agrawal

Numerade Educator

03:14

Problem 107

Determine the moment of inertia of the assembly about an axis that is perpendicular to the page and passes through point $O .$ The material has a specific weight of $\gamma=90 \mathrm{lb} / \mathrm{ft}^{3}$

Khoobchandra Agrawal

Numerade Educator

04:29

Problem 108

The pendulum consists of two slender rods $A B$ and $O C$ which have a mass of $3 \mathrm{kg} / \mathrm{m}$. The thin plate has a mass of $12 \mathrm{kg} / \mathrm{m}^{2} .$ Determine the location $\bar{y}$ of the center of mass $G$ of the pendulum, then calculate the moment of inertia of the pendulum about an axis perpendicular to the page and passing through $G$

Khoobchandra Agrawal

Numerade Educator

04:20

Problem 109

Determine the moment of inertia $I_{z}$ of the frustum of the cone which has a conical depression. The material has a density of $200 \mathrm{kg} / \mathrm{m}^{3}$.

Khoobchandra Agrawal

Numerade Educator