chapter 10 moments of inertia static textbook solution 12th edition
TRANSCRIPT
927
•10–1. Determine the moment of inertia of the area aboutthe axis.x
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y
x
2 m
2 m
y � 0.25 x3
10 Solutions 44918 1/28/09 4:21 PM Page 927
928
10–2. Determine the moment of inertia of the area aboutthe axis.y
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y
x
2 m
2 m
y � 0.25 x3
10 Solutions 44918 1/28/09 4:21 PM Page 928
929
10–3. Determine the moment of inertia of the area aboutthe axis.x
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y
x
y2 � x3 1 m
1 m
10 Solutions 44918 1/28/09 4:21 PM Page 929
930
*10–4. Determine the moment of inertia of the area aboutthe axis.y
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y
x
y2 � x3 1 m
1 m
10 Solutions 44918 1/28/09 4:21 PM Page 930
931
•10–5. Determine the moment of inertia of the area aboutthe axis.x
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y
x
y2 � 2x
2 m
2 m
10 Solutions 44918 1/28/09 4:21 PM Page 931
932
10–6. Determine the moment of inertia of the area aboutthe axis.y
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y
x
y2 � 2x
2 m
2 m
10 Solutions 44918 1/28/09 4:21 PM Page 932
933
10–7. Determine the moment of inertia of the area aboutthe axis.x
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y
xO
y � 2x4 2 m
1 m
10 Solutions 44918 1/28/09 4:21 PM Page 933
934
*10–8. Determine the moment of inertia of the area aboutthe axis.y
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y
xO
y � 2x4 2 m
1 m
10 Solutions 44918 1/28/09 4:21 PM Page 934
935
•10–9. Determine the polar moment of inertia of the areaabout the axis passing through point .Oz
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y
xO
y � 2x4 2 m
1 m
10 Solutions 44918 1/28/09 4:21 PM Page 935
936
10–10. Determine the moment of inertia of the area aboutthe x axis.
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y
x
2 in.
8 in.
y � x3
10–11. Determine the moment of inertia of the area aboutthe y axis.
y
x
2 in.
8 in.
y � x3
10 Solutions 44918 1/28/09 4:21 PM Page 936
937
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•10–13. Determine the moment of inertia of the areaabout the y axis.
x
y
1 in.
2 in.
y � 2 – 2 x 3
*10–12. Determine the moment of inertia of the areaabout the x axis.
x
y
1 in.
2 in.
y � 2 – 2 x 3
10 Solutions 44918 1/28/09 4:21 PM Page 937
938
10–14. Determine the moment of inertia of the area aboutthe x axis. Solve the problem in two ways, using rectangulardifferential elements: (a) having a thickness of dx, and (b) having a thickness of dy.
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1 in. 1 in.
4 in.
y � 4 – 4x2
x
y
10 Solutions 44918 1/28/09 4:21 PM Page 938
939
10–15. Determine the moment of inertia of the area aboutthe y axis. Solve the problem in two ways, using rectangulardifferential elements: (a) having a thickness of dx, and (b) having a thickness of dy.
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1 in. 1 in.
4 in.
y � 4 – 4x2
x
y
10 Solutions 44918 1/28/09 4:21 PM Page 939
940
*10–16. Determine the moment of inertia of the triangulararea about the x axis.
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y � (b � x)h––b
y
x
b
h
•10–17. Determine the moment of inertia of the triangulararea about the y axis.
y � (b � x)h––b
y
x
b
h
10 Solutions 44918 1/28/09 4:21 PM Page 940
941
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10–19. Determine the moment of inertia of the area aboutthe y axis.
x
y
b
h
y � x2 h—b2
10–18. Determine the moment of inertia of the area aboutthe x axis.
x
y
b
h
y � x2 h—b2
10 Solutions 44918 1/28/09 4:21 PM Page 941
942
*10–20. Determine the moment of inertia of the areaabout the x axis.
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y
x
y3 � x2 in.
8 in.
10 Solutions 44918 1/28/09 4:21 PM Page 942
943
•10–21. Determine the moment of inertia of the areaabout the y axis.
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y
x
y3 � x2 in.
8 in.
10 Solutions 44918 1/28/09 4:21 PM Page 943
944
10–22. Determine the moment of inertia of the area aboutthe x axis.
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y
x
y � 2 cos ( x)––8
2 in.
4 in.4 in.
π
10–23. Determine the moment of inertia of the area aboutthe y axis.
y
x
y � 2 cos ( x)––8
2 in.
4 in.4 in.
π
10 Solutions 44918 1/28/09 4:21 PM Page 944
945
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*10–24. Determine the moment of inertia of the areaabout the axis.x
y
x
x2 � y2 � r2
r0
0
10 Solutions 44918 1/28/09 4:21 PM Page 945
946
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•10–25. Determine the moment of inertia of the areaabout the axis.y
y
x
x2 � y2 � r2
r0
0
10 Solutions 44918 1/28/09 4:21 PM Page 946
947
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10–26. Determine the polar moment of inertia of the areaabout the axis passing through point O.z
y
x
x2 � y2 � r2
r0
0
10–27. Determine the distance to the centroid of thebeam’s cross-sectional area; then find the moment of inertiaabout the axis.x¿
y
2 in.
4 in.
1 in.1 in.
Cx¿
x
y
y
6 in.
10 Solutions 44918 1/28/09 4:21 PM Page 947
948
*10–28. Determine the moment of inertia of the beam’scross-sectional area about the x axis.
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2 in.
4 in.
1 in.1 in.
Cx¿
x
y
y
6 in.
•10–29. Determine the moment of inertia of the beam’scross-sectional area about the y axis.
2 in.
4 in.
1 in.1 in.
Cx¿
x
y
y
6 in.
10 Solutions 44918 1/28/09 4:21 PM Page 948
949
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10–30. Determine the moment of inertia of the beam’scross-sectional area about the axis.x
y
x
15 mm15 mm60 mm60 mm
100 mm
100 mm
50 mm
50 mm
15 mm
15 mm
10 Solutions 44918 1/28/09 4:22 PM Page 949
950
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10–31. Determine the moment of inertia of the beam’scross-sectional area about the axis.y
y
x
15 mm15 mm60 mm60 mm
100 mm
100 mm
50 mm
50 mm
15 mm
15 mm
10 Solutions 44918 1/28/09 4:22 PM Page 950
951
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*10–32. Determine the moment of inertia of thecomposite area about the axis.x
y
x
150 mm
300 mm
150 mm
100 mm
100 mm
75 mm
10 Solutions 44918 1/28/09 4:22 PM Page 951
952
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•10–33. Determine the moment of inertia of thecomposite area about the axis.y
y
x
150 mm
300 mm
150 mm
100 mm
100 mm
75 mm
10 Solutions 44918 1/28/09 4:22 PM Page 952
953
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10–34. Determine the distance to the centroid of thebeam’s cross-sectional area; then determine the moment ofinertia about the axis.x¿
y
x
x¿C
y
50 mm 50 mm75 mm
25 mm
25 mm
75 mm
100 mm
_y
25 mm
25 mm
100 mm
10 Solutions 44918 1/28/09 4:22 PM Page 953
954
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10–35. Determine the moment of inertia of the beam’scross-sectional area about the y axis.
x
x¿C
y
50 mm 50 mm75 mm
25 mm
25 mm
75 mm
100 mm
_y
25 mm
25 mm
100 mm
*10–36. Locate the centroid of the composite area, thendetermine the moment of inertia of this area about thecentroidal axis.x¿
y y
1 in.1 in.
2 in.
3 in.
5 in.x¿
xy
3 in.
C
10 Solutions 44918 1/28/09 4:22 PM Page 954
955
•10–37. Determine the moment of inertia of thecomposite area about the centroidal axis.y
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y
1 in.1 in.
2 in.
3 in.
5 in.x¿
xy
3 in.
C
10–38. Determine the distance to the centroid of thebeam’s cross-sectional area; then find the moment of inertiaabout the axis.x¿
y
300 mm
100 mm
200 mm
50 mm 50 mm
y
C
x
y
x¿
10 Solutions 44918 1/28/09 4:22 PM Page 955
956
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10–39. Determine the moment of inertia of the beam’scross-sectional area about the x axis.
300 mm
100 mm
200 mm
50 mm 50 mm
y
C
x
y
x¿
10 Solutions 44918 1/28/09 4:22 PM Page 956
957
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*10–40. Determine the moment of inertia of the beam’scross-sectional area about the y axis.
300 mm
100 mm
200 mm
50 mm 50 mm
y
C
x
y
x¿
10 Solutions 44918 1/28/09 4:22 PM Page 957
958
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•10–41. Determine the moment of inertia of the beam’scross-sectional area about the axis.x
y
50 mm 50 mm
15 mm115 mm
115 mm
7.5 mmx
15 mm
10 Solutions 44918 1/28/09 4:22 PM Page 958
959
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10–42. Determine the moment of inertia of the beam’scross-sectional area about the axis.y
y
50 mm 50 mm
15 mm115 mm
115 mm
7.5 mmx
15 mm
10 Solutions 44918 1/28/09 4:22 PM Page 959
960
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10–43. Locate the centroid of the cross-sectional areafor the angle. Then find the moment of inertia about the
centroidal axis.x¿
Ix¿
y
6 in.2 in.
6 in.
x 2 in.
C x¿
y¿y
–x
–y
*10–44. Locate the centroid of the cross-sectional areafor the angle. Then find the moment of inertia about the
centroidal axis.y¿
Iy¿
x
6 in.2 in.
6 in.
x 2 in.
C x¿
y¿y
–x
–y
10 Solutions 44918 1/28/09 4:22 PM Page 960
961
•10–45. Determine the moment of inertia of thecomposite area about the axis.x
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y
x
150 mm 150 mm
150 mm
150 mm
10 Solutions 44918 1/28/09 4:22 PM Page 961
962
10–46. Determine the moment of inertia of the compositearea about the axis.y
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y
x
150 mm 150 mm
150 mm
150 mm
10 Solutions 44918 1/28/09 4:22 PM Page 962
963
10–47. Determine the moment of inertia of the compositearea about the centroidal axis.y
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x
x¿
y
C
400 mm
240 mm
50 mm
150 mm 150 mm
50 mm
50 mm
y
10 Solutions 44918 1/28/09 4:22 PM Page 963
964
*10–48. Locate the centroid of the composite area, thendetermine the moment of inertia of this area about the
axis.x¿
y
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x
x¿
y
C
400 mm
240 mm
50 mm
150 mm 150 mm
50 mm
50 mm
y
10 Solutions 44918 1/28/09 4:22 PM Page 964
965
•10–49. Determine the moment of inertia of thesection. The origin of coordinates is at the centroid C.
Ix¿
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200 mm600 mm
20 mm
C
y¿
x¿
200 mm
20 mm
20 mm
10–50. Determine the moment of inertia of the section.The origin of coordinates is at the centroid C.
Iy¿
200 mm600 mm
20 mm
C
y¿
x¿
200 mm
20 mm
20 mm
10 Solutions 44918 1/28/09 4:22 PM Page 965
966
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10–51. Determine the beam’s moment of inertia aboutthe centroidal axis.x
Ix y
x50 mm
50 mm
100 mm
15 mm15 mm
10 mm
100 mm
C
10 Solutions 44918 1/28/09 4:22 PM Page 966
967
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*10–52. Determine the beam’s moment of inertia aboutthe centroidal axis.y
Iy y
x50 mm
50 mm
100 mm
15 mm15 mm
10 mm
100 mm
C
10 Solutions 44918 1/28/09 4:22 PM Page 967
968
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•10–53. Locate the centroid of the channel’s cross-sectional area, then determine the moment of inertia of thearea about the centroidal axis.x¿
y
6 in.
0.5 in.
0.5 in.
0.5 in.6.5 in. 6.5 in.
y
Cx¿
x
y
10 Solutions 44918 1/28/09 4:22 PM Page 968
969
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10–54. Determine the moment of inertia of the area of thechannel about the axis.y
6 in.
0.5 in.
0.5 in.
0.5 in.6.5 in. 6.5 in.
y
Cx¿
x
y
10 Solutions 44918 1/28/09 4:22 PM Page 969
970
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10–55. Determine the moment of inertia of the cross-sectional area about the axis.x
100 mm10 mm
10 mm
180 mm x
y¿y
C
100 mm
10 mm
x
10 Solutions 44918 1/28/09 4:22 PM Page 970
971
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*10–56. Locate the centroid of the beam’s cross-sectional area, and then determine the moment of inertia ofthe area about the centroidal axis.y¿
x
100 mm10 mm
10 mm
180 mm x
y¿y
C
100 mm
10 mm
x
10 Solutions 44918 1/28/09 4:22 PM Page 971
972
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•10–57. Determine the moment of inertia of the beam’scross-sectional area about the axis.x
y
100 mm12 mm
125 mm
75 mm12 mm
75 mmx
12 mm
25 mm
125 mm
12 mm
10 Solutions 44918 1/28/09 4:22 PM Page 972
973
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10–58. Determine the moment of inertia of the beam’scross-sectional area about the axis.y
y
100 mm12 mm
125 mm
75 mm12 mm
75 mmx
12 mm
25 mm
125 mm
12 mm
10 Solutions 44918 1/28/09 4:22 PM Page 973
974
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10–59. Determine the moment of inertia of the beam’scross-sectional area with respect to the axis passingthrough the centroid C of the cross section. .y = 104.3 mm
x¿
x¿C
A
B–y
150 mm
15 mm
35 mm
50 mm
*10–60. Determine the product of inertia of the parabolicarea with respect to the x and y axes.
y
x
y � 2x22 in.
1 in.
10 Solutions 44918 1/28/09 4:22 PM Page 974
975
•10–61. Determine the product of inertia of the righthalf of the parabolic area in Prob. 10–60, bounded by thelines . and .x = 0y = 2 in
Ixy
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y
x
y � 2x22 in.
1 in.
10 Solutions 44918 1/28/09 4:22 PM Page 975
976
10–62. Determine the product of inertia of the quarterelliptical area with respect to the and axes.yx
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y
a
b
x
� � 1x2––a2
y2––b2
10 Solutions 44918 1/28/09 4:22 PM Page 976
977
10–63. Determine the product of inertia for the area withrespect to the x and y axes.
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y
x
8 in.
2 in.y3 � x
10 Solutions 44918 1/28/09 4:22 PM Page 977
978
*10–64. Determine the product of inertia of the area withrespect to the and axes.yx
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y
x
y � x––4
4 in.
4 in.
(x � 8)
10 Solutions 44918 1/28/09 4:22 PM Page 978
979
•10–65. Determine the product of inertia of the area withrespect to the and axes.yx
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y
x2 m
3 m
8y � x3 � 2x2 � 4x
10 Solutions 44918 1/28/09 4:22 PM Page 979
980
10–66. Determine the product of inertia for the area withrespect to the x and y axes.
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y
x
2 m
1 m
y2 � 1 � 0.5x
10 Solutions 44918 1/28/09 4:22 PM Page 980
981
10–67. Determine the product of inertia for the area withrespect to the x and y axes.
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y
x
y3 � xbh3
h
b
10 Solutions 44918 1/28/09 4:22 PM Page 981
982
*10–68. Determine the product of inertia for the area ofthe ellipse with respect to the x and y axes.
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y
x
4 in.
2 in.
x2 � 4y2 � 16
•10–69. Determine the product of inertia for the parabolicarea with respect to the x and y axes.
y
4 in.
2 in.
x
y2 � x
10 Solutions 44918 1/28/09 4:22 PM Page 982
983
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10–70. Determine the product of inertia of the compositearea with respect to the and axes.yx
1.5 in.
y
x
2 in.
2 in.
2 in. 2 in.
10 Solutions 44918 1/28/09 4:22 PM Page 983
984
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10–71. Determine the product of inertia of the cross-sectional area with respect to the x and y axes that havetheir origin located at the centroid C.
4 in.
4 in.
x
y
5 in.
1 in.
1 in.
3.5 in.
0.5 in.
C
*10–72. Determine the product of inertia for the beam’scross-sectional area with respect to the x and y axes thathave their origin located at the centroid C.
x
y
5 mm
30 mm
5 mm
50 mm 7.5 mm
C
17.5 mm
10 Solutions 44918 1/28/09 4:22 PM Page 984
985
•10–73. Determine the product of inertia of the beam’scross-sectional area with respect to the x and y axes.
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x
y
300 mm
100 mm
10 mm
10 mm
10 mm
10–74. Determine the product of inertia for the beam’scross-sectional area with respect to the x and y axes thathave their origin located at the centroid C.
1 in.
5 in.5 in.
5 in.
1 in.
C
5 in.
x
y
1 in.0.5 in.
10 Solutions 44918 1/28/09 4:22 PM Page 985
986
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10–75. Locate the centroid of the beam’s cross-sectionalarea and then determine the moments of inertia and theproduct of inertia of this area with respect to the and
axes. The axes have their origin at the centroid C.vu
x y
x
u
x
200 mm
200 mm
175 mm
20 mm
20 mm
20 mm
C
60�
v
10 Solutions 44918 1/28/09 4:22 PM Page 986
987
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*10–76. Locate the centroid ( , ) of the beam’s cross-sectional area, and then determine the product of inertia ofthis area with respect to the centroidal and axes.y¿x¿
yx
x¿
y¿
x
y
300 mm
200 mm
10 mm
10 mm
Cy
x
10 mm
100 mm
10 Solutions 44918 1/28/09 4:22 PM Page 987
988
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•10–77. Determine the product of inertia of the beam’scross-sectional area with respect to the centroidal and
axes.yx
xC
150 mm
100 mm
100 mm
10 mm
10 mm
10 mm
y
150 mm
5 mm
10 Solutions 44918 1/28/09 4:22 PM Page 988
989
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10–78. Determine the moments of inertia and the productof inertia of the beam’s cross-sectional area with respect tothe and axes.vu
3 in.
1.5 in.
3 in.
y
u
x
1.5 in.
C
v
30�
10 Solutions 44918 1/28/09 4:22 PM Page 989
990
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10–79. Locate the centroid of the beam’s cross-sectionalarea and then determine the moments of inertia and theproduct of inertia of this area with respect to the and
axes.vu
y y
x
u
8 in.
4 in.
0.5 in.
0.5 in.
4.5 in.
0.5 in.
y
4.5 in.
C
v
60�
10 Solutions 44918 1/28/09 4:22 PM Page 990
991
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10 Solutions 44918 1/28/09 4:22 PM Page 991
992
*10–80. Locate the centroid and of the cross-sectionalarea and then determine the orientation of the principalaxes, which have their origin at the centroid C of the area.Also, find the principal moments of inertia.
yx
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y
x6 in.
0.5 in.
6 in.
y
x
0.5 in.
C
10 Solutions 44918 1/28/09 4:22 PM Page 992
993
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10 Solutions 44918 1/28/09 4:22 PM Page 993
994
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•10–81. Determine the orientation of the principal axes,which have their origin at centroid C of the beam’s cross-sectional area. Also, find the principal moments of inertia.
y
Cx
100 mm
100 mm
20 mm
20 mm
20 mm
150 mm
150 mm
10 Solutions 44918 1/28/09 4:22 PM Page 994
995
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10 Solutions 44918 1/28/09 4:22 PM Page 995
996
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10–82. Locate the centroid of the beam’s cross-sectionalarea and then determine the moments of inertia of this areaand the product of inertia with respect to the and axes.The axes have their origin at the centroid C.
vu
y
200 mm
25 mm
y
u
Cx
y
60�
75 mm75 mm
25 mm25 mm v
10 Solutions 44918 1/28/09 4:22 PM Page 996
997
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10 Solutions 44918 1/28/09 4:22 PM Page 997
998
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10–83. Solve Prob. 10–75 using Mohr’s circle.
10 Solutions 44918 1/28/09 4:22 PM Page 998
999
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*10–84. Solve Prob. 10–78 using Mohr’s circle.
10 Solutions 44918 1/28/09 4:22 PM Page 999
1000
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•10–85. Solve Prob. 10–79 using Mohr’s circle.
10 Solutions 44918 1/28/09 4:22 PM Page 1000
1001
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10–86. Solve Prob. 10–80 using Mohr’s circle.
10 Solutions 44918 1/28/09 4:22 PM Page 1001
1002
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10–87. Solve Prob. 10–81 using Mohr’s circle.
10 Solutions 44918 1/28/09 4:22 PM Page 1002
1003
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*10–88. Solve Prob. 10–82 using Mohr’s circle.
10 Solutions 44918 1/28/09 4:22 PM Page 1003
1004
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•10–89. Determine the mass moment of inertia of thecone formed by revolving the shaded area around the axis.The density of the material is . Express the result in termsof the mass of the cone.m
r
zIz z
z � (r0 � y)h––
y
h
xr0
r0
10 Solutions 44918 1/28/09 4:22 PM Page 1004
1005
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10–90. Determine the mass moment of inertia of theright circular cone and express the result in terms of thetotal mass m of the cone. The cone has a constant density .r
Ix
h
y
x
r
r–h xy �
10 Solutions 44918 1/28/09 4:22 PM Page 1005
1006
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10–91. Determine the mass moment of inertia of theslender rod. The rod is made of material having a variabledensity , where is constant. The cross-sectional area of the rod is . Express the result in terms ofthe mass m of the rod.
Ar0r = r0(1 + x>l)
Iy
x
y
l
z
10 Solutions 44918 1/28/09 4:22 PM Page 1006
1007
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*10–92. Determine the mass moment of inertia of thesolid formed by revolving the shaded area around the axis. The density of the material is . Express the result interms of the mass of the solid.m
r
yIy
z � y2
x
y
z
14
2 m
1 m
10 Solutions 44918 1/28/09 4:22 PM Page 1007
1008
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•10–93. The paraboloid is formed by revolving the shadedarea around the x axis. Determine the radius of gyration .The density of the material is .r = 5 Mg>m3
kx
y
x
100 mm
y2 � 50 x
200 mm
10 Solutions 44918 1/28/09 4:22 PM Page 1008
1009
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10–94. Determine the mass moment of inertia of thesolid formed by revolving the shaded area around the axis.The density of the material is . Express the result in termsof the mass of the semi-ellipsoid.m
r
yIy
y
a
b
z
x
� � 1y2––a2
z2––b2
10 Solutions 44918 1/28/09 4:22 PM Page 1009
1010
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10–95. The frustum is formed by rotating the shaded areaaround the x axis. Determine the moment of inertia andexpress the result in terms of the total mass m of thefrustum. The material has a constant density .r
Ix
y
x
2b
b–a x � by �
a
b
10 Solutions 44918 1/28/09 4:22 PM Page 1010
1011
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*10–96. The solid is formed by revolving the shaded areaaround the y axis. Determine the radius of gyration Thespecific weight of the material is g = 380 lb>ft3.
ky.
y3 � 9x3 in.
x3 in.
y
10 Solutions 44918 1/28/09 4:22 PM Page 1011
1012
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•10–97. Determine the mass moment of inertia of thesolid formed by revolving the shaded area around the axis.The density of the material is .r = 7.85 Mg>m3
zIz
2 m
4 mz2 � 8y
z
y
x
10 Solutions 44918 1/28/09 4:22 PM Page 1012
1013
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10–98. Determine the mass moment of inertia of thesolid formed by revolving the shaded area around the axis.The solid is made of a homogeneous material that weighs400 lb.
zIz
4 ft
8 ft
y
x
z � y3––2
z
10 Solutions 44918 1/28/09 4:22 PM Page 1013
1014
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10–99. Determine the mass moment of inertia of thesolid formed by revolving the shaded area around the axis.The total mass of the solid is .1500 kg
yIy
y
x
z
4 m
2 mz2 � y31––16
O
10 Solutions 44918 1/28/09 4:22 PM Page 1014
1015
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*10–100. Determine the mass moment of inertia of thependulum about an axis perpendicular to the page andpassing through point O.The slender rod has a mass of 10 kgand the sphere has a mass of 15 kg.
450 mm
A
O
B
100 mm
10 Solutions 44918 1/28/09 4:22 PM Page 1015
1016
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•10–101. The pendulum consists of a disk having a mass of 6 kg and slender rods AB and DC which have a mass per unitlength of . Determine the length L of DC so that thecenter of mass is at the bearing O. What is the moment ofinertia of the assembly about an axis perpendicular to thepage and passing through point O?
2 kg>m
O
0.2 mL
A B
C
D0.8 m 0.5 m
10 Solutions 44918 1/28/09 4:22 PM Page 1016
1017
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10–102. Determine the mass moment of inertia of the 2-kg bent rod about the z axis.
300 mm
300 mm
z
yx
10 Solutions 44918 1/28/09 4:22 PM Page 1017
1018
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10–103. The thin plate has a mass per unit area of. Determine its mass moment of inertia about the
y axis.10 kg>m2
200 mm
200 mm
200 mm
200 mm
200 mm
200 mm
200 mm
200 mm
z
yx
100 mm
100 mm
10 Solutions 44918 1/28/09 4:22 PM Page 1018
1019
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*10–104. The thin plate has a mass per unit area of. Determine its mass moment of inertia about the
z axis.10 kg>m2
200 mm
200 mm
200 mm
200 mm
200 mm
200 mm
200 mm
200 mm
z
yx
100 mm
100 mm
10 Solutions 44918 1/28/09 4:22 PM Page 1019
1020
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•10–105. The pendulum consists of the 3-kg slender rodand the 5-kg thin plate. Determine the location of thecenter of mass G of the pendulum; then find the massmoment of inertia of the pendulum about an axisperpendicular to the page and passing through G.
y
G
2 m
1 m
0.5 m
y
O
10 Solutions 44918 1/28/09 4:22 PM Page 1020
1021
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10–106. The cone and cylinder assembly is made ofhomogeneous material having a density of .Determine its mass moment of inertia about the axis.z
7.85 Mg>m3
300 mm
300 mm
z
xy
150 mm
150 mm
10 Solutions 44918 1/28/09 4:22 PM Page 1021
1022
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10–107. Determine the mass moment of inertia of theoverhung crank about the x axis. The material is steelhaving a density of .r = 7.85 Mg>m3
90 mm
50 mm
20 mm
20 mm
20 mm
x
x¿
50 mm30 mm
30 mm
30 mm
180 mm
10 Solutions 44918 1/28/09 4:22 PM Page 1022
1023
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*10–108. Determine the mass moment of inertia of theoverhung crank about the axis. The material is steelhaving a density of .r = 7.85 Mg>m3
x¿
90 mm
50 mm
20 mm
20 mm
20 mm
x
x¿
50 mm30 mm
30 mm
30 mm
180 mm
10 Solutions 44918 1/28/09 4:22 PM Page 1023
1024
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•10–109. If the large ring, small ring and each of the spokesweigh 100 lb, 15 lb, and 20 lb, respectively, determine the massmoment of inertia of the wheel about an axis perpendicularto the page and passing through point A.
A
O
1 ft
4 ft
10 Solutions 44918 1/28/09 4:22 PM Page 1024
1025
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10–110. Determine the mass moment of inertia of the thinplate about an axis perpendicular to the page and passingthrough point O. The material has a mass per unit area of
.20 kg>m2
400 mm
150 mm
400 mm
O
50 mm
50 mm150 mm
150 mm 150 mm
10 Solutions 44918 1/28/09 4:22 PM Page 1025
1026
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10–111. Determine the mass moment of inertia of the thinplate about an axis perpendicular to the page and passingthrough point O. The material has a mass per unit area of
.20 kg>m2
200 mm
200 mm
O
200 mm
10 Solutions 44918 1/28/09 4:22 PM Page 1026
1027
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*10–112. Determine the moment of inertia of the beam’scross-sectional area about the x axis which passes throughthe centroid C.
Cx
y
d2
d2
d2
d2 60�
60�
•10–113. Determine the moment of inertia of the beam’scross-sectional area about the y axis which passes throughthe centroid C.
Cx
y
d2
d2
d2
d2 60�
60�
10 Solutions 44918 1/28/09 4:22 PM Page 1027
1028
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10–114. Determine the moment of inertia of the beam’scross-sectional area about the x axis.
a a
a a
a––2
y � – x
y
x
10 Solutions 44918 1/28/09 4:22 PM Page 1028
1029
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10–115. Determine the moment of inertia of the beam’scross-sectional area with respect to the axis passingthrough the centroid C.
x¿
0.5 in.
0.5 in.
4 in.
2.5 in.C x¿
0.5 in.
_y
*10–116. Determine the product of inertia for the angle’scross-sectional area with respect to the and axeshaving their origin located at the centroid C. Assume allcorners to be right angles.
y¿x¿
C
57.37 mm
x¿
y¿
200 mm
20 mm57.37 mm
200 mm
20 mm
10 Solutions 44918 1/28/09 4:22 PM Page 1029
1030
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10–118. Determine the moment of inertia of the areaabout the x axis.
y
4y � 4 – x2
1 ft
x2 ft
•10–117. Determine the moment of inertia of the areaabout the y axis.
y
4y � 4 – x2
1 ft
x2 ft
10 Solutions 44918 1/28/09 4:22 PM Page 1030
1031
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10–119. Determine the moment of inertia of the areaabout the x axis. Then, using the parallel-axis theorem, findthe moment of inertia about the axis that passes throughthe centroid C of the area. .y = 120 mm
x¿
1–––200
200 mm
200 mm
y
x
x¿–y
Cy � x2
10 Solutions 44918 1/28/09 4:22 PM Page 1031
1032
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*10–120. The pendulum consists of the slender rod OA,which has a mass per unit length of . The thin diskhas a mass per unit area of . Determine thedistance to the center of mass G of the pendulum; thencalculate the moment of inertia of the pendulum about anaxis perpendicular to the page and passing through G.
y12 kg>m2
3 kg>m
G
1.5 m
A
y
O
0.3 m
0.1 m
10 Solutions 44918 1/28/09 4:22 PM Page 1032
1033
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•10–121. Determine the product of inertia of the areawith respect to the x and y axes.
y � x 3
y
1 m
1 m
x
10 Solutions 44918 1/28/09 4:22 PM Page 1033