awesomech9solutionmanual
TRANSCRIPT
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815
•9–1. Determine the mass and the location of the center ofmass of the uniform parabolic-shaped rod. The massper unit length of the rod is .2 kg>m
(x, y)
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y
x
4 m
4 m
y2 � 4x
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816
9–2. The uniform rod is bent into the shape of a parabolaand has a weight per unit length of . Determine thereactions at the fixed support A.
6 lb>ft
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y
x
3 ft
3 ft
A
y2 � 3x
9 Solutions 44918 1/28/09 2:34 PM Page 816
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817
9–3. Determine the distance to the center of mass of thehomogeneous rod bent into the shape shown. If the rod hasa mass per unit length of , determine the reactionsat the fixed support O.
0.5 kg>m
x
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1 m
1 m
y
x
y2 � x
3
O
9 Solutions 44918 1/28/09 2:34 PM Page 817
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818
*9–4. Determine the mass and locate the center of mass of the uniform rod. The mass per unit length of the
rod is .3 kg>m(x, y)
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y
x
2 m
4 m
y � 4 � x2
9 Solutions 44918 1/28/09 2:34 PM Page 818
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819
•9–5. Determine the mass and the location of the center ofmass of the rod if its mass per unit length is
.m = m0(1 + x>L)x
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y
x
L
9 Solutions 44918 1/28/09 2:34 PM Page 819
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820
9–6. Determine the location ( , ) of the centroid of the wire.yx
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y
x
y � x
2
2 ft
4 ft
9–7. Locate the centroid of the circular rod. Express theanswer in terms of the radius r and semiarc angle .a
x y
xC
r
r
–x
a
a
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821
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*9–8. Determine the area and the centroid of the area.(x, y) y
x
4 m
4 m
y2 � 4x
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822
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•9–9. Determine the area and the centroid of the area.(x, y) y
1 m
1 m
y2 � x3
x
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823
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9–10. Determine the area and the centroid of the area.(x, y) y
x
3 ft
3 ft y � x3 1––9
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824
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
9–11. Determine the area and the centroid of the area.(x, y) y
x
2 ab
b
y2 � 4ax
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825
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*�9–12. Locate the centroid of the area.x y
x
2 ft
x1/2 � 2x5/3y �
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826
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•�9–13. Locate the centroid of the area.y y
x
2 ft
x1/2 � 2x5/3y �
9 Solutions 44918 1/28/09 2:34 PM Page 826
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827
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9–14. Determine the area and the centroid of the area.(x, y) y
x
a
b
xy � c2
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828
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
9–15. Determine the area and the centroid of the area.(x, y) y
x
a
h y � x2 h––a2
9 Solutions 44918 1/28/09 2:34 PM Page 828
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829
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*9–16. Locate the centroid ( , ) of the area.yx y
x2 m
1 m
y � 1 – x21–4
9 Solutions 44918 1/28/09 2:34 PM Page 829
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830
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•9–17. Determine the area and the centroid of the area.(x, y)
x
h
a
y � x2 h––a2
y
9 Solutions 44918 1/28/09 2:34 PM Page 830
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831
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9–18. The plate is made of steel having a density of. If the thickness of the plate is 10 mm, determine
the horizontal and vertical components of reaction at the pinA and the tension in cable BC.
7850 kg>m3y
A
B
C
x
2 m
4 m
y3 � 2x
9 Solutions 44918 1/28/09 2:34 PM Page 831
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832
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
9–19. Determine the location to the centroid C of theupper portion of the cardioid, .r = a(1 - cos u)
x
r
r � a (1 � cos u)
C
_x
x
y
u
9 Solutions 44918 1/28/09 2:34 PM Page 832
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833
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*9–20. The plate has a thickness of 0.5 in. and is made ofsteel having a specific weight of . Determine thehorizontal and vertical components of reaction at the pin Aand the force in the cord at B.
490 lb>ft3y
A
B
x
3 ft
3 ft
y � x2––3
9 Solutions 44918 1/28/09 2:34 PM Page 833
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834
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•9–21. Locate the centroid of the shaded area.x y
x
a
ka
y � 2k(x � )x2—2a
9–22. Locate the centroid of the area.x y
x
2 in.
2 in.
y � 1
0.5 in.
0.5 in.
x
9 Solutions 44918 1/28/09 2:34 PM Page 834
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835
9–23. Locate the centroid of the area.y
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y
x
2 in.
2 in.
y � 1
0.5 in.
0.5 in.
x
*9–24. Locate the centroid ( , ) of the area.yx y
x
9 ft
3 ft
y � 9 � x2
9 Solutions 44918 1/28/09 2:34 PM Page 835
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836
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•9–25. Determine the area and the centroid of thearea.
(x, y) y
x
y �
y � x
3 ft
3 ft
x3––9
9 Solutions 44918 1/28/09 2:34 PM Page 836
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837
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
9–26. Locate the centroid of the area.x y
x1 m
y � x2
1 m
y2 � x
9–27. Locate the centroid of the area.y y
x1 m
y � x2
1 m
y2 � x
9 Solutions 44918 1/28/09 2:34 PM Page 837
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838
*9–28. Locate the centroid of the area.x
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y
x
h
a
y � xnh––an
•9–29. Locate the centroid of the area.y y
x
h
a
y � xnh––an
9 Solutions 44918 1/28/09 2:34 PM Page 838
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839
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
9–30. The steel plate is 0.3 m thick and has a density of. Determine the location of its center of mass.
Also determine the horizontal and vertical reactions at thepin and the reaction at the roller support. Hint: The normalforce at B is perpendicular to the tangent at B, which isfound from tan .u = dy>dx
7850 kg>m3y
A
B
x
2 m
2 m
2 m
y2 � 2x
9 Solutions 44918 1/28/09 2:34 PM Page 839
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840
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
9–31. Locate the centroid of the area. Hint: Chooseelements of thickness dy and length .[(2 - y) - y2]
y
x
1 m
1 m
1 m
y � x � 2y2 � x
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841
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*9–32. Locate the centroid of the area.x y
x1 ft
y � 2x
2 ft
y2 � 4x
•9–33. Locate the centroid of the area.y y
x1 ft
y � 2x
2 ft
y2 � 4x
9 Solutions 44918 1/28/09 2:34 PM Page 841
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842
9–34. If the density at any point in the rectangular plate isdefined by , where is a constant,determine the mass and locate the center of mass of theplate.The plate has a thickness t.
xr0r = r0(1 + x>a)
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
y
a
x
b––2
b––2
9 Solutions 44918 1/28/09 2:34 PM Page 842
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843
9–35. Locate the centroid of the homogeneous solidformed by revolving the shaded area about the y axis.
y
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y
x
z
y2 � (z � a)2 � a2
a
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844
*9–36. Locate the centroid of the solid.z
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y
z
x
a
z� a1
a a � y( )2
9 Solutions 44918 1/28/09 2:34 PM Page 844
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845
•9–37. Locate the centroid of the homogeneous solidformed by revolving the shaded area about the y axis.
y
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z
y
x
z2 � y31––16
2 m
4 m
9 Solutions 44918 1/28/09 2:34 PM Page 845
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846
9–38. Locate the centroid of the homogeneous solidfrustum of the paraboloid formed by revolving the shadedarea about the z axis.
z
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a
z � (a2 � y2)h–a2
h–2
h–2
z
x
y
9 Solutions 44918 1/28/09 2:34 PM Page 846
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847
9–39. Locate the centroid of the homogeneous solidformed by revolving the shaded area about the y axis.
y
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y
x
z
z2 � y2 � 9
3 ft
5 ft
4 ft
9 Solutions 44918 1/28/09 2:34 PM Page 847
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848
*9–40. Locate the center of mass of the circular coneformed by revolving the shaded area about the y axis. Thedensity at any point in the cone is defined by ,where is a constant.r0
r = (r0 >h)y
y
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y
x
z
h
a
z � � y � aa––h
9 Solutions 44918 1/28/09 2:34 PM Page 848
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849
•9–41. Determine the mass and locate the center of mass of the hemisphere formed by revolving the shaded areaabout the y axis. The density at any point in the hemispherecan be defined by , where is a constant.r0r = r0(1 + y>a)
y
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y
x
z
y2 � z2 � a2
r
9 Solutions 44918 1/28/09 2:34 PM Page 849
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850
9–42. Determine the volume and locate the centroid of the homogeneous conical wedge.
(y, z)
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z
x
y
a
z � ya––h
h
9 Solutions 44918 1/28/09 2:34 PM Page 850
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851
9–43. The hemisphere of radius r is made from a stack ofvery thin plates such that the density varies with height,
, where k is a constant. Determine its mass and thedistance to the center of mass G.zr = kz
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z
y
z
G
x
_r
9 Solutions 44918 1/28/09 2:34 PM Page 851
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852
*9–44. Locate the centroid ( , ) of the uniform wire bentin the shape shown.
yx
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x
y
50 mm
150 mm
100 mm
20 mm
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853
•9–45. Locate the centroid of the wire.(x, y, z)
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x
z
400 mm
200 mm
y
9 Solutions 44918 1/28/09 2:34 PM Page 853
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854
9–46. Locate the centroid ( , , ) of the wire.zyx
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z
6 in.
4 in.
yx
9 Solutions 44918 1/28/09 2:34 PM Page 854
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855
9–47. Locate the centroid ( , , ) of the wire which is bentin the shape shown.
zyx
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x y
z
4 in.
2 in.
2 in.
9 Solutions 44918 1/28/09 2:34 PM Page 855
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856
*9–48. The truss is made from seven members, each havinga mass per unit length of 6 kg/m. Locate the position ( , )of the center of mass. Neglect the mass of the gusset platesat the joints.
yx
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3 m
3 m
C
D
BA
E
y
x
3 m
9 Solutions 44918 1/28/09 2:34 PM Page 856
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857
•9–49. Locate the centroid of the wire. If the wire issuspended from A, determine the angle segment AB makeswith the vertical when the wire is in equilibrium.
(x, y)
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y
x
A
CB
200 mm200 mm
60�
9 Solutions 44918 1/28/09 2:34 PM Page 857
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858
9–50. Each of the three members of the frame has a massper unit length of 6 kg/m. Locate the position ( , ) of thecenter of mass. Neglect the size of the pins at the joints andthe thickness of the members. Also, calculate the reactionsat the pin A and roller E.
yx
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y
xA
B
C DE
4 m
6 m
7 m
4 m
9 Solutions 44918 1/28/09 2:34 PM Page 858
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859
9–51. Locate the centroid of the cross-sectional areaof the channel.
(x, y)
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x
y
9 in.1 in. 1 in.
22 in.
1 in.
9 Solutions 44918 1/28/09 2:34 PM Page 859
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860
*9–52. Locate the centroid of the cross-sectional area ofthe concrete beam.
y
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x
y
3 in.
6 in.
3 in.
27 in.
3 in.
12 in. 12 in.
9 Solutions 44918 1/28/09 2:34 PM Page 860
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861
•9–53. Locate the centroid of the cross-sectional area ofthe built-up beam.
y
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y
x
6 in.1 in.
1 in.
1 in.1 in.
3 in.3 in.
6 in.
9 Solutions 44918 1/28/09 2:34 PM Page 861
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862
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9–55. Locate the distance to the centroid of themember’s cross-sectional area.
y
x
y
0.5 in.
6 in.
0.5 in.
1.5 in.
1 in.
3 in. 3 in.
9–54. Locate the centroid of the channel’s cross-sectional area.
y
2 in.
4 in.
2 in.12 in.
2 in.
C
y
9 Solutions 44918 1/28/09 2:34 PM Page 862
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863
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*9–56. Locate the centroid of the cross-sectional area ofthe built-up beam.
y y
x
1.5 in.
1.5 in.
11.5 in.
1.5 in.
3.5 in.
4in. 1.5 in.4 in.
9 Solutions 44918 1/28/09 2:34 PM Page 863
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864
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•9–57. The gravity wall is made of concrete. Determine thelocation ( , ) of the center of mass G for the wall.yx
y
1.2 m
x
_x
_y
0.6 m 0.6 m2.4 m
3 mG
0.4 m
9 Solutions 44918 1/28/09 2:34 PM Page 864
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865
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9–58. Locate the centroid of the composite area.x
x
y
ri
r0
9 Solutions 44918 1/28/09 2:34 PM Page 865
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866
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
9–59. Locate the centroid of the composite area.(x, y)
x
y
4 in.
3 in.
3 in.
3 in.
9 Solutions 44918 1/28/09 2:34 PM Page 866
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867
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*9–60. Locate the centroid of the composite area.(x, y)
3 ft3 ft
1.5 ft
1 ft
y
x
9 Solutions 44918 1/28/09 2:34 PM Page 867
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868
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•9–61. Divide the plate into parts, and using the grid formeasurement, determine approximately the location ( , )of the centroid of the plate.
yxy
x
200 mm
200 mm
9 Solutions 44918 1/28/09 2:34 PM Page 868
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869
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9–62. To determine the location of the center of gravity ofthe automobile it is first placed in a level position, with thetwo wheels on one side resting on the scale platform P. Inthis position the scale records a reading of . Then, oneside is elevated to a convenient height c as shown. The newreading on the scale is . If the automobile has a totalweight of W, determine the location of its center of gravityG .(x, y)
W2
W1
b
Pc
G–y
–x
W2
9 Solutions 44918 1/28/09 2:34 PM Page 869
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870
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9–63. Locate the centroid of the cross-sectional area ofthe built-up beam.
y y
x
450 mm
150 mm150 mm
200 mm
20 mm
20 mm
9 Solutions 44918 1/28/09 2:34 PM Page 870
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871
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*9–64. Locate the centroid of the cross-sectional area ofthe built-up beam.
y
200 mm
20 mm50 mm
150 mm
y
x
200 mm
300 mm
10 mm
20 mm 20 mm
10 mm
9 Solutions 44918 1/28/09 2:34 PM Page 871
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872
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•9–65. The composite plate is made from both steel (A)and brass (B) segments. Determine the mass and location
of its mass center G. Take andrbr = 8.74 Mg>m3.
rst = 7.85 Mg>m31x, y, z2
y
x
z
G
B
A225 mm
150 mm
150 mm
30 mm
9 Solutions 44918 1/28/09 2:34 PM Page 872
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873
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9–66. The car rests on four scales and in this position thescale readings of both the front and rear tires are shown by
and . When the rear wheels are elevated to a height of3 ft above the front scales, the new readings of the frontwheels are also recorded. Use this data to compute thelocation and to the center of gravity G of the car. Thetires each have a diameter of 1.98 ft.
yx
FBFA
FA � 1129 lb � 1168 lb � 2297 lb
FA � 1269 lb � 1307 lb � 2576 lb
FB � 975 lb � 984 lb � 1959 lb
A
_x
B
9.40 ft
3.0 ft
G_y
BG
A
9 Solutions 44918 1/28/09 2:34 PM Page 873
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874
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9–67. Uniform blocks having a length L and mass m arestacked one on top of the other, with each block overhangingthe other by a distance d, as shown. If the blocks are gluedtogether, so that they will not topple over, determine thelocation of the center of mass of a pile of n blocks.x
L
d
2d
y
x
9 Solutions 44918 1/28/09 2:34 PM Page 874
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875
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*9–68. Uniform blocks having a length L and mass m arestacked one on top of the other, with each blockoverhanging the other by a distance d, as shown. Show thatthe maximum number of blocks which can be stacked inthis manner is .n 6 L>d
L
d
2d
y
x
9 Solutions 44918 1/28/09 2:34 PM Page 875
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876
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•9–69. Locate the center of gravity ( , ) of the sheet-metal bracket if the material is homogeneous and has aconstant thickness. If the bracket is resting on the horizontalx–y plane shown, determine the maximum angle of tilt which it can have before it falls over, i.e., begins to rotateabout the y axis.
u
zx z
y
x
60 mm
60 mm
20 mm
20 mm
20 mm
20 mm
60 mm
10 mm dia. holes
80 mm
u
9 Solutions 44918 1/28/09 2:34 PM Page 876
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877
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9–70. Locate the center of mass for the compressorassembly.The locations of the centers of mass of the variouscomponents and their masses are indicated and tabulated inthe figure.What are the vertical reactions at blocks A and Bneeded to support the platform?
x
y
1
2
34
Instrument panel
Filter system
Piping assembly
Liquid storage
Structural framework
230 kg
183 kg
120 kg
85 kg
468 kg
1
2
34
5
5
2.30 m1.80 m
3.15 m
4.83 m
3.26 m
A B
2.42 m 2.87 m1.64 m1.19m
1.20 m
3.68 m
9 Solutions 44918 1/28/09 2:34 PM Page 877
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878
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9–71. Major floor loadings in a shop are caused by theweights of the objects shown. Each force acts through itsrespective center of gravity G. Locate the center of gravity( , ) of all these components.yx
z
y
G2
G4G3
G1
x
600 lb9 ft
7 ft
12 ft
6 ft
8 ft4 ft 3 ft
5 ft
1500 lb
450 lb
280 lb
9 Solutions 44918 1/28/09 2:34 PM Page 878
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879
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*9–72. Locate the center of mass of thehomogeneous block assembly.
(x, y, z)
y
z
x 150 mm
250 mm
200 mm
150 mm150 mm100 mm
9 Solutions 44918 1/28/09 2:34 PM Page 879
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880
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•9–73. Locate the center of mass of the assembly. Thehemisphere and the cone are made from materials havingdensities of and , respectively.4 Mg>m38 Mg>m3
z
y
z
x
100 mm 300 mm
9 Solutions 44918 1/28/09 2:34 PM Page 880
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881
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9–74. Locate the center of mass of the assembly. Thecylinder and the cone are made from materials havingdensities of and , respectively.9 Mg>m35 Mg>m3
z z
x
0.8 m
0.6 m0.4 m
0.2 m
y
9 Solutions 44918 1/28/09 2:34 PM Page 881
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882
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9–75. Locate the center of gravity of thehomogeneous block assembly having a hemispherical hole.
(x, y, z)
y
z
x
1 in.
3 in.
2.25 in.
2.25 in.
2.5 in.
2.5 in.
1 in.
3 in.
9 Solutions 44918 1/28/09 2:34 PM Page 882
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883
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*9–76. Locate the center of gravity of theassembly. The triangular and the rectangular blocks aremade from materials having specific weights of and , respectively.0.1 lb>in3
0.25 lb>in3
(x, y, z)
y
z
x
1 in.
3 in.
2.25 in.
2.25 in.
2.5 in.
2.5 in.
1 in.
3 in.
9 Solutions 44918 1/28/09 2:34 PM Page 883
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884
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•9–77. Determine the distance to the centroid of thesolid which consists of a cylinder with a hole of length
bored into its base.h = 50 mm
x y
x
h
120 mm
40 mm
20 mm
9 Solutions 44918 1/28/09 2:34 PM Page 884
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885
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9–78. Determine the distance h to which a hole must bebored into the cylinder so that the center of mass of theassembly is located at . The material has adensity of .8 Mg>m3
x = 64 mm
y
x
h
120 mm
40 mm
20 mm
9 Solutions 44918 1/28/09 2:34 PM Page 885
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886
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9–79. The assembly is made from a steel hemisphere,, and an aluminum cylinder,. Determine the mass center of the
assembly if the height of the cylinder is .h = 200 mmral = 2.70 Mg>m3rst = 7.80 Mg>m3
160 mm
h
z
y
x
80 mm
z
G
_
9 Solutions 44918 1/28/09 2:34 PM Page 886
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887
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*9–80. The assembly is made from a steel hemisphere,, and an aluminum cylinder,. Determine the height h of the cylinder
so that the mass center of the assembly is located at.z = 160 mm
ral = 2.70 Mg>m3rst = 7.80 Mg>m3
160 mm
h
z
y
x
80 mm
z
G
_
9 Solutions 44918 1/28/09 2:35 PM Page 887
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888
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•9–81. The elevated water storage tank has a conical topand hemispherical bottom and is fabricated using thin steelplate. Determine how many square feet of plate is neededto fabricate the tank.
6 ft
8 ft
8 ft
10 ft
9 Solutions 44918 1/28/09 2:35 PM Page 888
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889
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9–82. The elevated water storage tank has a conical topand hemispherical bottom and is fabricated using thin steelplate. Determine the volume within the tank. 6 ft
8 ft
8 ft
10 ft
9 Solutions 44918 1/28/09 2:35 PM Page 889
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890
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9–83. Determine the volume of the solid formed byrevolving the shaded area about the x axis using the secondtheorem of Pappus–Guldinus.The area and centroid of theshaded area should first be obtained by using integration.
y
x
y
4 ft
4 ft
y2 � 4x
9 Solutions 44918 1/28/09 2:35 PM Page 890
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891
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*9–84. Determine the surface area from A to B of the tank.
A
1 m
z
B
1.5 m
3 m
•9–85. Determine the volume within the thin-walled tankfrom A to B.
A
1 m
z
B
1.5 m
3 m
9 Solutions 44918 1/28/09 2:35 PM Page 891
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892
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9–86. Determine the surface area of the roof of thestructure if it is formed by rotating the parabola about the y axis.
16 m
y
x
16 m
y � 16 � (x2/16)
9 Solutions 44918 1/28/09 2:35 PM Page 892
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893
9–87. Determine the surface area of the solid formed byrevolving the shaded area about the z axis.360°
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
2 in.
3 in.
z
0.75 in.0.75 in.
1 in.
0.5 in.
9 Solutions 44918 1/28/09 2:35 PM Page 893
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894
*9–88. Determine the volume of the solid formed byrevolving the shaded area about the z axis.360°
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2 in.
3 in.
z
0.75 in.0.75 in.
1 in.
0.5 in.
•9–89. Determine the volume of the solid formed byrevolving the shaded area about the z axis.360°
75 mm75 mm
75 mm
75 mm
250 mm
z
300 mm
9 Solutions 44918 1/28/09 2:35 PM Page 894
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895
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
9–90. Determine the surface area and volume of the solidformed by revolving the shaded area about the z axis.360°
1 in.
z
2 in.1 in.
9 Solutions 44918 1/28/09 2:35 PM Page 895
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896
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9–91. Determine the surface area and volume of the solidformed by revolving the shaded area about the z axis.360°
z
75 mm 50 mm
400 mm
300 mm
50 mm
75 mm
*9–92. The process tank is used to store liquids duringmanufacturing. Estimate both the volume of the tank andits surface area. The tank has a flat top and a thin wall.
3 m 3 m
6 m
4 m
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897
•9–93. The hopper is filled to its top with coal. Estimatethe volume of coal if the voids (air space) are 35 percent ofthe volume of the hopper.
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
0.2 m
4 m
z
1.2 m
1.5 m
9–94. The thin-wall tank is fabricated from a hemisphereand cylindrical shell. Determine the vertical reactions thateach of the four symmetrically placed legs exerts on thefloor if the tank contains water which is 12 ft deep inthe tank. The specific gravity of water is . Neglectthe weight of the tank.
62.4 lb>ft3 watersurface
8 ft
4 ft
6 ft
8 ft
9 Solutions 44918 1/28/09 2:35 PM Page 897
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898
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9–95. Determine the approximate amount of paint neededto cover the outside surface of the open tank.Assume that agallon of paint covers .400 ft2
watersurface
8 ft
4 ft
6 ft
8 ft
*9–96. Determine the surface area of the tank, whichconsists of a cylinder and hemispherical cap.
8 m
4 m
9 Solutions 44918 1/28/09 2:35 PM Page 898
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899
•9–97. Determine the volume of the thin-wall tank, whichconsists of a cylinder and hemispherical cap.
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8 m
4 m
9 Solutions 44918 1/28/09 2:35 PM Page 899
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900
9–98. The water tank AB has a hemispherical top and isfabricated from thin steel plate. Determine the volumewithin the tank.
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1.5 m
1.6 m
0.2 m
B
A
1.6 m
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901
9–99. The water tank AB has a hemispherical roof and isfabricated from thin steel plate. If a liter of paint can cover
of the tank’s surface, determine how many liters arerequired to coat the surface of the tank from A to B.3 m2
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
1.5 m
1.6 m
0.2 m
B
A
1.6 m
9 Solutions 44918 1/28/09 2:35 PM Page 901
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902
*9–100. Determine the surface area and volume of thewheel formed by revolving the cross-sectional area about the z axis.
360°
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4 in.2 in.
1 in.1 in.1.5 in.
z
9 Solutions 44918 1/28/09 2:35 PM Page 902
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903
•9–101. Determine the outside surface area of thestorage tank.
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15 ft
4 ft
30 ft
9–102. Determine the volume of the thin-wall storage tank. 15 ft
4 ft
30 ft
9 Solutions 44918 1/28/09 2:35 PM Page 903
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904
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9–103. Determine the height h to which liquid should bepoured into the conical paper cup so that it contacts half thesurface area on the inside of the cup.
100 mm
h
150 mm
*9–104. The tank is used to store a liquid having a specificweight of . If it is filled to the top, determine themagnitude of the force the liquid exerts on each of its twosides ABDC and BDFE.
80 lb>ft3A
B
E
C
D
F
12 ft
8 ft
4 ft
6 ft6 ft
9 Solutions 44918 1/28/09 2:35 PM Page 904
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905
•9–105. The concrete “gravity” dam is held in place by itsown weight. If the density of concrete is ,and water has a density of , determine thesmallest dimension d that will prevent the dam fromoverturning about its end A.
rw = 1.0 Mg>m3rc = 2.5 Mg>m3
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A
6 m
d
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906
9–106. The symmetric concrete “gravity” dam is held inplace by its own weight. If the density of concrete is
, and water has a density of, determine the smallest distance d at its
base that will prevent the dam from overturning about itsend A. The dam has a width of 8 m.
rw = 1.0 Mg>m3rc = 2.5 Mg>m3
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A
d
1.5 m
9 m
9 Solutions 44918 1/28/09 2:35 PM Page 906
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907
9–107. The tank is used to store a liquid having a specificweight of . If the tank is full, determine themagnitude of the hydrostatic force on plates CDEF andABDC.
60 lb>ft3
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x
B
A
D
E
y
z
5 ft
2 ft
2 ft
1.5 ft1.5 ft
1.5 ft1.5 ft
C
F
9 Solutions 44918 1/28/09 2:35 PM Page 907
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908
*9–108. The circular steel plate A is used to seal theopening on the water storage tank. Determine themagnitude of the resultant hydrostatic force that acts on it.The density of water is .rw = 1 Mg>m3
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45�
1 m
2 m
0.5 m0.5 m
1 m
A
B
1 m
9 Solutions 44918 1/28/09 2:35 PM Page 908
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909
•9–109. The elliptical steel plate B is used to seal theopening on the water storage tank. Determine themagnitude of the resultant hydrostatic force that acts on it.The density of water is .rw = 1 Mg>m3
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45�
1 m
2 m
0.5 m0.5 m
1 m
A
B
1 m
9 Solutions 44918 1/28/09 2:35 PM Page 909
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910
9–110. Determine the magnitude of the hydrostatic forceacting on the glass window if it is circular, A. The specificweight of seawater is .gw = 63.6 lb>ft3
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4 ft0.5 ft
0.5 ft1 ft 1 ft
1 ftA B
9 Solutions 44918 1/28/09 2:35 PM Page 910
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911
9–111. Determine the magnitude and location of theresultant hydrostatic force acting on the glass window if it iselliptical, B. The specific weight of seawater is
.gw = 63.6 lb>ft3
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4 ft0.5 ft
0.5 ft1 ft 1 ft
1 ftA B
9 Solutions 44918 1/28/09 2:35 PM Page 911
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912
*9–112. Determine the magnitude of the hydrostatic forceacting per foot of length on the seawall. .gw = 62.4 lb>ft3
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x
y
8 ft
2 ft
y � �2x2
9 Solutions 44918 1/28/09 2:35 PM Page 912
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913
•9–113. If segment AB of gate ABC is long enough, thegate will be on the verge of opening. Determine the lengthL of this segment in order for this to occur. The gate ishinged at B and has a width of 1 m. The density of water is
.rw = 1 Mg>m3
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C
BA
L
2 m
4 m
9 Solutions 44918 1/28/09 2:35 PM Page 913
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914
9–114. If L = 2 m, determine the force the gate ABC exertson the smooth stopper at C. The gate is hinged at B, free atA, and is 1 m wide. The density of water is .rw = 1 Mg>m3
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C
BA
L
2 m
4 m
9 Solutions 44918 1/28/09 2:35 PM Page 914
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915
9–115. Determine the mass of the counterweight A if the1-m-wide gate is on the verge of opening when the water isat the level shown. The gate is hinged at B and held by thesmooth stop at C. The density of water is .rw = 1 Mg>m3
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AB
C
1 m
45�
2 m
2 m
9 Solutions 44918 1/28/09 2:35 PM Page 915
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916
*9–116. If the mass of the counterweight at A is 6500 kg,determine the force the gate exerts on the smooth stop at C.The gate is hinged at B and is 1-m wide. The density ofwater is .rw = 1 Mg>m3
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AB
C
1 m
45�
2 m
2 m
9 Solutions 44918 1/28/09 2:35 PM Page 916
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917
•9–117. The concrete gravity dam is designed so that it is heldin position by its own weight. Determine the factor of safetyagainst overturning about point A if . The factor ofsafety is defined as the ratio of the stabilizing moment dividedby the overturning moment. The densities of concreteand water are and ,respectively.Assume that the dam does not slide.
rw = 1 Mg>m3rconc = 2.40 Mg>m3
x = 2 m
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y
x
x
3––2y � � x2
6 m
2 m
A
9 Solutions 44918 1/28/09 2:35 PM Page 917
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918
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
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919
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y
x
x
3––2y � � x2
6 m
2 m
A
9–118. The concrete gravity dam is designed so that it isheld in position by its own weight. Determine the minimumdimension x so that the factor of safety against overturningabout point A of the dam is 2.The factor of safety is definedas the ratio of the stabilizing moment divided by theoverturning moment. The densities of concrete and waterare and , respectively.Assume that the dam does not slide.
rw = 1 Mg>m3rconc = 2.40 Mg>m3
9 Solutions 44918 1/28/09 2:35 PM Page 919
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920
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9 Solutions 44918 1/28/09 2:35 PM Page 920
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921
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
9–119. The underwater tunnel in the aquatic center isfabricated from a transparent polycarbonate materialformed in the shape of a parabola. Determine the magnitudeof the hydrostatic force that acts per meter length along thesurface AB of the tunnel. The density of the water is
.rw = 1000 kg/m3
y
x
2 m 2 m
2 m
4 m
y � 4 � x2 A
B
9 Solutions 44918 1/28/09 2:35 PM Page 921
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922
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*9–120. Locate the centroid of the shaded area.x y
x
1 in. 1 in.
4 in.
1 in.
y � x2
•9–121. Locate the centroid of the shaded area.y y
x
1 in. 1 in.
4 in.
1 in.
y � x2
9 Solutions 44918 1/28/09 2:35 PM Page 922
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923
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
9–122. Locate the centroid of the beam’s cross-sectionalarea.
y
100 mm
25 mm
25 mm
x
25 mm
y
50 mm 50 mm
y
75 mm75 mm
C
9–123. Locate the centroid of the solid.z z
x
2a
y
y2 � a a – z–2
a
9 Solutions 44918 1/28/09 2:35 PM Page 923
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924
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*9–124. The steel plate is 0.3 m thick and has a density ofDetermine the location of its center of mass. Also
compute the reactions at the pin and roller support.7850 kg>m3.
A
B
x
y
y2 � 2x
y � �x
2 m
2 m
2 m
•9–125. Locate the centroid ( , ) of the area.yx y
x
3 in.1 in.
3 in.6 in.
9 Solutions 44918 1/28/09 2:35 PM Page 924
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925
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9–126. Determine the location ( , ) of the centroid forthe structural shape. Neglect the thickness of the member.
yx
1.5 in. 1.5 in. 1.5 in. 1.5 in.1 in.1 in.
3 in.
x
y
9–127. Locate the centroid of the shaded area.y
x
y
a—2
a—2
a
aa
9 Solutions 44918 1/28/09 2:35 PM Page 925
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926
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*9–128. The load over the plate varies linearly along thesides of the plate such that . Determinethe resultant force and its position ( , ) on the plate.yx
p =23 [x(4 - y)] kPa
p
3 m
4 m
y
x
8 kPa
•9–129. The pressure loading on the plate is described bythe function . Determinethe magnitude of the resultant force and coordinates of thepoint where the line of action of the force intersectsthe plate.
p = 5-240>(x + 1) + 3406 Pap
x
y
6 m
5 m
100 Pa
300 Pa
9 Solutions 44918 1/28/09 2:35 PM Page 926