•10–1. Determine the moment of inertia of the area about y ...
Transcript of •10–1. Determine the moment of inertia of the area about y ...
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•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
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928
10–2. Determine the moment of inertia of the area aboutthe axis.y
© 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
x
2 m
2 m
y � 0.25 x3
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929
10–3. Determine the moment of inertia of the area aboutthe axis.x
© 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
x
y2 � x3 1 m
1 m
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930
*10–4. Determine the moment of inertia of the area aboutthe axis.y
© 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
x
y2 � x3 1 m
1 m
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931
•10–5. Determine the moment of inertia of the area aboutthe axis.x
© 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
x
y2 � 2x
2 m
2 m
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932
10–6. Determine the moment of inertia of the area aboutthe axis.y
© 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
x
y2 � 2x
2 m
2 m
10 Solutions 44918 1/28/09 4:21 PM Page 932
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933
10–7. Determine the moment of inertia of the area aboutthe axis.x
© 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
xO
y � 2x4 2 m
1 m
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934
*10–8. Determine the moment of inertia of the area aboutthe axis.y
© 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
xO
y � 2x4 2 m
1 m
10 Solutions 44918 1/28/09 4:21 PM Page 934
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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
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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
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937
© 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.
•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
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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.
© 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 in. 1 in.
4 in.
y � 4 – 4x2
x
y
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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.
© 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 in. 1 in.
4 in.
y � 4 – 4x2
x
y
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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
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941
© 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.
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
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942
*10–20. Determine the moment of inertia of the areaabout the x axis.
© 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
x
y3 � x2 in.
8 in.
10 Solutions 44918 1/28/09 4:21 PM Page 942
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943
•10–21. Determine the moment of inertia of the areaabout the y axis.
© 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
x
y3 � x2 in.
8 in.
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944
10–22. Determine the moment of inertia of the area aboutthe x axis.
© 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
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
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945
© 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.
*10–24. Determine the moment of inertia of the areaabout the axis.x
y
x
x2 � y2 � r2
r0
0
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946
© 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.
•10–25. Determine the moment of inertia of the areaabout the axis.y
y
x
x2 � y2 � r2
r0
0
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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
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948
*10–28. Determine the moment of inertia of the beam’scross-sectional area about the x axis.
© 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.
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
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949
© 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.
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
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950
© 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.
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
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951
© 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.
*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
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952
© 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.
•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
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953
© 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.
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
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954
© 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.
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
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955
•10–37. Determine the moment of inertia of thecomposite area about the centroidal axis.y
© 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
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¿
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956
© 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.
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
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957
© 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.
*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
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958
© 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.
•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
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959
© 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.
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
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960
© 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.
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
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961
•10–45. Determine the moment of inertia of thecomposite area about the axis.x
© 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
x
150 mm 150 mm
150 mm
150 mm
10 Solutions 44918 1/28/09 4:22 PM Page 961
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962
10–46. Determine the moment of inertia of the compositearea about the axis.y
© 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
x
150 mm 150 mm
150 mm
150 mm
10 Solutions 44918 1/28/09 4:22 PM Page 962
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963
10–47. Determine the moment of inertia of the compositearea about the centroidal axis.y
© 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.
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
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964
*10–48. Locate the centroid of the composite area, thendetermine the moment of inertia of this area about the
axis.x¿
y
© 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.
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
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965
•10–49. Determine the moment of inertia of thesection. The origin of coordinates is at the centroid C.
Ix¿
© 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.
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
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966
© 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.
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
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967
© 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.
*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
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968
© 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.
•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
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969
© 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.
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
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970
© 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.
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
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971
© 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.
*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
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972
© 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.
•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
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973
© 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.
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
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974
© 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.
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
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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
© 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
x
y � 2x22 in.
1 in.
10 Solutions 44918 1/28/09 4:22 PM Page 975
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976
10–62. Determine the product of inertia of the quarterelliptical area with respect to the and axes.yx
© 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
b
x
� � 1x2––a2
y2––b2
10 Solutions 44918 1/28/09 4:22 PM Page 976
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977
10–63. Determine the product of inertia for the area withrespect to the x and y axes.
© 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
x
8 in.
2 in.y3 � x
10 Solutions 44918 1/28/09 4:22 PM Page 977
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978
*10–64. Determine the product of inertia of the area withrespect to the and axes.yx
© 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
x
y � x––4
4 in.
4 in.
(x � 8)
10 Solutions 44918 1/28/09 4:22 PM Page 978
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979
•10–65. Determine the product of inertia of the area withrespect to the and axes.yx
© 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
x2 m
3 m
8y � x3 � 2x2 � 4x
10 Solutions 44918 1/28/09 4:22 PM Page 979
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980
10–66. Determine the product of inertia for the area withrespect to the x and y axes.
© 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
x
2 m
1 m
y2 � 1 � 0.5x
10 Solutions 44918 1/28/09 4:22 PM Page 980
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981
10–67. Determine the product of inertia for the area withrespect to the x and y axes.
© 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
x
y3 � xbh3
h
b
10 Solutions 44918 1/28/09 4:22 PM Page 981
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982
*10–68. Determine the product of inertia for the area ofthe ellipse with respect to the x and y axes.
© 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
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
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983
© 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.
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
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984
© 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.
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
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985
•10–73. Determine the product of inertia of the beam’scross-sectional area with respect to the x and y axes.
© 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.
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
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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
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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
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988
© 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.
•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
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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
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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
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991
© 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.
10 Solutions 44918 1/28/09 4:22 PM Page 991
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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
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993
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10 Solutions 44918 1/28/09 4:22 PM Page 993
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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
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995
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10 Solutions 44918 1/28/09 4:22 PM Page 995
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996
© 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.
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
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997
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10 Solutions 44918 1/28/09 4:22 PM Page 997
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998
© 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.
10–83. Solve Prob. 10–75 using Mohr’s circle.
10 Solutions 44918 1/28/09 4:22 PM Page 998
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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