10 Solutions 44918
1/28/09
4:21 PM
Page 927
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–1. Determine the moment of inertia of the area about
the x axis.
y
2m
y ⫽ 0.25 x3
x
2m
927
10 Solutions 44918
1/28/09
4:21 PM
Page 928
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–2. Determine the moment of inertia of the area about
the y axis.
y
2m
y ⫽ 0.25 x3
x
2m
928
10 Solutions 44918
1/28/09
4:21 PM
Page 929
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–3. Determine the moment of inertia of the area about
the x axis.
y
1m
y2 ⫽ x3
x
1m
929
10 Solutions 44918
1/28/09
4:21 PM
Page 930
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–4. Determine the moment of inertia of the area about
the y axis.
y
1m
y2 ⫽ x3
x
1m
930
10 Solutions 44918
1/28/09
4:21 PM
Page 931
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–5. Determine the moment of inertia of the area about
the x axis.
y
y2 ⫽ 2x
2m
x
2m
931
10 Solutions 44918
1/28/09
4:21 PM
Page 932
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–6. Determine the moment of inertia of the area about
the y axis.
y
y2 ⫽ 2x
2m
x
2m
932
10 Solutions 44918
1/28/09
4:21 PM
Page 933
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–7. Determine the moment of inertia of the area about
the x axis.
y
y ⫽ 2x4
2m
O
933
x
1m
10 Solutions 44918
1/28/09
4:21 PM
Page 934
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–8. Determine the moment of inertia of the area about
the y axis.
y
y ⫽ 2x4
2m
O
934
x
1m
10 Solutions 44918
1/28/09
4:21 PM
Page 935
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–9. Determine the polar moment of inertia of the area
about the z axis passing through point O.
y
y ⫽ 2x4
2m
O
935
x
1m
10 Solutions 44918
1/28/09
4:21 PM
Page 936
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–10. Determine the moment of inertia of the area about
the x axis.
y
y ⫽ x3
8 in.
x
2 in.
10–11. Determine the moment of inertia of the area about
the y axis.
y
y ⫽ x3
8 in.
x
2 in.
936
10 Solutions 44918
1/28/09
4:21 PM
Page 937
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–12. Determine the moment of inertia of the area
about the x axis.
y
y ⫽ 2 – 2x 3
2 in.
x
1 in.
•10–13. Determine the moment of inertia of the area
about the y axis.
y
y ⫽ 2 – 2x 3
2 in.
x
1 in.
937
10 Solutions 44918
1/28/09
4:21 PM
Page 938
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–14. Determine the moment of inertia of the area about
the x axis. Solve the problem in two ways, using rectangular
differential elements: (a) having a thickness of dx, and
(b) having a thickness of dy.
y
y ⫽ 4 – 4x 2
4 in.
x
1 in. 1 in.
938
10 Solutions 44918
1/28/09
4:21 PM
Page 939
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–15. Determine the moment of inertia of the area about
the y axis. Solve the problem in two ways, using rectangular
differential elements: (a) having a thickness of dx, and
(b) having a thickness of dy.
y
y ⫽ 4 – 4x 2
4 in.
x
1 in. 1 in.
939
10 Solutions 44918
1/28/09
4:21 PM
Page 940
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–16. Determine the moment of inertia of the triangular
area about the x axis.
y
h (b ⫺ x)
y ⫽ ––
b
h
x
b
y
•10–17. Determine the moment of inertia of the triangular
area about the y axis.
h (b ⫺ x)
y ⫽ ––
b
h
x
b
940
10 Solutions 44918
1/28/09
4:21 PM
Page 941
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–18. Determine the moment of inertia of the area about
the x axis.
y
h
h x2
y ⫽—
b2
x
b
10–19. Determine the moment of inertia of the area about
the y axis.
y
h
h x2
y ⫽—
b2
x
b
941
10 Solutions 44918
1/28/09
4:21 PM
Page 942
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–20. Determine the moment of inertia of the area
about the x axis.
y
2 in.
y3 ⫽ x
x
8 in.
942
10 Solutions 44918
1/28/09
4:21 PM
Page 943
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–21. Determine the moment of inertia of the area
about the y axis.
y
2 in.
y3 ⫽ x
x
8 in.
943
10 Solutions 44918
1/28/09
4:21 PM
Page 944
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–22. Determine the moment of inertia of the area about
the x axis.
y
π x)
y ⫽ 2 cos (––
8
2 in.
x
4 in.
10–23. Determine the moment of inertia of the area about
the y axis.
4 in.
y
π x)
y ⫽ 2 cos (––
8
2 in.
x
4 in.
944
4 in.
10 Solutions 44918
1/28/09
4:21 PM
Page 945
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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 area
about the x axis.
y
x2 ⫹ y2 ⫽ r02
r0
x
945
10 Solutions 44918
1/28/09
4:21 PM
Page 946
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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 area
about the y axis.
y
x2 ⫹ y2 ⫽ r02
r0
x
946
10 Solutions 44918
1/28/09
4:21 PM
Page 947
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–26. Determine the polar moment of inertia of the area
about the z axis passing through point O.
y
x2 ⫹ y2 ⫽ r02
r0
x
10–27. Determine the distance y to the centroid of the
beam’s cross-sectional area; then find the moment of inertia
about the x¿ axis.
y
6 in.
x
2 in.
y
x¿
C
1 in.
947
4 in.
1 in.
10 Solutions 44918
1/28/09
4:21 PM
Page 948
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–28. Determine the moment of inertia of the beam’s
cross-sectional area about the x axis.
y
6 in.
x
2 in.
y
x¿
C
1 in.
•10–29. Determine the moment of inertia of the beam’s
cross-sectional area about the y axis.
4 in.
1 in.
y
6 in.
x
2 in.
y
x¿
C
1 in.
948
4 in.
1 in.
10 Solutions 44918
1/28/09
4:22 PM
Page 949
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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’s
cross-sectional area about the x axis.
y
60 mm
15 mm
60 mm
15 mm
100 mm 15 mm
50 mm
x
50 mm
100 mm 15 mm
949
10 Solutions 44918
1/28/09
4:22 PM
Page 950
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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’s
cross-sectional area about the y axis.
y
60 mm
15 mm
60 mm
15 mm
100 mm 15 mm
50 mm
x
50 mm
100 mm 15 mm
950
10 Solutions 44918
1/28/09
4:22 PM
Page 951
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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 the
composite area about the x axis.
y
150 mm 150 mm
100 mm
100 mm
x
300 mm
951
75 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 952
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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 the
composite area about the y axis.
y
150 mm 150 mm
100 mm
100 mm
x
300 mm
952
75 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 953
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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 y to the centroid of the
beam’s cross-sectional area; then determine the moment of
inertia about the x¿ axis.
y
25 mm
25 mm
100 mm
C
x¿
_
y
50 mm
100 mm
25 mm
x
75 mm
75 mm
25 mm
953
50 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 954
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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’s
cross-sectional area about the y axis.
y
25 mm
25 mm
100 mm
C
x¿
_
y
25 mm
x
50 mm
100 mm
75 mm
75 mm
50 mm
25 mm
*10–36. Locate the centroid y of the composite area, then
determine the moment of inertia of this area about the
centroidal x¿ axis.
y
1 in.
1 in.
5 in.
2 in.
x¿
C
y
x
3 in.
954
3 in.
10 Solutions 44918
1/28/09
4:22 PM
Page 955
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–37. Determine the moment of inertia of the
composite area about the centroidal y axis.
y
1 in.
1 in.
5 in.
2 in.
x¿
C
y
x
3 in.
10–38. Determine the distance y to the centroid of the
beam’s cross-sectional area; then find the moment of inertia
about the x¿ axis.
3 in.
y
50 mm 50 mm
300 mm
C
x¿
y
100 mm
x
200 mm
955
10 Solutions 44918
1/28/09
4:22 PM
Page 956
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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’s
cross-sectional area about the x axis.
y
50 mm 50 mm
300 mm
C
x¿
y
100 mm
x
200 mm
956
10 Solutions 44918
1/28/09
4:22 PM
Page 957
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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’s
cross-sectional area about the y axis.
y
50 mm 50 mm
300 mm
C
x¿
y
100 mm
x
200 mm
957
10 Solutions 44918
1/28/09
4:22 PM
Page 958
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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’s
cross-sectional area about the x axis.
y
15 mm
115 mm
7.5 mm
x
115 mm
15 mm
50 mm 50 mm
958
10 Solutions 44918
1/28/09
4:22 PM
Page 959
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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’s
cross-sectional area about the y axis.
y
15 mm
115 mm
7.5 mm
x
115 mm
15 mm
50 mm 50 mm
959
10 Solutions 44918
1/28/09
4:22 PM
Page 960
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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 y of the cross-sectional area
for the angle. Then find the moment of inertia Ix¿ about the
x¿ centroidal axis.
y
y¿
–x
6 in.
C
x¿
2 in.
–y
x
6 in.
2 in.
*10–44. Locate the centroid x of the cross-sectional area
for the angle. Then find the moment of inertia Iy¿ about the
y¿ centroidal axis.
y
y¿
–x
6 in.
C
x¿
2 in.
–y
x
2 in.
960
6 in.
10 Solutions 44918
1/28/09
4:22 PM
Page 961
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–45. Determine the moment of inertia of the
composite area about the x axis.
y
150 mm
x
150 mm
150 mm
961
150 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 962
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–46. Determine the moment of inertia of the composite
area about the y axis.
y
150 mm
x
150 mm
150 mm
962
150 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 963
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–47. Determine the moment of inertia of the composite
area about the centroidal y axis.
y
240 mm
50 mm
x¿
C
50 mm
400 mm
y
x
150 mm 150 mm
963
50 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 964
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–48. Locate the centroid y of the composite area, then
determine the moment of inertia of this area about the
x¿ axis.
y
240 mm
50 mm
x¿
C
50 mm
400 mm
y
x
150 mm 150 mm
964
50 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 965
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–49. Determine the moment of inertia Ix¿ of the
section. The origin of coordinates is at the centroid C.
y¿
200 mm
x¿
C
600 mm
20 mm
200 mm
20 mm
20 mm
10–50. Determine the moment of inertia Iy¿ of the section.
The origin of coordinates is at the centroid C.
y¿
200 mm
x¿
C
600 mm
20 mm
200 mm
20 mm
20 mm
965
10 Solutions 44918
1/28/09
4:22 PM
Page 966
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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 Ix about
the centroidal x axis.
y
15 mm
15 mm
50 mm
50 mm
C
100 mm
966
x
10 mm
100 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 967
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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 Iy about
the centroidal y axis.
y
15 mm
15 mm
50 mm
50 mm
C
100 mm
967
x
10 mm
100 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 968
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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 y of the channel’s crosssectional area, then determine the moment of inertia of the
area about the centroidal x¿ axis.
y
0.5 in.
x¿
C
6 in.
y
x
6.5 in.
0.5 in.
968
6.5 in.
0.5 in.
10 Solutions 44918
1/28/09
4:22 PM
Page 969
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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 the
channel about the y axis.
y
0.5 in.
x¿
C
6 in.
y
x
6.5 in.
0.5 in.
969
6.5 in.
0.5 in.
10 Solutions 44918
1/28/09
4:22 PM
Page 970
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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 crosssectional area about the x axis.
y
10 mm
y¿
x
180 mm
x
C
100 mm
10 mm
10 mm
970
100 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 971
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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 x of the beam’s crosssectional area, and then determine the moment of inertia of
the area about the centroidal y¿ axis.
y
10 mm
y¿
x
180 mm
x
C
100 mm
10 mm
10 mm
971
100 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 972
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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’s
cross-sectional area about the x axis.
y
125 mm
12 mm
100 mm
25 mm
12 mm
972
125 mm
12 mm
12 mm
75 mm
x
75 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 973
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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’s
cross-sectional area about the y axis.
y
125 mm
12 mm
100 mm
25 mm
12 mm
973
125 mm
12 mm
12 mm
75 mm
x
75 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 974
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. 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’s
cross-sectional area with respect to the x¿ axis passing
through the centroid C of the cross section. y = 104.3 mm.
35 mm
A
150 mm
C
x¿
15 mm
–y
*10–60. Determine the product of inertia of the parabolic
area with respect to the x and y axes.
B
50 mm
y
1 in.
2 in.
y ⫽ 2x2
x
974
10 Solutions 44918
1/28/09
4:22 PM
Page 975
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–61. Determine the product of inertia Ixy of the right
half of the parabolic area in Prob. 10–60, bounded by the
lines y = 2 in. and x = 0.
y
1 in.
2 in.
y ⫽ 2x2
x
975
10 Solutions 44918
1/28/09
4:22 PM
Page 976
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–62. Determine the product of inertia of the quarter
elliptical area with respect to the x and y axes.
y
2
2
y ⫽1
x ⫹ ––
––
a2
b2
b
x
a
976
10 Solutions 44918
1/28/09
4:22 PM
Page 977
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–63. Determine the product of inertia for the area with
respect to the x and y axes.
y
2 in.
y3 ⫽ x
x
8 in.
977
10 Solutions 44918
1/28/09
4:22 PM
Page 978
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–64. Determine the product of inertia of the area with
respect to the x and y axes.
y
4 in.
x
4 in.
x (x ⫺ 8)
y ⫽ ––
4
978
10 Solutions 44918
1/28/09
4:22 PM
Page 979
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–65. Determine the product of inertia of the area with
respect to the x and y axes.
y
8y ⫽ x3 ⫹ 2x2 ⫹ 4x
3m
x
2m
979
10 Solutions 44918
1/28/09
4:22 PM
Page 980
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–66. Determine the product of inertia for the area with
respect to the x and y axes.
y
y2 ⫽ 1 ⫺ 0.5x
1m
x
2m
980
10 Solutions 44918
1/28/09
4:22 PM
Page 981
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–67. Determine the product of inertia for the area with
respect to the x and y axes.
y
y3 ⫽
h3
x
b
h
x
b
981
10 Solutions 44918
1/28/09
4:22 PM
Page 982
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–68. Determine the product of inertia for the area of
the ellipse with respect to the x and y axes.
y
x2 ⫹ 4y2 ⫽ 16
2 in.
x
4 in.
•10–69. Determine the product of inertia for the parabolic
area with respect to the x and y axes.
y
y2 ⫽ x
2 in.
x
4 in.
982
10 Solutions 44918
1/28/09
4:22 PM
Page 983
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exist. 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 composite
area with respect to the x and y axes.
y
2 in.
2 in.
2 in.
1.5 in.
2 in.
x
983
10 Solutions 44918
1/28/09
4:22 PM
Page 984
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exist. 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 crosssectional area with respect to the x and y axes that have
their origin located at the centroid C.
y
4 in.
1 in.
0.5 in.
5 in.
x
C
3.5 in.
1 in.
4 in.
*10–72. Determine the product of inertia for the beam’s
cross-sectional area with respect to the x and y axes that
have their origin located at the centroid C.
y
5 mm
50 mm
7.5 mm
C
x
17.5 mm
5 mm
30 mm
984
10 Solutions 44918
1/28/09
4:22 PM
Page 985
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–73. Determine the product of inertia of the beam’s
cross-sectional area with respect to the x and y axes.
y
10 mm
300 mm
10 mm
x
10 mm
100 mm
10–74. Determine the product of inertia for the beam’s
cross-sectional area with respect to the x and y axes that
have their origin located at the centroid C.
y
5 in.
0.5 in.
1 in.
x
C
1 in.
5 in.
5 in.
1 in.
985
5 in.
10 Solutions 44918
1/28/09
4:22 PM
Page 986
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–75. Locate the centroid x of the beam’s cross-sectional
area and then determine the moments of inertia and the
product of inertia of this area with respect to the u and
v axes. The axes have their origin at the centroid C.
y
x
20 mm
v
200 mm
C
x
60⬚
200 mm
20 mm
20 mm
175 mm
986
u
10 Solutions 44918
1/28/09
4:22 PM
Page 987
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–76. Locate the centroid (x, y) of the beam’s crosssectional area, and then determine the product of inertia of
this area with respect to the centroidal x¿ and y¿ axes.
y¿
y
x
10 mm
100 mm
10 mm
300 mm
x¿
C
y
10 mm
x
200 mm
987
10 Solutions 44918
1/28/09
4:22 PM
Page 988
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exist. 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’s
cross-sectional area with respect to the centroidal x and
y axes.
y
100 mm
5 mm
10 mm
150 mm
10 mm
x
C
150 mm
100 mm
988
10 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 989
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–78. Determine the moments of inertia and the product
of inertia of the beam’s cross-sectional area with respect to
the u and v axes.
y
v
1.5 in.
u
1.5 in.
3 in.
30⬚
C
3 in.
989
x
10 Solutions 44918
1/28/09
4:22 PM
Page 990
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–79. Locate the centroid y of the beam’s cross-sectional
area and then determine the moments of inertia and the
product of inertia of this area with respect to the u and
v axes.
y
u
v
0.5 in.
4.5 in.
4.5 in.
0.5 in.
60⬚
4 in.
x
C
0.5 in.
8 in.
990
y
10 Solutions 44918
1/28/09
4:22 PM
Page 991
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
991
10 Solutions 44918
1/28/09
4:22 PM
Page 992
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–80. Locate the centroid x and y of the cross-sectional
area and then determine the orientation of the principal
axes, which have their origin at the centroid C of the area.
Also, find the principal moments of inertia.
y
x
0.5 in.
6 in.
C
x
0.5 in.
6 in.
992
y
10 Solutions 44918
1/28/09
4:22 PM
Page 993
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
993
10 Solutions 44918
1/28/09
4:22 PM
Page 994
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–81. Determine the orientation of the principal axes,
which have their origin at centroid C of the beam’s crosssectional area. Also, find the principal moments of inertia.
y
100 mm
20 mm
20 mm
150 mm
x
C
150 mm
100 mm
994
20 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 995
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
995
10 Solutions 44918
1/28/09
4:22 PM
Page 996
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exist. 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 y of the beam’s cross-sectional
area and then determine the moments of inertia of this area
and the product of inertia with respect to the u and v axes.
The axes have their origin at the centroid C.
y
25 mm
200 mm
v
25 mm
x
C
60⬚
25 mm
75 mm 75 mm
996
u
y
10 Solutions 44918
1/28/09
4:22 PM
Page 997
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
997
10 Solutions 44918
1/28/09
4:22 PM
Page 998
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exist. 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.
998
10 Solutions 44918
1/28/09
4:22 PM
Page 999
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–84.
Solve Prob. 10–78 using Mohr’s circle.
999
10 Solutions 44918
1/28/09
4:22 PM
Page 1000
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–85.
Solve Prob. 10–79 using Mohr’s circle.
1000
10 Solutions 44918
1/28/09
4:22 PM
Page 1001
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–86.
Solve Prob. 10–80 using Mohr’s circle.
1001
10 Solutions 44918
1/28/09
4:22 PM
Page 1002
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–87.
Solve Prob. 10–81 using Mohr’s circle.
1002
10 Solutions 44918
1/28/09
4:22 PM
Page 1003
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–88.
Solve Prob. 10–82 using Mohr’s circle.
1003
10 Solutions 44918
1/28/09
4:22 PM
Page 1004
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
z
•10–89. Determine the mass moment of inertia Iz of the
cone formed by revolving the shaded area around the z axis.
The density of the material is r. Express the result in terms
of the mass m of the cone.
h
z ⫽ ––
r0 (r0 ⫺ y)
h
y
x
1004
r0
10 Solutions 44918
1/28/09
4:22 PM
Page 1005
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–90. Determine the mass moment of inertia Ix of the
right circular cone and express the result in terms of the
total mass m of the cone. The cone has a constant density r.
y
y ⫽ –hr x
r
x
h
1005
10 Solutions 44918
1/28/09
4:22 PM
Page 1006
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
z
10–91. Determine the mass moment of inertia Iy of the
slender rod. The rod is made of material having a variable
density r = r0(1 + x>l), where r0 is constant. The crosssectional area of the rod is A. Express the result in terms of
the mass m of the rod.
l
y
x
1006
10 Solutions 44918
1/28/09
4:22 PM
Page 1007
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
z
*10–92. Determine the mass moment of inertia Iy of the
solid formed by revolving the shaded area around the y
axis. The density of the material is r. Express the result in
terms of the mass m of the solid.
z ⫽ 1 y2
4
1m
y
x
2m
1007
10 Solutions 44918
1/28/09
4:22 PM
Page 1008
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–93. The paraboloid is formed by revolving the shaded
area around the x axis. Determine the radius of gyration kx.
The density of the material is r = 5 Mg>m3.
y
y 2 ⫽ 50 x
100 mm
x
200 mm
1008
10 Solutions 44918
1/28/09
4:22 PM
Page 1009
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
z
10–94. Determine the mass moment of inertia Iy of the
solid formed by revolving the shaded area around the y axis.
The density of the material is r. Express the result in terms
of the mass m of the semi-ellipsoid.
a
2
z2 ⫽ 1
y ⫹ ––
––
2
a
b2
b
y
x
1009
10 Solutions 44918
1/28/09
4:22 PM
Page 1010
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–95. The frustum is formed by rotating the shaded area
around the x axis. Determine the moment of inertia Ix and
express the result in terms of the total mass m of the
frustum. The material has a constant density r.
y
y ⫽ –ba x ⫹ b
2b
b
x
a
1010
10 Solutions 44918
1/28/09
4:22 PM
Page 1011
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–96. The solid is formed by revolving the shaded area
around the y axis. Determine the radius of gyration ky. The
specific weight of the material is g = 380 lb>ft3.
y
3 in.
y3 ⫽ 9x
x
3 in.
1011
10 Solutions 44918
1/28/09
4:22 PM
Page 1012
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
z
•10–97. Determine the mass moment of inertia Iz of the
solid formed by revolving the shaded area around the z axis.
The density of the material is r = 7.85 Mg>m3.
2m
z2 ⫽ 8y
4m
y
x
1012
10 Solutions 44918
1/28/09
4:22 PM
Page 1013
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
z
10–98. Determine the mass moment of inertia Iz of the
solid formed by revolving the shaded area around the z axis.
The solid is made of a homogeneous material that weighs
400 lb.
4 ft
8 ft
z⫽
3
––
y2
y
x
1013
10 Solutions 44918
1/28/09
4:22 PM
Page 1014
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
z
10–99. Determine the mass moment of inertia Iy of the
solid formed by revolving the shaded area around the y axis.
The total mass of the solid is 1500 kg.
4m
1 y3
z2 ⫽ ––
16
O
x
1014
2m
y
10 Solutions 44918
1/28/09
4:22 PM
Page 1015
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–100. Determine the mass moment of inertia of the
pendulum about an axis perpendicular to the page and
passing through point O. The slender rod has a mass of 10 kg
and the sphere has a mass of 15 kg.
O
450 mm
A
100 mm
B
1015
10 Solutions 44918
1/28/09
4:22 PM
Page 1016
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•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 unit
length of 2 kg>m. Determine the length L of DC so that the
center of mass is at the bearing O. What is the moment of
inertia of the assembly about an axis perpendicular to the
page and passing through point O?
0.8 m
0.5 m
D
0.2 m
L
A
O
B
C
1016
10 Solutions 44918
1/28/09
4:22 PM
Page 1017
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
z
10–102. Determine the mass moment of inertia of the
2-kg bent rod about the z axis.
300 mm
x
1017
300 mm
y
10 Solutions 44918
1/28/09
4:22 PM
Page 1018
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
z
10–103. The thin plate has a mass per unit area of
10 kg>m2. Determine its mass moment of inertia about the
y axis.
200 mm
200 mm
100 mm
200 mm
100 mm
200 mm
200 mm
x
1018
200 mm
200 mm
200 mm
y
10 Solutions 44918
1/28/09
4:22 PM
Page 1019
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
z
*10–104. The thin plate has a mass per unit area of
10 kg>m2. Determine its mass moment of inertia about the
z axis.
200 mm
200 mm
100 mm
200 mm
100 mm
200 mm
200 mm
x
1019
200 mm
200 mm
200 mm
y
10 Solutions 44918
1/28/09
4:22 PM
Page 1020
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–105. The pendulum consists of the 3-kg slender rod
and the 5-kg thin plate. Determine the location y of the
center of mass G of the pendulum; then find the mass
moment of inertia of the pendulum about an axis
perpendicular to the page and passing through G.
O
y
2m
G
0.5 m
1m
1020
10 Solutions 44918
1/28/09
4:22 PM
Page 1021
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
z
10–106. The cone and cylinder assembly is made of
homogeneous material having a density of 7.85 Mg>m3.
Determine its mass moment of inertia about the z axis.
150 mm
300 mm
150 mm
300 mm
x
1021
y
10 Solutions 44918
1/28/09
4:22 PM
Page 1022
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–107. Determine the mass moment of inertia of the
overhung crank about the x axis. The material is steel
having a density of r = 7.85 Mg>m3.
20 mm
30 mm
90 mm
50 mm
x
180 mm
20 mm
x¿
30 mm
20 mm
1022
50 mm
30 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 1023
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–108. Determine the mass moment of inertia of the
overhung crank about the x¿ axis. The material is steel
having a density of r = 7.85 Mg>m3.
20 mm
30 mm
90 mm
50 mm
x
180 mm
20 mm
x¿
30 mm
20 mm
1023
50 mm
30 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 1024
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–109. If the large ring, small ring and each of the spokes
weigh 100 lb, 15 lb, and 20 lb, respectively, determine the mass
moment of inertia of the wheel about an axis perpendicular
to the page and passing through point A.
4 ft
1 ft
O
A
1024
10 Solutions 44918
1/28/09
4:22 PM
Page 1025
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–110. Determine the mass moment of inertia of the thin
plate about an axis perpendicular to the page and passing
through point O. The material has a mass per unit area of
20 kg>m2.
O
50 mm
150 mm
50 mm
400 mm
400 mm
150 mm 150 mm
1025
150 mm
10 Solutions 44918
1/28/09
4:22 PM
Page 1026
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–111. Determine the mass moment of inertia of the thin
plate about an axis perpendicular to the page and passing
through point O. The material has a mass per unit area of
20 kg>m2.
O
200 mm
200 mm
200 mm
1026
10 Solutions 44918
1/28/09
4:22 PM
Page 1027
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–112. Determine the moment of inertia of the beam’s
cross-sectional area about the x axis which passes through
the centroid C.
y
d
2
d
2
60⬚
x
C
60⬚
d
2
•10–113. Determine the moment of inertia of the beam’s
cross-sectional area about the y axis which passes through
the centroid C.
d
2
y
d
2
d
2
60⬚
d
2
1027
x
C
60⬚
d
2
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© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–114. Determine the moment of inertia of the beam’s
cross-sectional area about the x axis.
y
y⫽
a
a –x
––
2
a
x
a
1028
a
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© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–115. Determine the moment of inertia of the beam’s
cross-sectional area with respect to the x¿ axis passing
through the centroid C.
4 in.
0.5 in.
_
y
2.5 in.
C
x¿
0.5 in.
0.5 in.
*10–116. Determine the product of inertia for the angle’s
cross-sectional area with respect to the x¿ and y¿ axes
having their origin located at the centroid C. Assume all
corners to be right angles.
y¿
57.37 mm
20 mm
200 mm
C
20 mm
200 mm
1029
x¿
57.37 mm
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–117. Determine the moment of inertia of the area
about the y axis.
y
4y ⫽ 4 – x 2
1 ft
x
2 ft
10–118. Determine the moment of inertia of the area
about the x axis.
y
4y ⫽ 4 – x 2
1 ft
x
2 ft
1030
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Page 1031
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
10–119. Determine the moment of inertia of the area
about the x axis. Then, using the parallel-axis theorem, find
the moment of inertia about the x¿ axis that passes through
the centroid C of the area. y = 120 mm.
y
200 mm
200 mm
C
–y
x¿
1
y ⫽ –––
x2
200
x
1031
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4:22 PM
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
*10–120. The pendulum consists of the slender rod OA,
which has a mass per unit length of 3 kg>m. The thin disk
has a mass per unit area of 12 kg>m2. Determine the
distance y to the center of mass G of the pendulum; then
calculate the moment of inertia of the pendulum about an
axis perpendicular to the page and passing through G.
O
y
1.5 m
G
A
0.1 m
0.3 m
1032
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Page 1033
© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
•10–121. Determine the product of inertia of the area
with respect to the x and y axes.
y
1m
y ⫽ x3
x
1m
1033