hibbeler 12 solucionario chapter10

107
927 •10–1. Determine the moment of inertia of the 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 currently exist. 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 x 3 10 Solutions 44918 1/28/09 4:21 PM Page 927

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927

•10–1. 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

2 m

2 m

y � 0.25 x3

10 Solutions 44918 1/28/09 4:21 PM Page 927

928

10–2. Determine the moment of inertia of the area aboutthe axis.y

© 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

10 Solutions 44918 1/28/09 4:21 PM Page 928

929

10–3. Determine the moment of inertia of the area aboutthe axis.x

© 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

10 Solutions 44918 1/28/09 4:21 PM Page 929

930

*10–4. Determine the moment of inertia of the area aboutthe axis.y

© 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

10 Solutions 44918 1/28/09 4:21 PM Page 930

931

•10–5. Determine the moment of inertia of the area aboutthe axis.x

© 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 931

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

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

10 Solutions 44918 1/28/09 4:21 PM Page 933

934

*10–8. Determine the moment of inertia of the area aboutthe axis.y

© 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

935

•10–9. Determine the polar moment of inertia of the areaabout the axis passing through point .Oz

© 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 935

936

10–10. 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

2 in.

8 in.

y � x3

10–11. Determine the moment of inertia of the area aboutthe y axis.

y

x

2 in.

8 in.

y � x3

10 Solutions 44918 1/28/09 4:21 PM Page 936

937

© 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

10 Solutions 44918 1/28/09 4:21 PM Page 937

938

10–14. Determine the moment of inertia of the area aboutthe x axis. Solve the problem in two ways, using rectangulardifferential elements: (a) having a thickness of dx, and (b) having a thickness of dy.

© 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

10 Solutions 44918 1/28/09 4:21 PM Page 938

939

10–15. Determine the moment of inertia of the area aboutthe y axis. Solve the problem in two ways, using rectangulardifferential elements: (a) having a thickness of dx, and (b) having a thickness of dy.

© 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

10 Solutions 44918 1/28/09 4:21 PM Page 939

940

*10–16. Determine the moment of inertia of the triangulararea about the x axis.

© 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 � (b � x)h––b

y

x

b

h

•10–17. Determine the moment of inertia of the triangulararea about the y axis.

y � (b � x)h––b

y

x

b

h

10 Solutions 44918 1/28/09 4:21 PM Page 940

941

© 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

10 Solutions 44918 1/28/09 4:21 PM Page 941

942

*10–20. Determine the moment of inertia of the areaabout the x axis.

© 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

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.

10 Solutions 44918 1/28/09 4:21 PM Page 943

944

10–22. Determine the moment of inertia of the area aboutthe x axis.

© 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

10 Solutions 44918 1/28/09 4:21 PM Page 944

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

10 Solutions 44918 1/28/09 4:21 PM Page 945

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

10 Solutions 44918 1/28/09 4:21 PM Page 946

947

© 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.

10–26. Determine the polar moment of inertia of the areaabout the axis passing through point O.z

y

x

x2 � y2 � r2

r0

0

10–27. Determine the distance to the centroid of thebeam’s cross-sectional area; then find the moment of inertiaabout the axis.x¿

y

2 in.

4 in.

1 in.1 in.

Cx¿

x

y

y

6 in.

10 Solutions 44918 1/28/09 4:21 PM Page 947

948

*10–28. Determine the moment of inertia of the beam’scross-sectional area about the x axis.

© 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

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

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

10 Solutions 44918 1/28/09 4:22 PM Page 950

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

10 Solutions 44918 1/28/09 4:22 PM Page 951

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

10 Solutions 44918 1/28/09 4:22 PM Page 952

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

10 Solutions 44918 1/28/09 4:22 PM Page 953

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

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¿

10 Solutions 44918 1/28/09 4:22 PM Page 955

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

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

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

10 Solutions 44918 1/28/09 4:22 PM Page 958

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

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

10 Solutions 44918 1/28/09 4:22 PM Page 960

961

•10–45. Determine the moment of inertia of thecomposite area about the axis.x

© 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

980

10–66. Determine the product of inertia for the area withrespect to the x and y axes.

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y

x

2 m

1 m

y2 � 1 � 0.5x

10 Solutions 44918 1/28/09 4:22 PM Page 980

981

10–67. Determine the product of inertia for the area withrespect to the x and y axes.

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y

x

y3 � xbh3

h

b

10 Solutions 44918 1/28/09 4:22 PM Page 981

982

*10–68. Determine the product of inertia for the area ofthe ellipse with respect to the x and y axes.

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y

x

4 in.

2 in.

x2 � 4y2 � 16

•10–69. Determine the product of inertia for the parabolicarea with respect to the x and y axes.

y

4 in.

2 in.

x

y2 � x

10 Solutions 44918 1/28/09 4:22 PM Page 982

983

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10–70. Determine the product of inertia of the compositearea with respect to the and axes.yx

1.5 in.

y

x

2 in.

2 in.

2 in. 2 in.

10 Solutions 44918 1/28/09 4:22 PM Page 983

984

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10–71. Determine the product of inertia of the cross-sectional area with respect to the x and y axes that havetheir origin located at the centroid C.

4 in.

4 in.

x

y

5 in.

1 in.

1 in.

3.5 in.

0.5 in.

C

*10–72. Determine the product of inertia for the beam’scross-sectional area with respect to the x and y axes thathave their origin located at the centroid C.

x

y

5 mm

30 mm

5 mm

50 mm 7.5 mm

C

17.5 mm

10 Solutions 44918 1/28/09 4:22 PM Page 984

985

•10–73. Determine the product of inertia of the beam’scross-sectional area with respect to the x and y axes.

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x

y

300 mm

100 mm

10 mm

10 mm

10 mm

10–74. Determine the product of inertia for the beam’scross-sectional area with respect to the x and y axes thathave their origin located at the centroid C.

1 in.

5 in.5 in.

5 in.

1 in.

C

5 in.

x

y

1 in.0.5 in.

10 Solutions 44918 1/28/09 4:22 PM Page 985

986

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10–75. Locate the centroid of the beam’s cross-sectionalarea and then determine the moments of inertia and theproduct of inertia of this area with respect to the and

axes. The axes have their origin at the centroid C.vu

x y

x

u

x

200 mm

200 mm

175 mm

20 mm

20 mm

20 mm

C

60�

v

10 Solutions 44918 1/28/09 4:22 PM Page 986

987

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*10–76. Locate the centroid ( , ) of the beam’s cross-sectional area, and then determine the product of inertia ofthis area with respect to the centroidal and axes.y¿x¿

yx

x¿

y¿

x

y

300 mm

200 mm

10 mm

10 mm

Cy

x

10 mm

100 mm

10 Solutions 44918 1/28/09 4:22 PM Page 987

988

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•10–77. Determine the product of inertia of the beam’scross-sectional area with respect to the centroidal and

axes.yx

xC

150 mm

100 mm

100 mm

10 mm

10 mm

10 mm

y

150 mm

5 mm

10 Solutions 44918 1/28/09 4:22 PM Page 988

989

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10–78. Determine the moments of inertia and the productof inertia of the beam’s cross-sectional area with respect tothe and axes.vu

3 in.

1.5 in.

3 in.

y

u

x

1.5 in.

C

v

30�

10 Solutions 44918 1/28/09 4:22 PM Page 989

990

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10–79. Locate the centroid of the beam’s cross-sectionalarea and then determine the moments of inertia and theproduct of inertia of this area with respect to the and

axes.vu

y y

x

u

8 in.

4 in.

0.5 in.

0.5 in.

4.5 in.

0.5 in.

y

4.5 in.

C

v

60�

10 Solutions 44918 1/28/09 4:22 PM Page 990

991

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10 Solutions 44918 1/28/09 4:22 PM Page 991

992

*10–80. Locate the centroid and of the cross-sectionalarea and then determine the orientation of the principalaxes, which have their origin at the centroid C of the area.Also, find the principal moments of inertia.

yx

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y

x6 in.

0.5 in.

6 in.

y

x

0.5 in.

C

10 Solutions 44918 1/28/09 4:22 PM Page 992

993

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10 Solutions 44918 1/28/09 4:22 PM Page 993

994

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•10–81. Determine the orientation of the principal axes,which have their origin at centroid C of the beam’s cross-sectional area. Also, find the principal moments of inertia.

y

Cx

100 mm

100 mm

20 mm

20 mm

20 mm

150 mm

150 mm

10 Solutions 44918 1/28/09 4:22 PM Page 994

995

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10 Solutions 44918 1/28/09 4:22 PM Page 995

996

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10–82. Locate the centroid of the beam’s cross-sectionalarea and then determine the moments of inertia of this areaand the product of inertia with respect to the and axes.The axes have their origin at the centroid C.

vu

y

200 mm

25 mm

y

u

Cx

y

60�

75 mm75 mm

25 mm25 mm v

10 Solutions 44918 1/28/09 4:22 PM Page 996

997

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10 Solutions 44918 1/28/09 4:22 PM Page 997

998

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10–83. Solve Prob. 10–75 using Mohr’s circle.

10 Solutions 44918 1/28/09 4:22 PM Page 998

999

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*10–84. Solve Prob. 10–78 using Mohr’s circle.

10 Solutions 44918 1/28/09 4:22 PM Page 999

1000

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•10–85. Solve Prob. 10–79 using Mohr’s circle.

10 Solutions 44918 1/28/09 4:22 PM Page 1000

1001

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10–86. Solve Prob. 10–80 using Mohr’s circle.

10 Solutions 44918 1/28/09 4:22 PM Page 1001

1002

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10–87. Solve Prob. 10–81 using Mohr’s circle.

10 Solutions 44918 1/28/09 4:22 PM Page 1002

1003

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*10–88. Solve Prob. 10–82 using Mohr’s circle.

10 Solutions 44918 1/28/09 4:22 PM Page 1003

1004

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•10–89. Determine the mass moment of inertia of thecone formed by revolving the shaded area around the axis.The density of the material is . Express the result in termsof the mass of the cone.m

r

zIz z

z � (r0 � y)h––

y

h

xr0

r0

10 Solutions 44918 1/28/09 4:22 PM Page 1004

1005

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10–90. Determine the mass moment of inertia of theright circular cone and express the result in terms of thetotal mass m of the cone. The cone has a constant density .r

Ix

h

y

x

r

r–h xy �

10 Solutions 44918 1/28/09 4:22 PM Page 1005

1006

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10–91. Determine the mass moment of inertia of theslender rod. The rod is made of material having a variabledensity , where is constant. The cross-sectional area of the rod is . Express the result in terms ofthe mass m of the rod.

Ar0r = r0(1 + x>l)

Iy

x

y

l

z

10 Solutions 44918 1/28/09 4:22 PM Page 1006

1007

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*10–92. Determine the mass moment of inertia of thesolid formed by revolving the shaded area around the axis. The density of the material is . Express the result interms of the mass of the solid.m

r

yIy

z � y2

x

y

z

14

2 m

1 m

10 Solutions 44918 1/28/09 4:22 PM Page 1007

1008

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•10–93. The paraboloid is formed by revolving the shadedarea around the x axis. Determine the radius of gyration .The density of the material is .r = 5 Mg>m3

kx

y

x

100 mm

y2 � 50 x

200 mm

10 Solutions 44918 1/28/09 4:22 PM Page 1008

1009

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10–94. Determine the mass moment of inertia of thesolid formed by revolving the shaded area around the axis.The density of the material is . Express the result in termsof the mass of the semi-ellipsoid.m

r

yIy

y

a

b

z

x

� � 1y2––a2

z2––b2

10 Solutions 44918 1/28/09 4:22 PM Page 1009

1010

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10–95. The frustum is formed by rotating the shaded areaaround the x axis. Determine the moment of inertia andexpress the result in terms of the total mass m of thefrustum. The material has a constant density .r

Ix

y

x

2b

b–a x � by �

a

b

10 Solutions 44918 1/28/09 4:22 PM Page 1010

1011

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*10–96. The solid is formed by revolving the shaded areaaround the y axis. Determine the radius of gyration Thespecific weight of the material is g = 380 lb>ft3.

ky.

y3 � 9x3 in.

x3 in.

y

10 Solutions 44918 1/28/09 4:22 PM Page 1011

1012

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•10–97. Determine the mass moment of inertia of thesolid formed by revolving the shaded area around the axis.The density of the material is .r = 7.85 Mg>m3

zIz

2 m

4 mz2 � 8y

z

y

x

10 Solutions 44918 1/28/09 4:22 PM Page 1012

1013

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10–98. Determine the mass moment of inertia of thesolid formed by revolving the shaded area around the axis.The solid is made of a homogeneous material that weighs400 lb.

zIz

4 ft

8 ft

y

x

z � y3––2

z

10 Solutions 44918 1/28/09 4:22 PM Page 1013

1014

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10–99. Determine the mass moment of inertia of thesolid formed by revolving the shaded area around the axis.The total mass of the solid is .1500 kg

yIy

y

x

z

4 m

2 mz2 � y31––16

O

10 Solutions 44918 1/28/09 4:22 PM Page 1014

1015

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*10–100. Determine the mass moment of inertia of thependulum about an axis perpendicular to the page andpassing through point O.The slender rod has a mass of 10 kgand the sphere has a mass of 15 kg.

450 mm

A

O

B

100 mm

10 Solutions 44918 1/28/09 4:22 PM Page 1015

1016

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•10–101. The pendulum consists of a disk having a mass of 6 kg and slender rods AB and DC which have a mass per unitlength of . Determine the length L of DC so that thecenter of mass is at the bearing O. What is the moment ofinertia of the assembly about an axis perpendicular to thepage and passing through point O?

2 kg>m

O

0.2 mL

A B

C

D0.8 m 0.5 m

10 Solutions 44918 1/28/09 4:22 PM Page 1016

1017

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10–102. Determine the mass moment of inertia of the 2-kg bent rod about the z axis.

300 mm

300 mm

z

yx

10 Solutions 44918 1/28/09 4:22 PM Page 1017

1018

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10–103. The thin plate has a mass per unit area of. Determine its mass moment of inertia about the

y axis.10 kg>m2

200 mm

200 mm

200 mm

200 mm

200 mm

200 mm

200 mm

200 mm

z

yx

100 mm

100 mm

10 Solutions 44918 1/28/09 4:22 PM Page 1018

1019

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*10–104. The thin plate has a mass per unit area of. Determine its mass moment of inertia about the

z axis.10 kg>m2

200 mm

200 mm

200 mm

200 mm

200 mm

200 mm

200 mm

200 mm

z

yx

100 mm

100 mm

10 Solutions 44918 1/28/09 4:22 PM Page 1019

1020

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•10–105. The pendulum consists of the 3-kg slender rodand the 5-kg thin plate. Determine the location of thecenter of mass G of the pendulum; then find the massmoment of inertia of the pendulum about an axisperpendicular to the page and passing through G.

y

G

2 m

1 m

0.5 m

y

O

10 Solutions 44918 1/28/09 4:22 PM Page 1020

1021

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10–106. The cone and cylinder assembly is made ofhomogeneous material having a density of .Determine its mass moment of inertia about the axis.z

7.85 Mg>m3

300 mm

300 mm

z

xy

150 mm

150 mm

10 Solutions 44918 1/28/09 4:22 PM Page 1021

1022

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10–107. Determine the mass moment of inertia of theoverhung crank about the x axis. The material is steelhaving a density of .r = 7.85 Mg>m3

90 mm

50 mm

20 mm

20 mm

20 mm

x

x¿

50 mm30 mm

30 mm

30 mm

180 mm

10 Solutions 44918 1/28/09 4:22 PM Page 1022

1023

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*10–108. Determine the mass moment of inertia of theoverhung crank about the axis. The material is steelhaving a density of .r = 7.85 Mg>m3

x¿

90 mm

50 mm

20 mm

20 mm

20 mm

x

x¿

50 mm30 mm

30 mm

30 mm

180 mm

10 Solutions 44918 1/28/09 4:22 PM Page 1023

1024

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•10–109. If the large ring, small ring and each of the spokesweigh 100 lb, 15 lb, and 20 lb, respectively, determine the massmoment of inertia of the wheel about an axis perpendicularto the page and passing through point A.

A

O

1 ft

4 ft

10 Solutions 44918 1/28/09 4:22 PM Page 1024

1025

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10–110. Determine the mass moment of inertia of the thinplate about an axis perpendicular to the page and passingthrough point O. The material has a mass per unit area of

.20 kg>m2

400 mm

150 mm

400 mm

O

50 mm

50 mm150 mm

150 mm 150 mm

10 Solutions 44918 1/28/09 4:22 PM Page 1025

1026

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10–111. Determine the mass moment of inertia of the thinplate about an axis perpendicular to the page and passingthrough point O. The material has a mass per unit area of

.20 kg>m2

200 mm

200 mm

O

200 mm

10 Solutions 44918 1/28/09 4:22 PM Page 1026

1027

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*10–112. Determine the moment of inertia of the beam’scross-sectional area about the x axis which passes throughthe centroid C.

Cx

y

d2

d2

d2

d2 60�

60�

•10–113. Determine the moment of inertia of the beam’scross-sectional area about the y axis which passes throughthe centroid C.

Cx

y

d2

d2

d2

d2 60�

60�

10 Solutions 44918 1/28/09 4:22 PM Page 1027

1028

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10–114. Determine the moment of inertia of the beam’scross-sectional area about the x axis.

a a

a a

a––2

y � – x

y

x

10 Solutions 44918 1/28/09 4:22 PM Page 1028

1029

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10–115. Determine the moment of inertia of the beam’scross-sectional area with respect to the axis passingthrough the centroid C.

x¿

0.5 in.

0.5 in.

4 in.

2.5 in.C x¿

0.5 in.

_y

*10–116. Determine the product of inertia for the angle’scross-sectional area with respect to the and axeshaving their origin located at the centroid C. Assume allcorners to be right angles.

y¿x¿

C

57.37 mm

x¿

y¿

200 mm

20 mm57.37 mm

200 mm

20 mm

10 Solutions 44918 1/28/09 4:22 PM Page 1029

1030

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10–118. Determine the moment of inertia of the areaabout the x axis.

y

4y � 4 – x2

1 ft

x2 ft

•10–117. Determine the moment of inertia of the areaabout the y axis.

y

4y � 4 – x2

1 ft

x2 ft

10 Solutions 44918 1/28/09 4:22 PM Page 1030

1031

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10–119. Determine the moment of inertia of the areaabout the x axis. Then, using the parallel-axis theorem, findthe moment of inertia about the axis that passes throughthe centroid C of the area. .y = 120 mm

x¿

1–––200

200 mm

200 mm

y

x

x¿–y

Cy � x2

10 Solutions 44918 1/28/09 4:22 PM Page 1031

1032

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*10–120. The pendulum consists of the slender rod OA,which has a mass per unit length of . The thin diskhas a mass per unit area of . Determine thedistance to the center of mass G of the pendulum; thencalculate the moment of inertia of the pendulum about anaxis perpendicular to the page and passing through G.

y12 kg>m2

3 kg>m

G

1.5 m

A

y

O

0.3 m

0.1 m

10 Solutions 44918 1/28/09 4:22 PM Page 1032

1033

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•10–121. Determine the product of inertia of the areawith respect to the x and y axes.

y � x 3

y

1 m

1 m

x

10 Solutions 44918 1/28/09 4:22 PM Page 1033