Applied static ch10

1.10 Solutions 449181/28/094:21 PMPage 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.y2my ϭ 0.25 x3x 2m9272. 10 Solutions 449181/28/094:21 PMPage 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.y2my ϭ 0.25 x3x 2m9283. 10 Solutions 449181/28/094:21 PMPage 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.y1my2 ϭ x3x 1m9294. 10 Solutions 449181/28/094:21 PMPage 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.y1my2 ϭ x3x 1m9305. 10 Solutions 449181/28/094:21 PMPage 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.yy2 ϭ 2x 2mx 2m9316. 10 Solutions 449181/28/094:21 PMPage 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.yy2 ϭ 2x 2mx 2m9327. 10 Solutions 449181/28/094:21 PMPage 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.yy ϭ 2x42mO933x 1m8. 10 Solutions 449181/28/094:21 PMPage 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.yy ϭ 2x42mO934x 1m9. 10 Solutions 449181/28/094:21 PMPage 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.yy ϭ 2x42mO935x 1m10. 10 Solutions 449181/28/094:21 PMPage 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.yy ϭ x38 in.x 2 in.10–11. Determine the moment of inertia of the area about the y axis.yy ϭ x38 in.x 2 in.93611. 10 Solutions 449181/28/094:21 PMPage 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.yy ϭ 2 – 2x 3 2 in.x 1 in.•10–13. Determine the moment of inertia of the area about the y axis.yy ϭ 2 – 2x 3 2 in.x 1 in.93712. 10 Solutions 449181/28/094:21 PMPage 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.yy ϭ 4 – 4x 2 4 in.x 1 in. 1 in.93813. 10 Solutions 449181/28/094:21 PMPage 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.yy ϭ 4 – 4x 2 4 in.x 1 in. 1 in.93914. 10 Solutions 449181/28/094:21 PMPage 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.yh y ϭ –– (b Ϫ x) b hx by•10–17. Determine the moment of inertia of the triangular area about the y axis.h y ϭ –– (b Ϫ x) b hx b94015. 10 Solutions 449181/28/094:21 PMPage 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.yhh y ϭ — x2 b2 x b10–19. Determine the moment of inertia of the area about the y axis.yhh y ϭ — x2 b2 x b94116. 10 Solutions 449181/28/094:21 PMPage 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.y2 in.y3 ϭ x x 8 in.94217. 10 Solutions 449181/28/094:21 PMPage 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.y2 in.y3 ϭ x x 8 in.94318. 10 Solutions 449181/28/094:21 PMPage 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.yπ y ϭ 2 cos (–– x) 8 2 in. x 4 in.10–23. Determine the moment of inertia of the area about the y axis.4 in.yπ y ϭ 2 cos (–– x) 8 2 in. x 4 in.9444 in.19. 10 Solutions 449181/28/094:21 PMPage 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 2 x2 ϩ y2 ϭ r0r0 x94520. 10 Solutions 449181/28/094:21 PMPage 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 2 x2 ϩ y2 ϭ r0r0 x94621. 10 Solutions 449181/28/094:21 PMPage 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 2 x2 ϩ y2 ϭ r0r0 x10–27. 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.yx¿C1 in.9474 in.1 in.22. 10 Solutions 449181/28/094:21 PMPage 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.yx¿C1 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.yx¿C1 in.9484 in.1 in.23. 10 Solutions 449181/28/094:22 PMPage 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 mm60 mm 15 mm100 mm 15 mm 50 mm x 50 mm 100 mm 15 mm94924. 10 Solutions 449181/28/094:22 PMPage 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 mm60 mm 15 mm100 mm 15 mm 50 mm x 50 mm 100 mm 15 mm95025. 10 Solutions 449181/28/094:22 PMPage 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.y150 mm 150 mm100 mm 100 mm x 300 mm95175 mm26. 10 Solutions 449181/28/094:22 PMPage 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.y150 mm 150 mm100 mm 100 mm x 300 mm95275 mm27. 10 Solutions 449181/28/094:22 PMPage 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 mm25 mm100 mmCx¿_ y50 mm 100 mm25 mm x 75 mm75 mm25 mm95350 mm28. 10 Solutions 449181/28/094:22 PMPage 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 mm25 mm100 mmCx¿_ y25 mm x50 mm 100 mm75 mm75 mm50 mm25 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.y1 in.1 in.5 in. 2 in.x¿Cy x3 in.9543 in.29. 10 Solutions 449181/28/094:22 PMPage 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.y1 in.1 in.5 in. 2 in.x¿Cy x3 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 mm300 mm Cx¿y 100 mm x 200 mm95530. 10 Solutions 449181/28/094:22 PMPage 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 mm300 mm Cx¿y 100 mm x 200 mm95631. 10 Solutions 449181/28/094:22 PMPage 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 mm300 mm Cx¿y 100 mm x 200 mm95732. 10 Solutions 449181/28/094:22 PMPage 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.y15 mm 115 mm 7.5 mm x 115 mm 15 mm50 mm 50 mm95833. 10 Solutions 449181/28/094:22 PMPage 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.y15 mm 115 mm 7.5 mm x 115 mm 15 mm50 mm 50 mm95934. 10 Solutions 449181/28/094:22 PMPage 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.yy¿ – x6 in.Cx¿ 2 in.– y x6 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.yy¿ – x6 in.Cx¿ 2 in.– y x2 in.9606 in.35. 10 Solutions 449181/28/094:22 PMPage 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.y150 mm x 150 mm150 mm961150 mm36. 10 Solutions 449181/28/094:22 PMPage 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.y150 mm x 150 mm150 mm962150 mm37. 10 Solutions 449181/28/094:22 PMPage 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.y240 mm50 mm x¿C 50 mm400 mmyx 150 mm 150 mm96350 mm38. 10 Solutions 449181/28/094:22 PMPage 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.y240 mm50 mm x¿C 50 mm400 mmyx 150 mm 150 mm96450 mm39. 10 Solutions 449181/28/094:22 PMPage 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 mm20 mm200 mm20 mm 20 mm10–50. Determine the moment of inertia Iy¿ of the section. The origin of coordinates is at the centroid C.y¿200 mm x¿ C 600 mm20 mm200 mm20 mm 20 mm96540. 10 Solutions 449181/28/094:22 PMPage 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 mm50 mm 50 mmC100 mm966x 10 mm 100 mm41. 10 Solutions 449181/28/094:22 PMPage 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 mm50 mm 50 mmC100 mm967x 10 mm 100 mm42. 10 Solutions 449181/28/094:22 PMPage 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.y0.5 in. x¿ C6 in.y x6.5 in. 0.5 in.9686.5 in. 0.5 in.43. 10 Solutions 449181/28/094:22 PMPage 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.y0.5 in. x¿ C6 in.y x6.5 in. 0.5 in.9696.5 in. 0.5 in.44. 10 Solutions 449181/28/094:22 PMPage 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.y10 mmy¿ x180 mmxC 100 mm 10 mm 10 mm970100 mm45. 10 Solutions 449181/28/094:22 PMPage 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.y10 mmy¿ x180 mmxC 100 mm 10 mm 10 mm971100 mm46. 10 Solutions 449181/28/094:22 PMPage 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 mm12 mm 100 mm 25 mm 12 mm972125 mm 12 mm 12 mm 75 mm x 75 mm47. 10 Solutions 449181/28/094:22 PMPage 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 mm12 mm 100 mm 25 mm 12 mm973125 mm 12 mm 12 mm 75 mm x 75 mm48. 10 Solutions 449181/28/094:22 PMPage 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 A150 mm Cx¿ 15 mm– y*10–60. Determine the product of inertia of the parabolic area with respect to the x and y axes.B 50 mmy1 in.2 in. y ϭ 2x2x97449. 10 Solutions 449181/28/094:22 PMPage 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.y1 in.2 in. y ϭ 2x2x97550. 10 Solutions 449181/28/094:22 PMPage 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 22 y x –– ϩ –– ϭ 1 a2 b2b x a97651. 10 Solutions 449181/28/094:22 PMPage 977© 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.10–63. Determine the product of inertia for the area with respect to the x and y axes.y2 in.y3 ϭ x x 8 in.97752. 10 Solutions 449181/28/094:22 PMPage 978© 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.*10–64. Determine the product of inertia of the area with respect to the x and y axes.y4 in. x4 in.x y ϭ ––(x Ϫ 8) 497853. 10 Solutions 449181/28/094:22 PMPage 979© 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.•10–65. Determine the product of inertia of the area with respect to the x and y axes.y8y ϭ x3 ϩ 2x2 ϩ 4x3mx 2m97954. 10 Solutions 449181/28/094:22 PMPage 980© 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.10–66. Determine the product of inertia for the area with respect to the x and y axes.y y2 ϭ 1 Ϫ 0.5x1mx 2m98055. 10 Solutions 449181/28/094:22 PMPage 981© 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.10–67. Determine the product of inertia for the area with respect to the x and y axes.yy3 ϭh3 x b h x b98156. 10 Solutions 449181/28/094:22 PMPage 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.yx2 ϩ 4y2 ϭ 162 in.x 4 in.•10–69. Determine the product of inertia for the parabolic area with respect to the x and y axes.yy2 ϭ x 2 in. x 4 in.98257. 10 Solutions 449181/28/094:22 PMPage 983© 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.10–70. Determine the product of inertia of the composite area with respect to the x and y axes.y2 in.2 in.2 in.1.5 in. 2 in. x98358. 10 Solutions 449181/28/094:22 PMPage 984© 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.10–71. 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.xC 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 mm50 mm7.5 mm Cx 17.5 mm 5 mm30 mm98459. 10 Solutions 449181/28/094:22 PMPage 985© 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.•10–73. Determine the product of inertia of the beam’s cross-sectional area with respect to the x and y axes.y 10 mm300 mm10 mm x 10 mm 100 mm10–74. 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.y5 in.0.5 in.1 in.xC 1 in.5 in.5 in.1 in.9855 in.60. 10 Solutions 449181/28/094:22 PMPage 986© 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.10–75. 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 Cx 60Њ200 mm 20 mm20 mm 175 mm986u61. 10 Solutions 449181/28/094:22 PMPage 987© 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.*10–76. 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 x10 mm100 mm 10 mm 300 mm x¿Cy 10 mm x 200 mm98762. 10 Solutions 449181/28/094:22 PMPage 988© 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.•10–77. 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 xC 150 mm100 mm98810 mm63. 10 Solutions 449181/28/094:22 PMPage 989© 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.10–78. 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.u1.5 in.3 in. 30Њ C 3 in.989x64. 10 Solutions 449181/28/094:22 PMPage 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 v0.5 in. 4.5 in.4.5 in.0.5 in.60Њ4 in.xC 0.5 in.8 in.990y65. 10 Solutions 449181/28/094:22 PMPage 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.99166. 10 Solutions 449181/28/094:22 PMPage 992© 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.*10–80. 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 x0.5 in. 6 in.Cx 0.5 in.6 in.992y67. 10 Solutions 449181/28/094:22 PMPage 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.99368. 10 Solutions 449181/28/094:22 PMPage 994© 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.•10–81. 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.y100 mm 20 mm 20 mm150 mm xC 150 mm100 mm99420 mm69. 10 Solutions 449181/28/094:22 PMPage 995© 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.99570. 10 Solutions 449181/28/094:22 PMPage 996© 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.10–82. 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.y25 mm200 mmv25 mmxC 60Њ25 mm 75 mm 75 mm996uy71. 10 Solutions 449181/28/094:22 PMPage 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.99772. 10 Solutions 449181/28/094:22 PMPage 998© 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.10–83.Solve Prob. 10–75 using Mohr’s circle.99873. 10 Solutions 449181/28/094:22 PMPage 999© 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.*10–84.Solve Prob. 10–78 using Mohr’s circle.99974. 10 Solutions 449181/28/094:22 PMPage 1000© 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.•10–85.Solve Prob. 10–79 using Mohr’s circle.100075. 10 Solutions 449181/28/094:22 PMPage 1001© 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.10–86.Solve Prob. 10–80 using Mohr’s circle.100176. 10 Solutions 449181/28/094:22 PMPage 1002© 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.10–87.Solve Prob. 10–81 using Mohr’s circle.100277. 10 Solutions 449181/28/094:22 PMPage 1003© 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.*10–88.Solve Prob. 10–82 using Mohr’s circle.100378. 10 Solutions 449181/28/094:22 PMPage 1004© 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•10–89. 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 ϭ –– (r0 Ϫ y) r0hy x1004r079. 10 Solutions 449181/28/094:22 PMPage 1005© 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.10–90. 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.yr y ϭ –x hr xh100580. 10 Solutions 449181/28/094:22 PMPage 1006© 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.z10–91. 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 yx100681. 10 Solutions 449181/28/094:22 PMPage 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 ϭ 1 y2 4 1m yx 2m100782. 10 Solutions 449181/28/094:22 PMPage 1008© 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.•10–93. 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 ϭ 50 x 100 mm x200 mm100883. 10 Solutions 449181/28/094:22 PMPage 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.z10–94. 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 y –– ϩ –– ϭ 1 2 a b2byx100984. 10 Solutions 449181/28/094:22 PMPage 1010© 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.10–95. 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 b y ϭ –x ϩ b a2b b xa101085. 10 Solutions 449181/28/094:22 PMPage 1011© 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.*10–96. 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.y3 in. y3 ϭ 9x x 3 in.101186. 10 Solutions 449181/28/094:22 PMPage 1012© 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•10–97. 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>m3.2mz2 ϭ 8y4myx101287. 10 Solutions 449181/28/094:22 PMPage 1013© 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.z10–98. 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 ft8 ft zϭ3 –– y2yx101388. 10 Solutions 449181/28/094:22 PMPage 1014© 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.z10–99. 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.4m1 z2 ϭ –– y3 16Ox10142my89. 10 Solutions 449181/28/094:22 PMPage 1015© 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.*10–100. 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.O450 mmA100 mm B101590. 10 Solutions 449181/28/094:22 PMPage 1016© 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.•10–101. 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 m0.5 mD0.2 mL AOB C101691. 10 Solutions 449181/28/094:22 PMPage 1017© 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.z10–102. Determine the mass moment of inertia of the 2-kg bent rod about the z axis.300 mmx1017300 mmy92. 10 Solutions 449181/28/094:22 PMPage 1018© 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.z10–103. 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 mm200 mm 100 mm 200 mm 200 mm x1018200 mm200 mm 200 mmy93. 10 Solutions 449181/28/094:22 PMPage 1019© 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*10–104. 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 mm200 mm 100 mm 200 mm 200 mm x1019200 mm200 mm 200 mmy94. 10 Solutions 449181/28/094:22 PMPage 1020© 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.•10–105. 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.Oy 2mG 0.5 m 1m102095. 10 Solutions 449181/28/094:22 PMPage 1021© 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.z10–106. 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 mm300 mm150 mm 300 mm x1021y96. 10 Solutions 449181/28/094:22 PMPage 1022© 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.10–107. 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 x180 mm20 mm x¿ 30 mm 20 mm102250 mm30 mm97. 10 Solutions 449181/28/094:22 PMPage 1023© 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.*10–108. 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 x180 mm20 mm x¿ 30 mm 20 mm102350 mm30 mm98. 10 Solutions 449181/28/094:22 PMPage 1024© 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.•10–109. 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 ft1 ft OA102499. 10 Solutions 449181/28/094:22 PMPage 1025© 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.10–110. 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.O50 mm150 mm 50 mm400 mm400 mm150 mm 150 mm1025150 mm100. 10 Solutions 449181/28/094:22 PMPage 1026© 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.10–111. 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.O200 mm200 mm200 mm1026101. 10 Solutions 449181/28/094:22 PMPage 1027© 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.*10–112. Determine the moment of inertia of the beam’s cross-sectional area about the x axis which passes through the centroid C.yd 2 d 260Њ xC60Њ 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 2yd 2 d 260Њd 21027xC60Њd 2102. 10 Solutions 449181/28/094:22 PMPage 1028© 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.yyϭ aa –– – x 2axa1028a103. 10 Solutions 449181/28/094:22 PMPage 1029© 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. Cx¿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 mm200 mm C 20 mm 200 mm1029x¿ 57.37 mm104. 10 Solutions 449181/28/094:22 PMPage 1030© 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.•10–117. Determine the moment of inertia of the area about the y axis.y4y ϭ 4 – x 2 1 ft x 2 ft10–118. Determine the moment of inertia of the area about the x axis.y4y ϭ 4 – x 2 1 ft x 2 ft1030105. 10 Solutions 449181/28/094:22 PMPage 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 mm200 mmC – yx¿ 1 y ϭ ––– x 2 200x1031106. 10 Solutions 449181/28/094:22 PMPage 1032© 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.*10–120. 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.Oy 1.5 mG A 0.1 m 0.3 m1032107. 10 Solutions 449181/28/094:22 PMPage 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.y1m y ϭ x3 x 1m1033