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9 Solutions 44918 1/28/09 2:34 PM Page 815 © 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 writi...

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9 Solutions 44918

1/28/09

2:34 PM

Page 815

© 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.

•9–1. Determine the mass and the location of the center of mass (x, y) of the uniform parabolic-shaped rod. The mass per unit length of the rod is 2 kg>m.

y

y2 ⫽ 4x

4m

x 4m

815

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9–2. The uniform rod is bent into the shape of a parabola and has a weight per unit length of 6 lb>ft. Determine the reactions at the fixed support A.

y

y2 ⫽ 3x

3 ft

A x 3 ft

816

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9–3. Determine the distance x to the center of mass of the homogeneous rod bent into the shape shown. If the rod has a mass per unit length of 0.5 kg>m, determine the reactions at the fixed support O.

y

1m

1m y2 ⫽ x 3 O x

817

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*9–4. Determine the mass and locate the center of mass (x, y) of the uniform rod. The mass per unit length of the rod is 3 kg>m.

y

y ⫽ 4 ⫺ x2

4m

x 2m

818

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•9–5. Determine the mass and the location of the center of mass x of the rod if its mass per unit length is m = m0(1 + x>L).

y

x L

819

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9–6. Determine the location (x, y) of the centroid of the wire.

y

2 ft

4 ft

y ⫽ x2

x

9–7. Locate the centroid x of the circular rod. Express the answer in terms of the radius r and semiarc angle a.

y

r

a C

a r

–x

820

x

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y

*9–8. Determine the area and the centroid (x, y) of the area.

4m

y2 ⫽ 4x

x 4m

821

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•9–9. Determine the area and the centroid (x, y) of the area.

y

1m

y2 ⫽ x3

x 1m

822

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y

9–10. Determine the area and the centroid (x, y) of the area.

3 ft

1 x3 y ⫽ –– 9

x 3 ft

823

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9–11. Determine the area and the centroid (x, y) of the area.

y

y2 ⫽ 4ax 2 ab

x b

824

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*쐍9–12. Locate the centroid x of the area.

y

y ⫽ x1/ 2 ⫹ 2x5/3 x 2 ft

825

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•쐍9–13. Locate the centroid y of the area.

y

y ⫽ x1/ 2 ⫹ 2x5/3 x 2 ft

826

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y

9–14. Determine the area and the centroid (x, y) of the area.

xy  c2 x a b

827

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9–15. Determine the area and the centroid (x, y) of the area.

y

h

h x2 y ⫽ –– a2

x a

828

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*9–16. Locate the centroid (x, y) of the area.

y 1 y  1 – – x2 4 1m x 2m

829

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•9–17. Determine the area and the centroid (x, y) of the area.

y

h h x2 y ⫽ –– a2 x a

830

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9–18. The plate is made of steel having a density of 7850 kg>m3. If the thickness of the plate is 10 mm, determine the horizontal and vertical components of reaction at the pin A and the tension in cable BC.

y C

B y3 ⫽ 2x 2m A

x

4m

831

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r  a (1  cos u) y

9–19. Determine the location x to the centroid C of the upper portion of the cardioid, r = a(1 - cos u).

C

r u _ x

832

x

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*9–20. The plate has a thickness of 0.5 in. and is made of steel having a specific weight of 490 lb>ft3. Determine the horizontal and vertical components of reaction at the pin A and the force in the cord at B.

y

B

3 ft

A

x2 y  –– 3 x 3 ft

833

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•9–21.

Locate the centroid x of the shaded area.

y x2 ) y  2k(x  — 2a

ka

x a

9–22. Locate the centroid x of the area.

y 0.5 in.

2 in.

y  1x

0.5 in. x 2 in.

834

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9–23. Locate the centroid y of the area.

y 0.5 in.

2 in.

y  1x

0.5 in. x 2 in.

*9–24. Locate the centroid (x, y) of the area.

y

y  9  x2 9 ft

x 3 ft

835

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•9–25. Determine the area and the centroid (x, y) of the area.

y

y⫽x 3 ft x3 y ⫽ –– 9 x 3 ft

836

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9–26. Locate the centroid x of the area.

y

y2  x 1m y  x2 x

1m

9–27. Locate the centroid y of the area.

y

y2  x 1m y  x2

1m

837

x

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*9–28. Locate the centroid x of the area.

y

h xn y  –– an h

x a

•9–29. Locate the centroid y of the area.

y

h xn y  –– an h

x a

838

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9–30. The steel plate is 0.3 m thick and has a density of 7850 kg>m3. Determine the location of its center of mass. Also determine the horizontal and vertical reactions at the pin and the reaction at the roller support. Hint: The normal force at B is perpendicular to the tangent at B, which is found from tan u = dy>dx.

y y2  2x

2m

A

x

2m

B 2m

839

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9–31. Locate the centroid of the area. Hint: Choose elements of thickness dy and length [(2 - y) - y2].

y

y2  x

yx2

1m

x 1m

840

1m

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*9–32. Locate the centroid x of the area.

y

y2  4x

2 ft y  2x

x 1 ft

•9–33. Locate the centroid y of the area.

y

y2  4x

2 ft y  2x

x 1 ft

841

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9–34. If the density at any point in the rectangular plate is defined by r = r0(1 + x>a), where r0 is a constant, determine the mass and locate the center of mass x of the plate. The plate has a thickness t.

y

b –– 2 x b –– 2

a

842

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z

9–35. Locate the centroid y of the homogeneous solid formed by revolving the shaded area about the y axis.

y2 ⫹ (z ⫺ a)2 ⫽ a2

a

y

x

843

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z

*9–36. Locate the centroid z of the solid.

z  a1 (a  y)2

a

y a

x

844

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z

•9–37. Locate the centroid y of the homogeneous solid formed by revolving the shaded area about the y axis.

1 y3 z2 ⫽ –– 16 2m y x 4m

845

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z

9–38. Locate the centroid z of the homogeneous solid frustum of the paraboloid formed by revolving the shaded area about the z axis. h z  – (a2  y2) a2

h – 2

h – 2 y

x

846

a

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z

9–39. Locate the centroid y of the homogeneous solid formed by revolving the shaded area about the y axis.

5 ft z2 ⫽ y2 ⫺ 9 4 ft 3 ft y

x

847

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