Pappus's Centroid Theorem.pptx

Pappu's Theorem;
Pappus-Guldin Theorem;
Pappus Centroid theorem;
Surface area from centroid;
Volume from Centroid;
Theorem-1 (Surface area);
Theorem-2 (volume);
Solved problems;
Solved numericals;
Practice problems;
Equations to remember;
Engineering Mechanics;
Basic Engineering mechanics;
Mechanics of Centroid;
B.Tech Semester-1

1. Pappus's Centroid Theorem CL-101 ENGINEERING MECHANICS B. Tech Semester-I Prof. Samirsinh P Parmar Mail: [email protected] Asst. Professor, Department of Civil Engineering, Faculty of Technology, Dharmsinh Desai University, Nadiad-387001 Gujarat, BHARAT

2. Content of the presentation Pappus- Guldin theorem Theorem-I (Surface) Theorem-II (Volume0 Solved Problems Unsolved problems Practice problems

3. Pappus– Guldin Theorems The Pappus– Guldin Theorems Suppose that a plane curve is rotated about an axis external to the curve. Then The resulting surface area of revolution is equal to the product of the length of the curve and the displacement of its centroid; In the case of a closed curve, the resulting volume of revolution is equal to the product of the plane area enclosed by the curve and the displacement of the centroid of this area.

4. GENERATION OF SURFACE OF REVOLUTION

5. Pappus's Theorem for Surface Area The first theorem of Pappus states that the surface area A of a surface of revolution obtained by rotating a plane curve C about a non-intersecting axis which lies in the same plane is equal to the product of the curve length L and the distance d traveled by the centroid of C: A = Ld

6. Pappus's Theorem for Volume The second theorem of Pappus states that the volume of a solid of revolution obtained by rotating a lamina F about a non-intersecting axis lying in the same plane is equal to the product of the area A of the lamina and the distance d traveled by the centroid of F: V = Ad

7. Surface Area and Volume of a Torus A torus is the solid of revolution obtained by rotating a circle about an external coplanar axis. We can easily find the surface area of a torus using the 1 st Theorem of Pappus. If the radius of the circle is r and the distance from the center of circle to the axis of revolution is R then the surface area of the torus is The volume inside the torus is given by the Theorem of Pappus: The Pappus's theorem can also be used in reverse to find the centroid of a curve or figure.

8. Theorem-1 (Surface generation) Ref: Engg . Mechanics – D.S.Kumar

9. Theorem-II (Volume generation) Ref: Engg . Mechanics – D.S.Kumar

10. Theorem-1 (Surface generation) Ref: Engg . Mechanics – A.K. Tayal

11. Theorem-II (Volume generation) Ref: Engg . Mechanics – A.K.Tayal

12. Solved Problems

13. Example 1. A regular hexagon of side length a is rotated about one of the sides. Find the volume of the solid of revolution. Solution. Given the side of the hexagon a ,  we can easily find the the apothem length m. Hence, the distance d traveled by the centroid C when rotating the hexagon is written in the form The area A of the hexagon is equal to Using the 2nd theorem of Pappus, we obtain the volume of the solid of revolution:

14. Example 2. Find the centroid of a uniform semicircle of radius R Solution. Let m be the distance between the centroid G and the axis of rotation. When the semicircle makes the full turn, the path d traversed by the centroid is equal to D =2 π m The solid of rotation is a ball of volume By the  2nd  theorem of Pappus, we have the relationship V = Ad where    is the area of the semicircle.

15. Example-3 3. An ellipse with the semimajor axis a and semi minor axis b is rotated about a straight line parallel to the axis and spaced from it at a distance m > b. Find the volume of the solid of revolution. Solution. The volume of the solid of revolution can be determined using the 2nd theorem of Pappus: V = Ad The path d traversed in one turn by the centroid of the ellipse is equal to d= 2 π m The area of the ellipse is given by the formula A = π ab Hence, the volume of the solid is V =Ad = π ab x 2 π m = 2 π 2 mab In particular, when m = 2b the volume is equal to  V =4 π 2 ab 2

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31. Solved Problem- 11

32. Solved Problem- 12

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34. Practice Problems

35. Practice Problems

36. Practice Problems

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38. Practice Problems

39. Practice Problems

40. Practice Problems

41. Equations to remember

42. The following table summarizes the surface areas calculated using Pappus's centroid theorem for various surfaces of revolution.

43. The following table summarizes the surface areas and volumes calculated using Pappus's centroid theorem for various solids and surfaces of revolution.

44. Contact: Prof. Samirsinh P Parmar Mail: [email protected] Dept. of Civil Engg . Dharmsinh Desai University, Nadiad, Gujarat Bharat.