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. CL-101 ENGINEERING MECHANICS
B. Tech Semester-I
Prof. Samirsinh P Parmar
Mail: spp.cl@ddu.ac.in
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
1. The resulting surface area of revolution is equal to the
product of the length of the curve and the displacement
of its centroid;
2. 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

7. 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

8. 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 1st
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.

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

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

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

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

13. Solved Problems

14. 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:

15. 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 =2πm
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.

16. 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= 2πm
The area of the ellipse is given by the formula
A = πab
Hence, the volume of the solid is
V =Ad = πab x 2πm = 2π2mab
In particular, when m = 2b the volume is equal to V =4π2ab2

17. Solution

18. Solution,

19. Solution.

20. Solution,

21. Solution.

22. Solution,

23. Solution.

24. Solution,

25. Solution.

26. Solution.

27. Solution.

28. Solution.

29. Solution.

30. Solution.

31. Solution.

32. Solved Problem- 11

33. Solved Problem- 12

34. Solution:

35. Practice Problems

36. Practice Problems

37. Practice Problems

38. Practice Problems

39. Practice Problems

40. Practice Problems

41. Practice Problems

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

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

45. CONTACT:
Prof. Samirsinh P Parmar
Mail: spp.cl@ddu.ac.in
Dept. of Civil Engg.
Dharmsinh Desai University, Nadiad,
Gujarat
Bharat.