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ENGINEERING MECHANICS
Theorems of Pappus-Guldinus
04/10/23 2
Theorems of Pappus-Guldinus
Also known as Pappus’s Theorem, Guldinus Theorem or Pappus’s Centroid Theorem.
Refers to either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.
Attributed to Pappus of Alexandria and Paul Guldin.
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Pappus of Alexandria (4th Century AD)
Known for his theorem in projective geometry.
Paul Guldin (1577-1643)Swiss mathematician and astronomer.
Associate of Johannes Kepler.
Independently rediscovered Pappus’s
Theorem
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Theorem I
“The area of a surface of revolution is the product of the length of the generating curve and the distance travelled by the centroid of the curve, while the surface is generated.”
Surface of Revolution:A surface generated by rotating a two-dimensional curve (straight line, arc etc.) about a fixed axis.
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Surfaces of Revolution
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Cylinder
Distance travelled by Centroid = 2 π r Length of the generating curve = h
Surface area generated = 2 π r h
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Sphere
Distance travelled by the Centroid = 2 π (2r/π) = 4r
Length of the generating curve = π r
Surface area generated = 4 π r2
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Cone
Distance travelled by the Centroid = 2 π (r/2) = π r
Length of the generating curve = √(h2+r2)
Surface area generated = π r √(h2+r2)
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Theorem II
“The volume of a body of revolution is obtained from the product of the generating area and the distance travelled by the centroid of the area, while the body is being generated.”
Solid/Body of Revolution
A solid generated by rotating a plane area about an axis that lies in the same plane.
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Cylinder
Generating Area = h r
Distance of CG from axis = r/2
Distance travelled by the centroid = 2 π (r/2) = πr
Volume generated = π r2 h
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Sphere
Generating Area = (π r2)/2
Distance of CG from axis = 4r/3π
Distance travelled by the centroid = 2 π (4r/3π) = 8r/3
Volume generated = (4/3) πr3
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Cone
Generating Area = hr/2
Distance of CG from axis = r/3
Distance travelled by the centroid = 2 π (r/3) = 2 π r/3
Volume generated = πr2h/3
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Torus
A torus is a surface of revolution generated by revolving a circle in 3-D space,
about an axis coplanar with the circle. E.g. inner tubes of tyres, lifebuoys…
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Torus
Surface Area
Length of the generating curve = 2 π rDistance of the CG from axis = RDistance travelled by the centroid = 2 π RSurface area of the Torus = 4 π2 Rr
Volume
Area of the generating lamina = π r2
Distance of the CG from axis = RDistance travelled by the centroid = 2 π RVolume of the Torus = 2 π2 Rr2
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References
Wolfram Mathworld
http://mathworld.wolfram.com/PappussCentroidTheorem.html
Engineering Mechanics: Dr. N Kottiswaran, Sri Balaji Publications, Tiruchengode.
Engineering Mechanics: Dr. D S Kumar, S K Kataria & Sons, New Delhi.
Engineering Mechanics: S Rajasekharan & G Sankarasubramanian, Vikas Publishing House, New Delhi.