ENGINEERING MECHANICS

Theorems of Pappus-Guldinus

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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.