Theorems of Pappus and Goldinus Mechanical Engineering Notes | EduRev

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Mechanical Engineering : Theorems of Pappus and Goldinus Mechanical Engineering Notes | EduRev

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Pappus's Centroid Theorem

Theorems of Pappus and Goldinus Mechanical Engineering Notes | EduRevTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRevTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRev

The first theorem of Pappus states that the surface area Sof a surface of revolution generated by the revolution of a curve about an external axis is equal to the product of the arc length of the generating curve and the distance d1traveled by the curve's geometric centroid Theorems of Pappus and Goldinus Mechanical Engineering Notes | EduRev

Theorems of Pappus and Goldinus Mechanical Engineering Notes | EduRev

(Kern and Bland 1948, pp. 110-111). The following table summarizes the surface areas calculated using Pappus's centroid theorem for various surfaces of revolution.

solidgenerating curvesTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRevS
coneinclined line segmentTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRevTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRevTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRev
cylinderparallel line segmenthr2πrh
spheresemicircleπr2π/π4πr2

Theorems of Pappus and Goldinus Mechanical Engineering Notes | EduRevTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRevTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRev

Similarly, the second theorem of Pappus states that the volume V of a solid of revolution generated by the revolution of a lamina about an external axis is equal to the product of the area A of the lamina and the distance d2 traveled by the lamina's geometric centroidTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRev

Theorems of Pappus and Goldinus Mechanical Engineering Notes | EduRev

(Kern and Bland 1948, pp. 110-111). The following table summarizes the surface areas and volumes calculated using Pappus's centroid theorem for various solids and surfaces of revolution.

solidgenerating laminaATheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRevV
coneright triangleTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRevTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRevTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRev
cylinderrectanglehrTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRevπr2h
spheresemicircleTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRevTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRevTheorems of Pappus and Goldinus Mechanical Engineering Notes | EduRev

 

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