Mechanics: Statics Distributed Force 3D Body Theorems of PappusGuldinus Draft for Information Only
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Theorems of PappusGuldinus
Theorems of PappusGuldinusThe theorems of PappusGuldinus were formulated by the Greek geometer Pappus of Alexandria during the 4th century A.D. (about 340 A.D.) and were restated by the Swiss mathematician Paul Guldinus (1640). The two theorems of PappusGuldinus describe the area of surface of revolution and the volume of body of revolution by the circular path traversed by their centroid during the revolution. Theorem 1: Surface of RevolutionFor the surface of a surface of revolution generated by the rotation of a plane curve about a nonintersecting axis, the surface area A of the surface of revolution is equal to the product of the curve length L of the generating curve and the travelled distance d of the centroid of the generating curve during the generation of the surface by revolution. And the travelled distance of the centroid by revolution can also be expressed in terms of the perpendicular distance y of the centroid away from the rotating axis. Surface of RevolutionThe area A of the surface of revolution can be determined by integration through the revolution of an elemental segment dL. Imply The area A can be rearranged in the form of the integral of the first moment of an elemental segment , which can also be expressed in terms of the centrod of the generating curve, Imply Theorem 2: Body of RevolutionFor the body of a body of revolution generated by the rotation of a plane region about a nonintersecting axis, the body volume V of the body of revolution is equal to the product of the area A of the generating plane region and the travelled distance d of the centroid of the generating region during the generation of the body by revolution. And the travelled distance of the centroid by revolution can also be expressed in terms of the perpendicular distance y of the centroid away from the rotating axis. Body of RevolutionThe volume V of the body of revolution can be determined by integration through the revolution of an elemental area dA. Imply The volume V can be rearranged in the form of the integral of the first moment of an elemental area, which can also be expressed in terms of the centrod of the generating area, Imply Applications of Theorems of PappusGuldinusThe Theorems of PappusGuldinus provides a simple relationship between the area of surface of revolution or the volume of body of revolution and the centroid of the generating plane curve or the centroid of the generating plane area. Therefore the Theorems of PappusGuldinus can be used to determine the area of surface of revolution and the volume of body of revolution from the generating curve and the generating area accordingly. And the centroid of a generating plane curve and the centroid of a generating plane area can also be determined from the surface of revolution and body of revolution accordingly. Surface of RevolutionCylinder Area of cylinder of surface of revolution is Area of cylinder of surface of revolution by theorem of PappusGuldinus is Cone Area of cone of surface of revolution is Area of cone of surface of revolution by theorem of PappusGuldinus is Sphere Area of sphere of surface of revolution is Area of sphere of surface of revolution by theorem of PappusGuldinus is Torus Area of torus of surface of revolution is Area of torus of surface of revolution by theorem of PappusGuldinus is Solid of RevolutionCylinder Volume of cylinder of body of revolution is Volume of cylinder of body of revolution by theorem of PappusGuldinus is Cone Volume of cone of surface of revolution is Volume of cone of surface of revolution by theorem of PappusGuldinus is Sphere Volume of sphere of surface of revolution is Volume of sphere of surface of revolution by theorem of PappusGuldinus is Torus Volume of torus of surface of revolution is Volume of torus of surface of revolution by theorem of PappusGuldinus is ©sideway ID: 120700003 Last Updated: 9/7/2012 Revision: 0 Ref: References
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