Engineering Mechanics : STATICS Lecture #07 By, Noraniah Kassim Hairul Mubarak b Hassim Universiti Tun Hussein Onn Malaysia (UTHM),

DAJ 21003 ( Statics & Dynamics )

THEOREMS OF PAPPUS AND GULDINUS Today’s Objective: Students will be able to : use the theorems of Pappus and Guldinus for finding the area and volume for a surface of revolution Learning Topics: Theorem of Pappus and Guldinus DAJ 21003 ( Statics & Dynamics )

THEOREM OF PAPPUS AND GULDINUS Theorems of Pappus and Guldinus are used to find the surface area and volume of any object of revolution Surface area of revolution is generated by revolving a planar curve about a nonintersecting fixed axis in the plane of the curve. Volume of revolution is generated by revolving a plane area about a nonintersecting fixed axis in the plane of the area. DAJ 21003 ( Statics & Dynamics )

DAJ 21003 ( Statics & Dynamics ) SURFACE AREA The area of a surface of revolution equals the product of the length of the generating curve and the distance traveled by the centroid of the curve in generating the surface area. where, A = surface area of revolution θ = angle of revolution measured in radians, θ ≤ 2π r = perpendicular distance from the axis of revolution to the centroid of the generating curve L = length of the generating curve DAJ 21003 ( Statics & Dynamics )

DAJ 21003 ( Statics & Dynamics ) VOLUME The volume of a body of revolution equals the product of the generating area and the distance traveled by the centroid of the area in generating the volume where, V = volume of revolution θ = angle of revolution measured in radians, θ ≤ 2π r = perpendicular distance from the axis of revolution to the centroid of the generating area A = generating area DAJ 21003 ( Statics & Dynamics )

DAJ 21003 ( Statics & Dynamics ) EXAMPLE Surface of revolution is generated by rotating a plane curve about a fixed axis. Area of a surface of revolution is equal to the length of the generating curve times the distance traveled by the centroid through the rotation. DAJ 21003 ( Statics & Dynamics )

DAJ 21003 ( Statics & Dynamics ) EXAMPLE Body of revolution is generated by rotating a plane area about a fixed axis. Volume of a body of revolution is equal to the generating area times the distance traveled by the centroid through the rotation. DAJ 21003 ( Statics & Dynamics )

DAJ 21003 ( Statics & Dynamics ) EXAMPLE Given: Homogeneous thin plate as shown Find: Volume of the solid obtained by rotating the area about (a) the x axis Plan: Find the centroid of the plate using the method of composite ANSWERS: 2. D DAJ 21003 ( Statics & Dynamics )

IN CLASS TUTORIAL (Continued) Solution : DAJ 21003 ( Statics & Dynamics )

IN CLASS TUTORIAL (Continued) DAJ 21003 ( Statics & Dynamics )

DAJ 21003 ( Statics & Dynamics ) EXAMPLE (9-86) Using integration, determine both the area and the distance yc to the centroid of the shaded area. Then using the second theorem of PappusGuldinus, determine the volume of the solid generated by revolving the shaded area about the x axis. DAJ 21003 ( Statics & Dynamics )