Volume and Pappus theorem

Centre of Gravity of 3-Dimensional Body

Determine the location of the centre of gravity of the homogenous body of , which was obtained by joining a hemisphere and a cylinder and carving out a cone.

Determine the y coordinate of the centroid of the body shown in figure.

• A surface of revolution is generated by revolving a given curve about an axis. The given curve is a profile curve while the axis is the axis of revolution.

• A body of revolution is a body which can be generated by rotating a plane area about fixed axis.

Theorems of Pappus –Guldinus

• Theorem I : The area of a surface of revolution is equal to the length of the generating curve times the distance travelled by the centroid of the curve while the surface is being generated.

• Theorem II : The volume of a body of revolution is equal to the generating area times the distance travelled by the centroid of the area while the body is being generated.

• Proof

• Determine the area of the surface of revolution shown, which is obtained by rotating a quarter-circular arc about a vertical axis.