Theorem of Pappus- Guldinus
Theorem of Pappus- Guldinus:
Consider a coplanar generating curve and an axis of revolution in the plane of this curve. The generating curve can touch but must not cross the axis of revolution. The surface of revolution developed by revolving the generating curve about the axis of revolution has an area equal to the product of the length of the genearating curve times the circumference of the circle formed by the centroid of the generating curve in the process of generating a surface of revolution.
Consider an element dl of the generating curve. For a single revolution of the generating curve about the x-axis , the line segment dl traces an area dA = 2pydl
For the entire curve, this area becomes the surface of revolution given as
where L is the length of the curve and yc is the centroidal coordinate of the curve. But 2pyc is the circumferential length of the circle formed by having the centroid of the curve rotate about the x-axis. This proves the theorem.
Consider a plane surface and an axis of revolution coplanar with the surface but oriented such that the axis such that the axis can intersect the surface only as a tangent at the bounding or have no interaction at all. The volume of the body of revolution developed by rotating the plane surface about the axis of revolution equals the product of the surface times the circumference of the circle formed by the centroid of the surface in the process of generating the body of revolution.
Consider a plane surface A as shown in the figure. The volume generated by revolving dA of this surface about the x-axis is
Teh volume of the body of revolution formed from A is then,
This completes the proof of the second theorem
More on Centre of Gravity:
Consider a suspended body as shown. The weights of variuos parts of this body are acting vertically downwards. The only upward force is the force in the string. To satisfy the equilibrium condition, the resultant of the weight W along the line of string. Now, if the orientation is changed and the body is suspended, the line of action must pass through 2-2. The interaction of two lines gives the centre of gravity.