Theorem of Pappu's states that if a plane area is rotated through any angle about an axis in its plane, the volume traced out by the area is equal to the area multiplied by length of the path followed by its centre of mass. The only condition is that the axis should not cross the area. For example, we take a semicircular disc and rotate it about its diameter as shown. If the disc is rotated by 180°, it will trace out a hemisphere. V= 2πR3/3 ; A = πR2/2 Suppose the center of mass of the semicircular disc is at distance x from the center. The length of the path traced out by it is πx. From the pappu's theorem, πx = V/A = 2πR3/3/ πR2/2. From this we can easily obtain the center of mass of the semicircular disc as 4R/3π.

Engineering

Physics

Center of Mass

Question

Theorem of Pappu's states that if a plane area is rotated through any angle about an axis in its plane, the volume traced out by the area is equal to the area multiplied by length of the path followed by its centre of mass. The only condition is that the axis should not cross the area. For example, we take a semicircular disc and rotate it about its diameter as shown. If the disc is rotated by 180°, it will trace out a hemisphere.

V= 2πR³/3 ; A = πR²/2 Suppose the center of mass of the semicircular disc is at distance x from the center. The length of the path traced out by it is πx. From the pappu's theorem, πx = V/A = 2πR³/3/ πR²/2. From this we can easily obtain the center of mass of the semicircular disc as 4R/3π.