Archive for November, 2009

Center of gravity of a trapezium

November 14, 2009

It is worth noting that the center of gravity of a trapezium ABCD does not coincide with the center of gravities of the set \lbrace A,B,C,D \rbrace. Indeed, if we consider the special case where the coordinates of A,B,C,D are (-\alpha,0),(\alpha,0),(\beta,h) and (-\beta,h), respectively, the coordinates of the first center of gravity (let’s call it G_1) are (0,\frac{(\alpha+2\beta)h}{3(\alpha+\beta)}) while the coordinates of the second (let’s call it G_2) are (0,\frac{h}{2}). If \alpha>\beta, then G_1 is below G_2. Also, G_1 satisifies (2\alpha+beta)(\overrightarrow{GA}+\overrightarrow{GB})+(\alpha+2\beta)(\overrightarrow{GC}+\overrightarrow{GD})=\overrightarrow{0} while G_2 satisifies \overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}+\overrightarrow{GD}=\overrightarrow{0}.