Question

2.Three particles each 1kg mass are placed at the corners of a right angled triangled triangle AOB, O being the origin of the coordiantes system (OAOB) along positive x-direction and positive y direction.The position vector of the centre of mass?If OA=OB =1m (in meters)

Answer 1

$$\begin{gathered} \mathrm{Given}, \\ \mathrm{OA}=\mathrm{OB}=1\mathrm{m} \\ {\mathrm{m}}_1={\mathrm{m}}_2={\mathrm{m}}_3=1\mathrm{kg} \\ \mathrm{Coordinates}\mathrm{of}\mathrm{O}=(0,0) \\ \mathrm{Coordinates}\mathrm{of}\mathrm{A}=(1,0) \\ \mathrm{Coordinates}\mathrm{of}\mathrm{B}=(0,1) \\ {\mathrm{x}}_{\mathrm{cm}}=\frac{{\mathrm{m}}_1{\mathrm{x}}_1+{\mathrm{m}}_2{\mathrm{x}}_2+{\mathrm{m}}_3{\mathrm{x}}_3}{{\mathrm{m}}_1+{\mathrm{m}}_2+{\mathrm{m}}_3}=\frac{1\times 0+1\times 1+1\times 0}{1+1+1}=\frac{1}{3} \\ {\mathrm{y}}_{\mathrm{cm}}=\frac{{\mathrm{m}}_1{\mathrm{y}}_1+{\mathrm{m}}_2{\mathrm{y}}_2+{\mathrm{m}}_3{\mathrm{y}}_3}{{\mathrm{m}}_1+{\mathrm{m}}_2+{\mathrm{m}}_3}=\frac{1\times 0+1\times 0+1\times 1}{1+1+1}=\frac{1}{3} \\ \mathrm{Thus},\mathrm{position}\mathrm{vector}\mathrm{of}\mathrm{center}\mathrm{of}\mathrm{mass}={\mathrm{x}}_{\mathrm{cm}}\hat{\mathrm{i}}+{\mathrm{y}}_{\mathrm{cm}}\hat{\mathrm{j}} \\ =\frac{\mathrm{i}+\mathrm{j}}{3} \end{gathered}$$