question_answer

 Four particles, each of mass 1 kg, are placed at the corners of a square of side 1m in the X- Y plane. If the point of intersection of the diagonals of the square, is taken as the origin, the co-ordinates of the centre of mass are:

A)                  (1, 1)                                    

B)                  (-1, 1)

C)                  (1,-1)                                   

D)                  (0, 0)

Correct Answer: D

Solution :

                 Given: \[{{m}_{1}}={{m}_{2}}={{m}_{3}}={{m}_{4}}=1\,kg\]                 \[AB=BC=CD=DA=1m\]                                 Hence, the co-ordinates of A, B, C, D are given in the figure, from the relation for\[{{x}_{cm}}\] is \[{{x}_{cm}}=\frac{{{m}_{1}}{{x}_{1}}+{{m}_{2}}{{x}_{2}}+{{m}_{3}}{{x}_{3}}+{{m}_{4}}{{x}_{4}}}{{{m}_{1}}+{{m}_{2}}+{{m}_{3}}+{{m}_{4}}}\]                 \[=\frac{1\times 0.5+1\times 0.5+1(-0.5)+1\times (-0.5)}{1+1+1+1}=0\]\[{{y}_{cm}}=\frac{{{m}_{1}}{{y}_{1}}+{{m}_{2}}{{y}_{2}}+{{m}_{3}}{{y}_{3}}+{{m}_{4}}{{y}_{4}}}{{{m}_{1}}+{{m}_{2}}+{{m}_{3}}+{{m}_{4}}}\]                 \[=\frac{1\times 0.5+1\times (-0.5)+1\times (-0.5)+1\times (0.5)}{1+1+1+1}\]                 Co-ordinates of centre of mass will be (0,0).