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(C) Let the positions of the four bodies be $a(-x, -y)$,$b(x, -y)$,$c(x, y)$,and $d(-x, y)$.The centre of mass $(X_{cm}, Y_{cm})$ of two bodies of equ

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$a$ and $c$

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(C) Let the positions of the four bodies be $a(-x, -y)$,$b(x, -y)$,$c(x, y)$,and $d(-x, y)$.The centre of mass $(X_{cm}, Y_{cm})$ of two bodies of equal mass $m$ at positions $(x_1, y_1)$ and $(x_2, y_2)$ is given by $X_{cm} = \frac{x_1 + x_2}{2}$ and $Y_{cm} = \frac{y_1 + y_2}{2}$.For the centre of mass to remain at the origin $(0, 0)$,we must have $x_1 + x_2 = 0$ and $y_1 + y_2 = 0$.$1$. For pair $a$ and $c$: $a$ is at $(-x, -y)$ and $c$ is at $(x, y)$. The sum of coordinates is $(-x+x, -y+y) = (0, 0)$. Thus,the centre of mass is at the origin.$2$. For pair $b$ and $d$: $b$ is at $(x, -y)$ and $d$ is at $(-x, y)$. The sum of coordinates is $(x-x, -y+y) = (0, 0)$. Thus,the centre of mass is at the origin.Looking at the options,the combination $a$ and $c$ ensures the centre of mass remains at the origin.

Four bodies of equal mass start moving with the same speed as shown in the figure. In which of the following combinations will the centre of mass remain at the origin?

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