If three point masses `m_(1),m_(2) and m_(3)` are situated at the vertices of an equilateral triangle of side a, then what will be the co-ordinates of the centre of mass of this system ?

Explanation

The following figure shows the location of the point masses in the co-ordinate system, where mass `m_(1)` is chosen to be located at the origin.
Coordinates of `m_(1):x_(1)=0,y_(1)=0`
Coordinates of `m_(2):x_(2)=s,y_(2)=0`
Coordinates of `m_(3):x_(3)=s//2,y_(3)=(sqrt3)/(2)s`
`:.X_(cm)=(m_(1)x_(1)+m_(2)x_(2)+m_(3)x_(3))/(m_(1)+m_(2)+m_(3))`
`=(m_(1)xx0+m_(2)xxs+m_(3)xx(s)/(2))/(m_(1)+m_(2)+m_(3))=(m_(2)s+m_(3)((s)/(2)))/(m_(1)+m_(2)+m_(3))`
`=(2m_(2)s+m_(3)8)/(2(m_(1)+m_(2)+m_(3)))`
`Y_(cm)=(m_(1)y_(1)+m_(2)y_(2)+m_(3)y_(3))/(m_(1)+m_(2)+m_(3))`
`=(sqrt3m_(3)s)/(2(m_(1)+m_(2)+m_(3)))`