If the centre of mass of three particles of masses of 1kg, 2 kg, 3 kg is at(2, 2, 2), then where should a fourth particle of mass 4 kg be placed so that the combined centre of mass may be at (0, 0, 0).

If the centre of mass of three particles of masses of 1kg, 2 kg, 3 kg

1.	If the centre of mass of three particles of masses of 1kg, 2 kg, 3 kg is at(2, 2, 2), then where should a fourth particle of mass 4 kg be placed so that the combined centre of mass may be at (0, 0, 0).
Answer» Solution :`(x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2))` and `(x_(3),y_(3),z_(3))` be the positions of masses 1kg, 2KG, 3KG and let the co-ordinates of centre of MASS of the three particle system is `(x_(cm),y_(cm),z_(cm))` respectively `x_(cm)=(m_(1)x_(1)+m_(2)x_(2)+m_(3)x_(3))/(m_(1)+m_(2)+m_(3))` `implies2=(1xxx_(1)+2xxx_(2)+3xxx_(3))/(1+2+3)`, (or) `x_(1)+2x_(2)+3x_(3)=12"".......(1)` Suppose the fourth particle of mass 4kg is placed at `(x_(4),y_(4),z_(4))` so that centre of mass of new system SHIFTS to (0,0,0). For X-co-ordinate of new centre of mass we have `0=(1xxx_(1)+2xxx_(2)+3xxx_(3)+4xxx_(4))/(1+2+3+4)` `impliesx_(1)+2x_(2)+3x_(3)+4x_(4)=0...................(2)` From equation (1) and (2) `12+4x_(4)=0impliesx_(4)=-3` Similarly, `y_(4)=-3andz_(4)=-3` Therefore 4 kg should be placed at (-3,-3,-3).