The centre of mass of three particles of masses 1 ~kg , 2 ~kg and 3 ~kg is at (2,2,2) . The position of the fourth mass of 4 ~kg to be placed in the system as that the new centre of mass is at (0,0,0) is.

Step-by-step solution

m₁=1 ~kg , m₂=2 ~kg , m₃=3 ~kg Position of centre of mass (2,2,2)m₄=4 ~kg New position of centre of mass (0,0,0) . For initial position, aligned & X_ CM = m₁ x₁+m₂ x₂+m₃ x₃ m₁+m₂+m₃ & 2= m₁ x₁+m₂ x₂+m₃ x₃ 1+2+3 & m₁ x₁+m₂ x₂+m₃ x₃=12 aligned Similarly, aligned & m₁ y₁+m₂ y₂+m₃ y₃=12 & m₁ z₁+m₂ z₂+m₃ z₃=12 aligned and m₁ z₁+m₂ z₂+m₃ z₃=12 For new position, gathered X_ CM ^ = m₁ x₁+m₂ x₂+m₃ x₃+m₄ x₄ m₁+m₂+m₃+m₄ 0= 12+4+x₄ 1+2+3+4 4 x₄=-12 x₄=-3 gathered Similarly, y₄=-3z₄=-3 Position of fourth mass (-3,-3,-3) .