The centre of mass of three particle 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 such that the new centre of mass is at (0,0,0) is

Options

A. `(-3,-3,-3)`
B. `(-3,3,-3)`
C. `(2,3,3-3)`
D. `(2,-3,3)`

Answer

A

Explanation

`m_(1)=kgm_(2)=2kg,m_(3)=3kg`
Position of centre of mass (2,2,2),
`m_(4)=4kg`
Now position of centre of mass (0,0,0)
For initial position
`X_(CM)=(m_(1)x_(1)+m_(2)x_(2)+m_(3)x_(3))/(m_(1)+m_(2)+m_(3))`
`2=(m_(1)x_(1)+m_(2)x_(2)+m_(3)x_(3))/(1+2+3)`
`m_(1)x_(1)+m_(2)x_(2)+m_(3)x_(3)=12`
Similarly `m_(1)y_(1)+m_(2)z_(2)+m_(3)z_(3)=12`
For new position
`X_(CM)^(.)=(m_(1)x_(1)+m_(2)x_(2)+m_(3)x_(3)+m_(4)x_(4))/(m_(1)+m_(2)+m_(3)+m_(4))`
`0=(12+4xx x_(4))/(1+2+3+4)`
`4x_(4)=-12`
`x_(4)=-3`
Similarly `y_(4)=-3`
`z_(4)=-3`
`:.` Position of fourth mass
`(-3,-3,-3)`