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

Options

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

Answer

(a)

Explanation

`X_(CM)=2.Y_(CM)=2,Z_(CM)=2` (Given)
`m_(1)=1,m_(2)=2,m_(3)=3`
`X_(CM)=(m_(1)x_(1)+m_(2)x_(2)+m_(3)x_(3))/(m_(1)+m_(2)+m_(3))`
`implies 2=(x_(1)+2x_(2)+3x_(3))/(6)`
`implies x_(1)+2x_(2)+3x_(3)=12`
`Y_(CM)=(m_(1)y_(1)+m_(2)y_(2)+m_(3)y_(3))/(m_(1)+m_(2)+m_(3))`
`implies y_(1)+2y_(2)+3y_(3)=12`
Similarly,
`z_(1)+2z_(2)+3z_(3)=12`
If `m_(4)=4,(X_(CM)^(1),Y_(CM)^(1),Z_(CM)^(1))`
`=(0,0,0)`
`implies (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`
` implies (x_(1)+2x_(2)+3x_(3)+4x_(4))/(1+2+3+4)=0` (because From equation (1))
`implies 12+4x_(4)=0`
`x_(4)=-12//4=-3`
Similarly, ` y_(4)= - 3,z_(4)=-3`