The centre of mass of a system of three particles of masses 1 g, 2 g and 3 g is taken as the origin of a coordinate system. The position vector of a fourth particle of mass 4 g such that the centre of mass of the four particle system lies at the point (1,2,3) is `alpha (hati +2hatj +3hatk)`, where `alpha` is a constant. The value of `alpha` is

Options

A. `(10)/(3)`
B. `(5)/(2)`
C. `(1)/(2)`
D. `(2)/(5)`

Answer

B

Explanation

Let (x , y , z) coordinates of three particles of masses 1 g , 2 g and 3g be `(x_(1), y_(1) , z_(1)) , (x_(2) , y_(2) , z_(2))` and `(x_(3) , y_(3) ,z_(3))`
or `0 = (1 x_(1) + 2x_(2) + 3 x_(3))/(1 + 2 + 3) or x_(1) + 2x_(2) + 3x_(3) = 0`
Let the fourth particle of mass 4 g be placed at `(x_(4) , y_(4) , z_(4))` so that the centre of mass of the four particles system lies at (1 , 2 , 3)
`therefore 1 = (x_(1) + 2x_(2) + 3x_(4) + 4 x_(4))/(1 +2 + 3 + 4) `
or `x_(1) + 2x_(2) + 3 x_(4) + 4 x_(4) = 10 " " .... (ii)`
Subtract (i) from (ii) , we get
`4x _(4) = 10 or x_(4) = (10)/(4) = (5)/(2) therefore alpha = (5)/(2)`