Seven particles, each of mass `m` are placed at the seven corners of a cube of side `'a'`, but one corner is vacant, as shown in figure. If the `x-`coordinates of the centre of mass of the system is `(na)/(m)`. Where `n` and `m` are least integer values. Then find vlaue of `m - n`.

Answer

4

Explanation

Place `+m` and `-m` particle at the vacant corner.

Assume the `+m` particle and the reast seven particles as system `(1)`
The total mass of system `(1)` is `m_(1) = 8m` and the position
vector of its com is `r_(1) = (a)/(2)hat(i)+(a)/(2)hat(j)+(a)/(2)hat(k)`
`rArr vecr_(cm) = (m_(1)vecr_(1)+m_(2)vecr_(2))/(m_(1) + m_(2))`
`=((8m)((a)/(2)hat(i)+(a)/(2)hat(j)+(a)/(2)ohat(k))+(-m)(ahat(i)+hat(j)+ahat(k)))/(8m+(-m))`
`rArrvecr_(cm) = (3a)/(7)hat(i)+(3a)/(7)hat(j)+(3a)/(7)hat(k)`