Derive an expression for the center of mass of two point masses.

Explanation

(i) Consider the point masses `m_(1) and m_(2)` which are positioned as `x_(1) and x_(2)` along X-axis. The center of mass can be found in this system in three ways as follows
(a) When the masses are on positive X-axis :
(i) The origin is taken arbitrarily as shown in figure.

(ii) The center of mass along X-axis is,
`x_(CM) = (m_(1)x_(1)+m_(2)x_(2))/(m_(1)+m_(2))`
(b) When the origin coincides with any one of the masses :
(i) If the orging coincide with mass `m_(1)` as shown in figure, its position `x_(1) = 0`

(ii) The center of mass along X-axis is,
`x_(CM) = (m_(1)(0) + m_(2) x_(2))/(m_(1)+m_(2)) = (m_(2)x_(2))/(m_(1)+m_(2))`
(c) When the origin coincides with the center of mass itself :
(i) If the origin coincide with center of mass as shown in figure, `X_(CM) = 0`. Hence, the position `x_(1)` is negative.

(ii) The center of mass along X-axis is,
`0 = (m_(1)(-x_(i))+m_(2)x_(2))/(m_(1)+m_(2))`
`m_(1)x_(1) = m_(2)x_(2)`
(iii) The above equation is known as principle of moments.