Discuss the center of mass of two point masses with pictorial representation.

Explanation

With the equations for center of mass, let us find the center of mass of two point masses `m_(1) and m_(2)`, which are at positions, and x, respectively on the X-axis. For this case, we can express the position of center of mass in the following three ways based on the choice of the coordinate system.
(i) When the masses are on positive X-axis: The origin is taken arbitrarily so that the masses `m_(1) and m_(2)` are at positions `x_(1) and x_(2)` on the positive X-axis as shown in figure (a). The center of mass will also be on the positive X-axis at `X_(CM)` as given by the equation.
`x_(CM)=(m_(1)x_(1)+m_(2)x_(2))/(m_(1)+m_(2))`
(ii) When the origin coincides with any one of the masses: The calculation could be minimised if the origin of the coordinate system is made to coincide with any one of the masses as shown in figure (b). When the origin coincides with the point mass `m_(1)` its position `x_(1)` is zero, (i.e.` x _(1)= 0)`. Then,
`x_(CM)=(m_(1)(0)+m_(2)x_(2))/(m_(1)+m_(2))`
The equation further simplifies as
`x_(CM)=(m_(2)x_(2))/(m_(1)+m_(2))`
(iii) When the origin coincides with the center of mass it self:
If the origin of the coordinate system is made to coincide with the center of mass, then, `X_(CM)=0`and the mass `m_(1)` is found to be on the negative X-axis as shown in figure (c). Hence, its position `X_(1)` is negative, (ie, `-x_(1))`
`0=(m_(1)(-x_(1))+m_(2)x_(2))/(m_(1)+m_(2))`
`0=m_(2)(-x_(1))+m_(2)x_(2)`
`m_(1)x_(1)=m_(2)x_(2)`
The equation given above is known as principle of moments.

Center of mass of two point masses determined by shifting the origin