With the equations for center of mass, let us find the center of mass of two point masses m1 and m2, which are at positions x1 and x2 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.
(1) When the masses are on positive X-axis:
The origin is taken arbitrarily so that the masses m1 and m2 are at positions x1 and x2 on the positive Xaxis as shown in figure (a). The center of mass will also be on the positive X- axis at xCM as given by the equation,
\(x_{CM} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}\)
(2) When the origin coincides with any one of the masses:
The calculation could be minimized 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 m1 , its position x1 is zero, (i.e. x1 = 0). Then,
\(x_{CM} = \frac{m_1(0) + m_2x_2}{m_1 + m_2}\)
The equation further simplifies as,
\(x_{CM} = \frac{m_2x_2}{m_1 + m_2}\)
(3) When the origin coincides with the center of mass itself:
If the origin of the coordinate system is made to coincide with the center of mass, then, xCM = O and the mass rn1 is found to be on the negative Xaxis as shown in figure (c). Hence, its position x1 is negative, (i.e. – x1 ).
The equation given above is known as principle of moments.
(c) When the origin coincides with the center of mass itself center of mass of two point masses determined by shifting the origin