# Center of Mass Calculation

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### Center of Mass Calculation

By definition, the center of mass of a system of masses is the location at which a mass equal to the sum of the masses of the system could be placed and have the same torque or moment with respect to some other fixed point. For example, with respect to the origin of coordinates, in a 3-dimensional reference frame, the $x$-coordinate of the center of mass calculation of a system of $N$ masses, $m_{i}$, goes as follows:

\[

\left(\sum_{i=0}^{N}{m_{i}}\right) x_{c} \ = \

\sum_{i=0}^{N}{m_{i} x_{i}}

\]

The center of mass can also be thought of as the average location of the masses weighted by their masses. For a rigid body with a uniform density, the center of mass coincides with the center of gravity.

**Two-body System**

In a simple one-dimensional coordinate system of two masses, $m_{1}$ and $m_{2}$, seprated by a distance $d$, moving with a speed $v$ relative to each other, suppose that the center of mass is located at a distance $x_{1}$ from $m_{1}$ and $x_{2}$ from $m_{2}$ so that

\[

x_{1} + x_{2} = d

\]

and by defintion

\[

x_{1}m_{1} + x_{2}m_{2} = (m_{1} + m_{2}) \times 0 = 0

\]

so

\[

x_{1} + \left(\frac{-m_{1}}{m_{2}}\right) x_{1} = d

\]

or

\[

x_{1} = \left(\frac{m_{2}}{m_{1}+m_{2}}\right)d \\

\\

x_{2} = \left(\frac{m_{1}}{m_{1}+m_{2}}\right)d

\]

Also, in the center-of-mass frame,

\[

v_{1} = -\left(\frac{m_{2}}{m_{1}+m_{2}}\right) v \\

\\

v_{2} = \left(\frac{m_{1}}{m_{1}+m_{2}}\right) v

\]

so that $v_{2} -v_{1}=v$ as required. Also, in the center-of-mass frame, the total momentum is divided in such as way that each mass gets an equal and opposite momentum.

\[

m_{1}v_{1} + m_{2}v_{2} = \left(\frac{m_{2}m_{1}}{m_{1}+m_{2}} \ – \

\frac{m_{1}m_{2}}{m_{1}+m_{2}}\right)v

\]

Note also that in the center-of-mass frame, by definition the center of mass point is stationary so does not experience any net force or torque. See also *reduced mass*

**Extended Continuous Mass Distribution**

For an extended mass distribution with a uniform density (i.e. constant density throughout), the coordinates of the center of mass in a cartesian system can be found from considering the mass of infiniesimal volumes and balancing moments about each axis, integrating over the entire structure. However, this can involve some tricky integrals and functions.