Moment of Mass and Center of Mass

1. The Moment of Mass for a Point Mass Object of Mass \(m\) having Location given by Position Vector \(\vec{R_p}\) with respect to any Reference Location given by Position Vector \(\vec{R_{ref}}\) is calculated as \(m(\vec{R_p} - \vec{R_{ref}})\). Hence, if the Reference Location \(\vec{R_{ref}}\) is Origin then the Moment of Mass is calculated as \(m\vec{R_p}\).
2. Center of Mass for a Point Mass Object is the Location of Mass.
   
   Center of Mass for a Collection of Point Mass Objects (also known as System of Particles) is the Location around which the Sum of Masses of the Collection of Point Mass Objects is considered to be Uniformly Distributed.
   
   The Center of Mass of any Rigid Body Object (which is a Continuous Distribution of Point Mass Objects over a Path, Surface Area or Volume) is the Location either within or outside of that Object around which the Entire Mass of the Object is considered to be Uniformly Distributed.
   
   For Any Rigid Body Object the Center of Mass is the Location around which the Object is Balanced against any kind of Rotational Motion i.e. Any Force applied on the Object along the Line that Passes through the Center of Mass of the Object causes only a Linear Acceleration of the Object along that line without causing any Angular Acceleration
3. The Moment of Mass for any Collection of Point Masses having Sum of Masses as \(M\) having Location of Center of Mass given by Position Vector \(\vec{R_c}\) with respect to any Reference Location given by Position Vector \(\vec{R_{ref}}\) is calculated as \(M(\vec{R_c} - \vec{R_{ref}})\). Hence, if the Reference Location \(\vec{R_{ref}}\) is Origin then the Moment of Mass is calculated as \(M\vec{R_c}\).
4. The Moment of Mass for any Collection of Point Masses is the Sum of Moment of Mass of the Individual Point Masses of the Collection. That is
   
   \(M(\vec{R_c}-\vec{R_{ref}}) = m_1(\vec{R_1}-\vec{R_{ref}}) + m_2(\vec{R_2}-\vec{R_{ref}}) + \cdots + m_n(\vec{R_n}-\vec{R_{ref}}) \)   ...(1)
   
   where
   
   \(m_1, m_2, \cdots, m_n=\) The Masses of Individual Point Mass Objects of the Collection
   
   \(M=\) The Sum of Masses of All the Point Mass Objects (i.e. \(m_1 + m_2 + \cdots + m_n)\)
   
   \(\vec{R_1}, \vec{R_2}, ... , \vec{R_n}=\) Position Vectors of Point Masses \(m_1, m_2, \cdots, m_n\) respectively
   
   \(\vec{R_c}=\) Position Vector of the Location of Center of Mass for the Collection of Point Mass Objects
   
   \(\vec{R_{ref}}=\) Position Vector of the Location with respect to which the Moments of Masses are calculated
   
   when \(\vec{R_{ref}}=\vec{R_c}\) then equation (1) becomes
   
   \(M(\vec{R_c}-\vec{R_c}) = m_1(\vec{R_1}-\vec{R_c}) + m_2(\vec{R_2}-\vec{R_c}) + \cdots + m_n(\vec{R_n}-\vec{R_c}) = 0\)   ...(2)
   
   Hence, Center of Mass for a Collection of Point Masses can be mathematically defined as the Reference Location for which the Sum of Moments of Masses of the Individual Point Masses of the Collection is 0.
5. The Center of Mass for any Collection of Point Masses can be calculated using equation (1) given above as follows
   
   \(\vec{R_c} ={\Large \frac{m_1(\vec{R_1}-\vec{R_{ref}}) + m_2(\vec{R_2}-\vec{R_{ref}}) + \cdots + m_n(\vec{R_n}-\vec{R_{ref}})}{M}} + \vec{R_{ref}} \)   ...(3)
   
   Hence, if the Reference Location \(\vec{R_{ref}}\) is Origin then the Center of Mass is calculated as
   
   \(\vec{R_c} ={\Large \frac{m_1\vec{R_1} + m_2\vec{R_2} + \cdots + m_n\vec{R_n}}{M}} \)   ...(4)
   
   
6. The Moment of Mass for any Rigid Body Object is the calculated as follows
   
   Moment of Mass for Rigid Body \(= {\Large \int} \vec{R} \hspace{2mm} dm\)   ...(5)
   
   where
   
   \(dm=\) The Infinitesimally Small Mass of Point Mass Object forming the Rigid Body
   
   \(\vec{R}=\) Vector Function for the Location of Point Mass Object forming the Rigid Body
7. The Center of Mass for any Rigid Body Object can be mathematically defined as the Reference Location for which the Moment of Mass for the Rigid Body is 0. It is the calculated as follows
   
   Center of Mass for Rigid Body \(\vec{R_c}= {\Large \frac {{\Large \int} \vec{R} \hspace{2mm} dm} {{\Large \int} dm}}\)   ...(6)
   
   
8. If the Rigid Body Object is in form of a Continuous Distribution of Point Mass Objects over a Path then the Integrals given in equations (5) and (6) above are evaluated as Line Integrals.
   
   If the Rigid Body Object is in form of a Continuous Distribution of Point Mass Objects over a Surface Area then the Integrals given in equations (5) and (6) above are evaluated as Surface Integrals.
   
   If the Rigid Body Object is in form of a Continuous Distribution of Point Mass Objects over a Volume then the Integrals given in equations (5) and (6) above are evaluated as Volume Integrals.