Centre of Mass

Centre of mass of a body or system of a particle is defined as, a point at which whole of the mass of the body or all the masses of a system of particle appeared to be concentrated.
When we are studying the dynamics of the motion of the system of a particle as a whole, then we need not bother about the dynamics of individual particles of the system. But only focus on the dynamic of a unique point corresponding to that system.
Motion of this unique point is identical to the motion of a single particle whose mass is equal to the sum of all individual particles of the system and the resultant of all the forces exerted on all the particles of the system by surrounding bodies or action of a field of force is exerted directly to that particle. This point is called the center of mass of the system of particles. The concept of center of mass (COM) is useful in analyzing the complicated motion of the system of objects, particularly When two and more objects collide or an object explodes into fragments.
==System of Particles==
The term system of particles means a well-defined collection of a large number of particles which may or may not interact with each other or connected to each other. They may be an actual particle of rigid bodies in translational motion. The particle which interacts with each other they apply force on each other.
The force of interaction \overrightarrow{F_{\widehat{i} \widehat{j}}} and \overrightarrow{F_{\widehat{j} \widehat{i}}} between a pair of i and j particle. These forces of mutual interaction between the particle of the system are called the internal force of the system.
These internal forces always exist as in pair of equal magnitude and opposite direction. Other than internal forces external forces may also act on all or some of the particles. Here the term external force means a force which is acting on any one of particle which is included in the system by some other body outside the system.
==Formula==
We can extend the formula to a system of particles.The equation can be applied individually to each axis,
 X_{com} = \sum_{i=0}^{n}_M m_i x_i 
 Y_{com} = \sum_{i=0}^{n}_M m_i y_i 
 Z_{com} = \sum_{i=0}^{n}_M m_i z_i 
The above formula can be used if we have point objects. But we have to take a different approach if we have to find the center of mass of an extended object like a rod. We have to consider a differential mass and its position and then integrate it over the entire length.
 X_{com} = \int _M x dm 
 Y_{com} = \int _M y dm 
 Z_{com} = \int _M z dm 
Category:Mechanics