Let us now look at a baseball bat closely. It has a peculiar shape. Where should the COM lie? By experience, we can say that it should lie on the heavier side. This perception comes from the realization that heavier part has most of the mass. It gives us the clue about COM that it lies on the heavier side of an irregularly shaped body. Now, let us consider the motion of spherical sphere of uniform density. Here, mass is evenly distributed. Where should the COM lie? Obviously, COM coincides with the center of spherical ball.

It emerges from the discussion that COM is a statement of spatial arrangement of mass i.e. distribution of mass within the system. The position of COM is given a mathematical formulation which involves distribution of mass in a volumetric space, assuming as if all mass is concentrated there :

COM for a system of particles |
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This relation is read as "position of COM is the mass weighted average of the positions of particles".

Further, COM is a geometric point in three-dimensional volume and it involves distribution of mass with respect to some reference system. This expression of the position of COM is description of a position in three dimensional volume. Now, substituting with appropriate symbols, we have :

It is important to note here that this formulation is not arbitrary. We shall find subsequently that this mathematical expression of COM, as a matter of fact, leads us to formulation of Newton's second law for a system of particles, while keeping its form intact.

We can express the position of COM in scalar component form as :

We know that there are only two directions involved with each of the axes. As such, we assign positive value for distances in reference direction and negative in opposite direction.

** Special cases **

(a) Particle system along a straight line :

COM of particle system along a straight line is given by the scalar expression in one dimension:

COM for a system of particles along a straight line |
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Evidently, COM of particles lie on the line containing particles.

(b) Two particles system

The expression of COM reduces for two particles as :

As only one coordinate is involved, it is imperative that COM lies on the line joining two particles. There are two additional simplified cases for two particles system as :

(i) Origin of coordinate coincides with position of one of the particles :

The expression of COM is simplified when position of one of the particles is the origin. For two particles system,

where "x" is the linear distance between two particles.

(ii) The masses of the particles are equal :

In this case,

When origin of coordinate coincides with position of one of the particles,

This result is on expected line. Center of mass (COM) of two particles of equal masses is midway between the particles.

#### Example 1

Problem : Two particles of 2 kg and 4 kg are placed at (1,4) and (3,-2) in x and y plane. If the coordinates are in meters, then find the position of COM.

Solution : Using expression in scalar component form :

Two particles system |
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and

Thus, position of COM lies on x - axis at x = 7/3 m.