Consider two particles of unequal mass connected by a massless rod (figure). If a force is applied between the lighter particle and the centre of mass, the system rotates clockwise figure (a). When a force is applied between heavier particle and centre of mass, the system rotates anticlockwise figure (b). When a force is applied at the centre of mass, the system moves in the direction of force without rotation figure (c).

The overall motion of a system can be described in terms of a point called centre of mass. The centre of mass of a system (M) represents translational motion of the system. It moves as if all the mass of the system is concentrated at this point.

The system behaves as if the resultant external force is applied to a single particle of mass *M* located at the centre of mass.

Consider first a system of two particles *m*1 and *m*2 having abscissa *x*1 and *x*2 respectively from some origin O. We define a point *C*, the centre of mass of the system, at a position *xCM* from origin O.

We define *xCM* as

… (i)

*xCM* can be treated as

mass – weighted means of *x*1 and *x*2 For particles in 2-D plane,

We define *yCM* as

**… (ii)**

**We can rewrite equation (i) as**

Now and are abscissa of particles with respect to C.M. of system.

So, if we choose our origin at C.M. of system

Similarly,

For a system of many particle in three dimensions the position of centre of mass w.r.t. any fixed axis is determined form

Similarly for *y*- and *z*-coordinates of the centre of mass, we may use the equations

and

In vector notation, the position vector of centre of mass is

where is the position vector of the *i*th particle

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