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).(a) (b) (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 m1 and m2 having abscissa x1 and x2 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
xCM can be treated as
mass – weighted means of x1 and x2 For particles in 2-D plane,
We define yCM as
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
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
In vector notation, the position vector of centre of mass is
where is the position vector of the ith particle
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