Centre of mass
Centre of mass of a system of particles: The centre of mass of a system of particles is a point in the system which moves in the same way as a single particle of mass equal to the total mass of the system would move when subjected to the same external forces as applied to the system.
Let us consider, a system of total mass M consisting of n number of particles each of mass m1, m2, —– mn whose position vectors are represented by 
, 
—– 
respectively. The centre of mass of the system is defined as the point whose position vector 
is given by = 
= 
= 
——— (1)
= M is the total mass of the system. If the centre of mass coincides with origin of the system, then, 
= 0 and 
= 0.
Therefore, the centre of mass is defined as a point in space such that, the vector sum of the moment of all the mass points around it is zero.
Velocity and acceleration of centre of mass:
Let us consider, the motion of a system consisting of n number of particles and the total mass is M. Assuming that the mass of the system remains constant that is no mass enter or leave the system. From equation (1) we get, M = 
.
Differentiating with respect to time we get, M
= 
+
+ —- + 
.
Here = 
represents the velocity of centre of mass. = 
, = 
, —– = 
represent the velocity of individual particles. Therefore, the velocity of centre of mass is given by = 
= 
——- (2).
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