# Centre of mass

The centre of mass of a system of particles can be regarded as the mass weighted average location of the constituent particles. When we deal with rigid bodies of continuous distribution of matter we need to replace the summation symbol with an integral symbol.

Consider a body be made up of bid number of particles which mass is equal to the total mass of all the particles. When such a body undergoes a translational motion the move is produced in each and every particle of the body with respect to their original position

### Centre of Mass Formula

$$ x_{com} = \frac{\sum_{n}^{i=0}m_ix_i}{M} $$ $$ y_{com} = \frac{\sum_{n}^{i=0}m_iy_i}{M} $$ $$ z_{com} = \frac{\sum_{n}^{i=0}m_iz_i}{M} $$To find the center of mass of an extended object like a rod, then consider a differential mass and its position and then integrate it over the entire length.

$$ x_{com} = \frac{\int xdm}{M} $$ $$ y_{com} = \frac{\int ydm}{M} $$ $$ z_{com} = \frac{\int zdm}{M} $$Where,

x_{com}, y_{com} and z_{com} = Center of mass along x, y, and z-axis,

M = The total mass of system,

n = Number of objects,

m_{i} = Mass of the i^{th} object and

x_{i} = Distance from the x-axis of i^{th} object

y_{i} = Distance from the y-axis of i^{th} object

z_{i} = Distance from the z-axis of i^{th} object