The centre of mass is a point in a system where the total mass of the system can be assumed to be concentrated.

It is the average position of all the parts of the system, weighted according to their masses.

For a system of particles in one dimension, the centre of mass can be calculated using the formula

where xcm is the position of the centre of mass and mx is the mass of the particle at position x.

For a system of particles in two dimensions, the centre of mass can be calculated using the formulas

where xcm and ycm are the x and y coordinates of the centre of mass, and mx and my are the masses of the particles at positions x and y.

For a system of particles in three dimensions, the centre of mass can be calculated using the formulas

where xcm, ycm, and zcm are the x, y, and z coordinates of the centre of mass, and mx, my, and mz are the masses of the particles at positions x, y, and z.

For a continuous body with a mass distribution, the centre of mass can be found using integration.

The formulas for centre of mass in one, two, and three dimensions still apply, but with the masses replaced by mass densities and the positions replaced by position vectors.

The centre of mass is an important concept in physics and engineering.

It is used in the study of motion, equilibrium, collisions, and stability of systems.

It is also used in fields such as astronomy, robotics, and biomechanics.

Understanding the concept of centre of mass is crucial for analyzing complex systems and optimizing designs.