Centre of mass of a system of particles in a line

Centre of Mass for a System of Particles

The centre of mass of a system of particles is the point at which the total mass of the system can be considered to be concentrated.
It is a key concept in mechanics, especially when dealing with systems of particles where each particle may have different mass and position.

For a system of particles lying on a straight line, the centre of mass is determined by balancing the moments about a certain point.
The centre of mass, x, can be calculated by the formula: x = Σ(mi * xi) / Σmi, where mi is the mass of particle i, xi is the position of particle i, and Σmi is the sum of all the masses.
The symbols Σmi and Σ(mi * xi) represent the summation of all the masses and the product of each mass by its position, respectively.

Imagine a rod with particles of different masses situated at different points. The point at which the rod would balance, assuming no other forces are acting, is the centre of mass.
If the system of particles can rotate about some axis without accelerating (ignoring any external forces), that axis must pass through the centre of mass.
The distance from the centre of mass to a specific particle is the product of the mass of the particle and its distance from a chosen point, divided by the total mass of the system.

For two point masses m1 and m2 at positions x1 and x2 on the x-axis, the centre of mass X can be calculated as X = (m1x1 + m2x2) / (m1 + m2).
For three point masses m1, m2, and m3 at positions x1, x2, and x3 on the x-axis, the centre of mass X can be calculated as X = (m1x1 + m2x2 + m3x3) / (m1 + m2 + m3).