Find the centre of mass of systems

Centre of Mass Fundamentals

Centre of mass is the point in an object where the sum of all forces act. It is considered an average of all the locations and masses in a system.
Overall, the centre of mass in a system is the weighted average of individual mass positions, with each mass acting as a weight.
For a system of particles, the equation to find the centre of mass is: (∑mixi) / (∑mi) for the x coordinate and (∑miyi) / (∑mi) for the y coordinate.
In these formulas, mi is the mass of an individual particle, and xi and yi are its x and y coordinates respectively.

To find the centre of mass in two dimensions for a system of particles, calculate the x and y coordinates separately using their respective formulas. The result is a coordinate pair.
It should be noted that the centre of mass does not have to be located on the object itself. For example, it is outside the object in the case of a boomerang or a doughnut.
Objects will rotate about the centre of mass, which is why an object’s centre of mass is also its balance point.

The centre of mass can be extended for a three-dimensional system of particles by adding a third coordinate z.
So, the z coordinate of the centre of mass will be (∑mizi) / (∑mi), where z denotes the z-coordinate of the individual particle.
It should be remembered that in three dimensions, the point where the three median lines of a triangle intersect is the centre of mass.

If a body is uniform, its density is uniform throughout. In these cases, the centre of mass is also the geometric centre.
Examples include a uniform ring, where the centre of mass is at the centre of the ring, or a uniform rod, where the centre of mass is at the midpoint of the rod.
For bodies with uniform density but irregular shape, it is useful to divide them into uniform components, find the centres of mass of those components, and then find the overall centre of mass.

The concept of centre of mass plays a crucial role in understanding physics, kinetic motions, dynamics, and even in engineering studies.
It helps simplify problems with complex shapes or systems of particles into a single point, making calculations and analyses easier and more manageable.
Understanding how centre of mass influences an object’s motion is critical to many practical applications, such as designing stable structures and predicting the movement of objects.