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In physics, the mass distribution has a point where the weighted relative position of the mass distribution is zero. This point is known as the centre of mass of the object. When force is applied to a body, it can produce angular or linear acceleration depending on where it is applied. The centre of mass of a body is the point on which force can be applied to have linear acceleration. The centre of mass is calculated through a centre of mass formula.
The weighted position vectors of a distribution of mass sum to zero about the unique point at the centre of the distribution. This point is known as the centre of mass. Different systems have a centre of mass formula. These are as follows:
A system of particles: When there is a system of particles Pi, i = 1, …, n , and each particle has a mass equal to mi and their location in space is given by the coordinates ri, i = 1, …, n , then the coordinates of the centre of mass denoted by R fulfil the following conditions:
Here
It is the total sum of the mass of all the particles in the system.
A system of continuous volume: If a solid Q has a uniform distribution of mass with a density ρ(r), then the integration of the weighted position coordinates with respect to the centre of mass R results in zero:
When this equation is solved for R the following formula is arrived at:
Here M is the total mass of the volume of the solid.
The point of note for solids of uniform density is that the centre of mass is the centroid of the volume in systems of uniform density.
Barycentric coordinates: A system that has two particles P1 and P2 whose masses are m1 and m2 will have the following centre of mass formula for the coordinates of the centre of mass R:
A system with periodic boundary conditions: In a system with periodic boundary conditions, the centre of mass formula has a generalised method. The coordinates are assumed to be on a circle instead of a line. Every particle’s x coordinate is taken, and it is mapped on to an angle in the following way:
xmax = size of the system in the direction of x
and
From this formula, two new points 
The averages of these points are calculated:
Here M is the total mass of all the particles in the system
These values are used to calculate the x coordinate of the centre of mass with the following centre of mass formula
Location of the centre of mass in an object
The gravity field near the earth’s surface is parallel. This makes the centre of gravity the same as the centre of mass. So to locate the centre of mass experimentally, the effect of the gravity force is studied on an object. A body with uniform density and an axis of symmetry will have its centre of mass on the axis of symmetry. For example, if there is a circular cylinder made up of material with uniform density, the centre of mass of the cylinder will lie on the cylinder’s axis.
Conclusion
The centre of mass has various applications in diverse fields in physics. That is why the centre of the mass formula is so important in these areas. The location of the centre of mass is of utmost importance when designing cars and automobiles. Especially in race cars, engineers keep the centre of mass as low as possible. This gives the car more excellent traction and keeps it stable even when it is turned sharply. In aircrafts, the centre of mass plays an important role. The location of the centre of mass in aircraft profoundly impacts the craft’s stability when it is in the air. So it is essential to understand how the formulae for the different systems are arrived at and to have informative JEE notes on the centre of the mass formula.
