Centre of the Mass of a Two-Particle System

The centre of mass is the unique spot where the entire mass of an object or system of particles is concentrated. It is the point where the force applied results in linear acceleration without an angular acceleration. 

The centre of mass of objects varies. For a single rigid body, it is fixed in relation to the body. In a body with uniform density, it will be located in the centroid. In hollow or open-shaped objects, the centre of mass may be located outside the body. A force applied to or against an object’s mass cannot rotate it. The centre of mass equation applied to the object’s left or right side will cause it to rotate. The gravitational pull is attributed to an object’s centre of mass. Due to its low centre of gravity, a mass rises when flipped. As the centre of mass is always below the point of hanging, we can hang a flat item to find its centre of mass.

System of Particles 

The objects that cannot be seen with the naked eye as they are extremely small are called microscopic objects, and the objects that can be observed easily are termed macroscopic objects. A macroscopic system consists of a large number of atoms and molecules that behave differently. Any system composed of multiple particles has two parts:

• The first part is the centre of mass, where the whole system is treated as one particle.
• The second part is a system that describes the internal motion of the system that is easily observed by an observer present at the centre of mass (as well as moving with it).

In other words, we can say that the term ‘system of particles’ is often used to describe the collection of particles that may or may not interact with each other, moving in a translational motion. The particles that interact with each other apply force on each other. So, the force of interaction between the ith and jth particles can be given by:

Fij and Fji 

This mutual force of interaction between the particles of a system is called the internal force of the system. These internal forces always occur in pairs having equal magnitude and opposite directions. 

Centre of Mass Formula

Even though the centre of mass and centre of gravity often coincide, they are different. When the entire system is subjected to uniform gravitational fields, both the centre of gravity and centre mass are the same. The shape of the object affects the centre of mass formula as well. 

The centre of mass formula for pointer objects:

zcom =i=1kmizii=1kmi

mi is mass of ith object 

zi is distance from the z-axis of ith object

Centre of Mass of a Two-Particle System

Let’s assume a system of two particles with masses m1 and m2. Both are present on points A and B, respectively. Assume r1 and r2 to be the position vectors with respect to the origin O.

Thus, the centre of mass C of that particular system with the position vector rcm is as follows: 

rcm = (m1 r1+m2 r2)/(m1+m2)

Centre of Gravity

The centre of gravity is a point in a system where the weight distribution is equal in all directions depending on the gravitational field.

It’s the point at which the resulting torque from gravity forces vanishes into thin air. The centre of gravity and centre of mass is always the same in a uniform gravitational field. When referring to the same point or location, the words ‘centre of gravity’ and ‘centre of mass’ are commonly used interchangeably.

Conclusion

An object’s centre of mass is a spot where the entire mass of the object or system of particles can be seen to be concentrated. The centre of mass and centre of gravity are different terms. However, they interconnect. The centre of gravity is a point in a system where the weight distribution is equal in all directions depending on the gravitational field.

The centre of gravity and centre of mass is always the same in a uniform gravitational field. The formula or equation of calculating varies based on a pointer or extended object. By using gravity forces, we can empirically discover a body’s centre of mass if necessary. The formula of the centre of mass for the two-particle system is 

rcm = (m1 r1+m2 r2)/(m1+m2)

Where m1 and m2 are masses of two objects and r1 and r2 are their positions respectively.