The Centre of Mass of a Two-Particle System

The study of physics relies on the observation of various systems. In order to understand the effect of multiple forces and how they interact with each other, the knowledge of the centre of mass of a system is also imperative.

The centre of mass of a system provides an easy correlation to how the body would react when subjected to a particular force. For example, if it was required to put a certain amount of force on a body to move it, what would be the point at which the least amount of force would produce the desired result? This point would be the centre of mass of the body. With the knowledge of the centre of mass of a system, various mechanical problems can be solved easily.

What is the Centre of Mass?

The centre of mass of a rigid body is a very unique point in physics.

A rigid body, by definition, is any body that does not undergo deformation when force is applied to it.

The centre of mass is a relative position that is defined with respect to an object or a system formed by various objects. Every system consists of parts. These parts not only allow the system to operate but also dictate how the different forces will act on the system. 

However, it is difficult to calculate the aggregate effect of how these parts would react in a system in a mechanical problem. What if all the mass in the system could be concentrated to a point, and the reaction of this point to a force would be similar to the reaction of the entire system? This is possible when you know the centre of mass of the body.

Most forces that act on a system are uniform in nature. The effect of a uniform force on the centre of mass of a system can be generalised to the entire system. The centre of mass gives the average position of any part that is in the system when all the parts are weighted according to their masses.

Centre of Mass of a Two particle System

The centre of mass of a system can also be defined as the exact point where Newton’s law of motion applies perfectly. Therefore being able to find out where it lies is very important. One of the most common systems observed in real life is a two-particle system. Let us try to find the centre of mass of a two-particle system.

Centre of Mass in One Dimension

Consider two masses, m1 and m2, placed on the x-axis. We need to find the centre of mass, c, of these two masses. Let us say the two masses m1 and m2 are located at a distance d1 and d2 from the origin O. In order to find the centre of mass of these two masses, we will have to find the mass-weighted mean of the two masses. We will find the distance of the centre of mass from the origin; let that distance be D.

D = (m1d1 + m2d2) / (m1 + m2)

This equation gives the distance of the centre of mass from the origin. What happens when the mass of both the masses is the same? Hence m1=m2=m

D = (md1 + md2) / (m + m)

⇒ D = m(d1 + d2) / 2m

⇒ D = (d1 + d2) / 2

Hence we can see that the centre of mass lies at the centre of the two masses when the masses are equal to each other. We need to find the centre of mass, c, of these two masses.

Centre of Mass in Two Dimensions

Consider two masses, m1 and m2, placed on the coordinate axis. Let us say the two masses, m1 and m2 are located at coordinates (x1,y1) and (x2,y2) from the origin O. In order to find the centre of mass of these two masses, we will have to find the mass-weighted mean of the two masses. We will find the coordinate of the centre of mass from the origin, let that distance be D(x,y).

D(x,y)=[(m1x1+m2x2)/(m1+m2), (m1y1+m2y2)/(m1+m2)]

When the masses of both objects are the same, we will find that the centre of mass lies at the centre of both coordinates.

Application of Centre of Mass

One of the most prominent uses of the centre of mass is in calculating the toppling angle and the toppling stability of a body. If we have a truck that is loaded with an uneven distribution of weight, then it is possible to know at what elevation the truck will tip and topple over.

The gravitational force on a body acts along the centre of mass of the body. For a given mass distribution, when the elevation is such that the line across the centre of mass of the body crosses the base of any of the wheels on the truck, it is guaranteed that the truck will topple over.

Conclusion

The centre of mass is a relative position that is defined with respect to an object or a system formed by various objects. Every system consists of parts. These parts not just allow the system to operate but also dictate how the different forces will act on the system. 

Understanding the concept is crucial in many manufacturing industries like the locomotive and automobile industries. One of the most prominent uses of the centre of mass is in calculating the toppling angle and the toppling stability of a body.