Centre of Mass of a Two-particle System

Introduction

The centre of mass is defined as the average of spatial arrangement of masses in a system or the distribution of masses in a system.

It can also be stated as a point where the mass of a body is concentrated. At this point, the applied force results in a linear acceleration without the involvement of angular acceleration. The motion of a system can be described with respect to the motion of the centre of mass. Let us look at some important terms related to this topic:

• Torque – A rotational effect of force. Its SI Unit is a Newton-meter.
• Point particle – A particle that lacks spatial extension.
• Rigid-body – A solid whose shape and size are fixed or unaffected by the application of any kind of force.
• Couple – Two equal and opposite forces having different lines of action acting on a body.

Centre of Gravity

The centre of gravity (COG) is defined as the point where the whole body weight is concentrated. It describes the behaviour of a moving body under the effect of gravity. It is used to make decisions related to the designs of buildings and bridges. The centre of gravity is somehow similar to the centre of mass. For an asymmetrically shaped object (made of homogeneous components), the centre of gravity matches the geometric centre of the body. Meanwhile, asymmetrical objects (composed of objects with different masses) have their centre of gravity located at some distance. For hollow or irregularly shaped bodies, the COG is located at a point that is present external to a physical material.

Calculation of the Centre of Gravity:

The centre of gravity can be determined in the following manner:

1. Determination of the COG for symmetrical bodies: When we balance objects using a string or weighing balance, the point at which the object balances is the centre of gravity.
2. Determination of COG for asymmetrical bodies: When the mass is not distributed uniformly, we integrate the continuous function with the weight using the symbol S dw. So the COG can be articulated as:

cg × W = SxdW

where x = distance,

where dw = weight,

and W = total weight of the object.

As we know

W = mg and m = ρV

So after combining both the equations we get:

W = gρV

On solving,

dW = (g)(ρ)(dV)

dW = (g)(ρ(x, y, z))( dxdydz)

System of Particles 

The objects that cannot be seen with the naked eye, i.e., when 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 have two parts:

• The first part is the centre of mass, where the whole system is treated as one particle, as the total mass of the system is supposed to be concentrated on that single point.
• 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).

Further, 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 particle can be given by:

Fi j and Fj i 

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. 

Rigid Body

A non-deformable body is referred to as a rigid body. It can also be defined as a body or an extended object where the location and separations of the particles remain constant in all conditions/circumstances. A simple rigid object having uniform density will have its centre of mass present at the centroid.

Definition of the Centre of Mass

Considering a system of particles, the centre of mass can be defined as a particular point where the entire mass of the system is supposed to be concentrated for its translational motion. In such a case, if external forces are applied at that point, the motion of the body/ bodies remains unaffected. It can also be referred to as a balancing point. The centre of mass of the two-particle system is found on the line that joins the two particles.

Assuming that two particles of masses m1 and m2 have their position vectors as r1 and r2, respectively. In this case, the position of the vector for the centre of mass will be given by:

rcm = (m1r1 + m2r2 )/ m1 + m2

The Motion of the Centre of Mass

As we know, the centre of mass of particles in a system moves like the entire mass of the system is concentrated on that point, and all the external forces are applied to it.

• The velocity of the centre of mass of two particles having masses m1 and m2 and velocity v1 and v2, respectively, is given by:

            vcm =( m1v1 + m2v2 )/ m1 + m2

• And the acceleration of the centre of mass will be given by:

           acm = (m1a1 + m2a2 )/ m1 + m2

• If no external force is applied, then the momentum of the centre of mass will be conserved, i.e., velocity will be constant, and acceleration will be zero.

So it will be written as mvcm = constant.

Conclusion

Every physical system has a point by which the motion of the whole system is related. When the system is subjected to an external force, then this point moves as if the mass of the entire system is concentrated on it and also as if the force is applied on it. This point is called the centre of mass of the particular system. The centre of mass of the human body is situated 10cm below the navel, near the hip bones. In the case of motion of a rigid body, the different parts of the body have different motions. In such cases, the motion is separated/divided into translational parts and the rotational part. The centre of mass can be measured by the linear and rotational equations. When there is no application of external force, the momentum of the centre of mass remains conserved.