​​The Motion of the Centre of Mass

According to the motion of the centre of mass, the centre of mass is an essential property of any rigid body system. Moreover, there is usually more than one particle in these systems, and analysing these systems as a whole becomes vital. These bodies should be treated as a single point mass for calculating the mechanics, and such a point is denoted by the centre of the mass. The movements of the mechanical systems often take place in a rotatory or a transitory manner, and here, the centre of mass moves as well, gaining acceleration and velocity at the same time.

Centre of Mass

Various problems can be simplified by assuming that the object’s mass is present at a particular point. If the position is chosen correctly, the equations of force and motion act the same way as they would if applied while the mass spreads out. This particular location is known as the centre of mass.

The position or place is said to be relative to the system of the object(s) in which the calculation of the centre of mass is to be done. The centre of mass of the symmetrical and uniform shapes is located at their centroid. In the case of a ring, the centre of mass is present inside the ring, implying that a body’s centre of mass need not lie in the body only.

Finding the Centre of Mass

The symmetrical and uniform bodies comprise their own centre of masses at their centroid. However, the solution is not so easy for the bodies that are not uniform, and the centre of mass for these bodies can be anywhere. Moreover, the locations of each mass of the body consider a weighted average for working out the centre of mass of a complex object.

The Motion of the Centre of Mass

Consider a system of numerous particles, where every particle of that system moves at a different velocity. Is it possible to allot a velocity to the system as a whole? Let us take a system of the particles m4, m5, m6…, etc. Here, the starting or very first position vectors of these particles are; r4, r5, r6…rn. Now, these particles start making moves in the directions of the position vectors which belong to them. The aim is to determine the velocity and direction of the system’s centre of mass.

According to the definition of the centre of mass,

Rcm = (m4r4 + m5r5 + …+mnrn)/(m4 + m5 +……+ mn)

As the particles are moving, they vary their position vectors, resulting in differentiating the equation from both ends.

Importance of Centre of Mass

The system’s centre of mass is a point where any uniform force is applied to the object. Moreover, it is essential to determine the object’s centre of mass because it becomes effortless to solve the mechanics’ problem to define the abnormally shaped and complicated object’s motion. During calculations, we suppose that all the mass belonging to an abnormally shaped and complex object is concentrated in a small-sized object present at the centre of mass, called the point mass.

Centre of Mass Formula

The vector addition of the weighted position vectors indicates the centre of mass of every single object of the system. Along every single axis, the calculation for the centre of mass is individually done for the components.

Centre of Mass of a two particle system is  rcm = (m1 r1+ m2r2)/( m1 + m2 ).

Centre of Mass of a Two-Particle System

Let’s consider a system with two particles with masses m3 and m4 situated at points C and D. Let r3 and r4 be the particle’s position vectors relative to a fixed origin, i.e. ‘O’. Subsequently, the position vector ‘rcm’ of the centre of mass ‘C’ that belongs to the system is defined by:

 rcm = (m3 r3 + m4 r4)/(m3+m4).

The product of the system’s total mass and the centre of mass’s position are equivalent to the complete summation of the products of the masses of two particles and their position vectors, respectively.

Conclusion

The motion of the centre of mass study material concludes that the centre of mass is the particle equal to the given object for the functioning of Newton’s laws of motion. Moreover, in the case of an individual rigid body, the centre of mass is present relevant to the body. If the body comprises an identical density, it will be present at the centroid.