Suppose we say that the centre of mass of a body or a system of bodies is a point at which the entire motion of mass of the body/system of bodies is supposed to be concentrated. We can also say that the average position of all the system parts is the mean location of a distribution of mass in space where the force is usually applied, resulting in a linear acceleration without any angular acceleration. If the force is applied externally acting on the body/system of bodies to be used at the centre of the mass, the state of rest/motion of the body/system shall remain unaffected.

The motion of Centre of Mass

The motion of the centre of mass of a body is a system that acts as a point where the whole of the mass of the body or object acts. At the centre of the mass, the weighted mass gives a sum equal to zero. It is the point where any uniform force is applied.

Centre of Mass for Two Particles

There are two particles with equal masses. COM is the point that lies exactly in the middle of both.

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

The motion of the centre of mass of a body or system of a particle is defined as a point where the whole of the body’s mass or all the masses of a set of particles appears to be concentrated.

  • The above derivation shows that the concept of mass, the motion of total mass of the system, is described under the effect of external forces only.
  • (m1+m2) i.e., the motion of the total mass is supposed to be concentrated at a single point called the centre of mass of the system.
  • It shows the position vector of the centre of mass can be taken as the weighted average of the position vectors of the two particles.

Centre of Mass for n Particles

For a system of n particles, the centre of mass, according to its definition is:

D = (m1d1 + m2d2 + m3d3 +…… mndn) / (m1 + m2 + m3 +….. mn ) = ∑midi / ∑ mi

mi here is the sum of the masses of the particles.

Things to Note:

  1. The position of the centre of mass of a system is independent of the choice of coordinate system.
  2. The position of the motion of the centre of mass depends on the shape and size of the body and its mass distribution. Hence, it may lie within or outside the material of the body.
  3. With a uniform distribution of mass, the centre of mass coincides with the body’s geometrical centre or centre of symmetry.
  4. The centre of mass changes its position only under the translatory motion. There is no effect of rotatory motion on the body’s center of mass.
  5. The concept of motion of the centre of mass proves that the laws of mechanics, which are true for a point mass, are equally valid for all macroscopic bodies.
  6. The centre of mass of different parts of the system and their masses can get the combined centre of mass by treating various parts as point objects whose masses are connected at their respective centres of masses.

Centre of Mass for Three Particles at Different Positions

The centre of mass lies on the same axis of a straight line, whereas there can be cases where particles lie on the Centre of Mass for Three Particles at Different Positions.

Equation of Motion of Centre of Mass

When there is a motion in an object, we always concentrate on the velocity of the object or the acceleration with which the object is moving.

mv =  m1 v1 + m2v2 +……….. mnvn

where vi =dri/dt  is the velocity of the ith particle and v =dr/dt is the velocity of the centre of mass.

V = Σmivi /  Σmi

This is an expression for the velocity of the centre of mass.

Centre of mass acceleration formula

If the velocity of the centre of mass changes, it’s accelerating. This equation shows the acceleration of the centre of mass: 

aCM = m1a1+m2a2+m3a3 /m1+m2+m3

There is something unique about the centre of mass acceleration equation: the numerator is part of Newton’s second law, which tells us Σ F = ma. We can substitute each mass times its velocity, with F, as shown in this third equation:

aCM = F1+F2+F3 /m1+m2+m3

  Example of Centre of Mass

  • A radioactive nucleus is initially at rest and decays. Its fragments fly off in different directions, obeying the principles of momentum and energy conservation. The fragments fly off in different directions with different velocities so that the vector sum of linear momenta of all fragments is zero. So, the momentum of the motion of the centre of mass of all fragments must also be zero after decay.
  • As we know, the moon revolves around the earth in a circular orbit & the earth revolves around the sun in an elliptical orbit. The earth and moon exert gravitational forces of attraction on each other, which are internal forces only.
  • When an object of finite size is thrown with some velocity at an angle with the horizontal, it follows a parabolic path. The center of mass of such an object also follows a parabolic path.

 Conclusion

If we take three masses in space, there is Gravitational attraction between the particles and is the only force acting on them.

Newton’s third law states that for every action, there is an equal and opposite reaction; therefore, the gravitational forces between them are identical in magnitude but opposite in direction. In other words, F12 = F21.

There is no net force on the centre of mass. The center of mass is not accelerating. External forces are the only forces that will cause the centre of mass to accelerate. So:

aCM = F1+F2+F3 /m1+m2+m3

aCM = 0+0+0/ m1+m2+m3

aCM = 0

We can take this equation and rearrange it to solve for force, giving us:

F1 + F2 + F3  = (m1+m2+m3 )(aCM)

Σ F = (m1+m2+m3)  aCM