System of Particles

An object of an ordinary size which we call a “macroscopic” system contains a huge number of atoms or molecules. It is out of the question to attempt to use the laws we have discussed for a single particle to describe separately each particle in such a system. There are nevertheless some relatively simple aspects of the behaviour of a macroscopic system that (as we will show) follow the basic laws for a particle. We will be able to show that in any multi-particle system each of the mechanical quantities relevant to the state of the system consists of two parts:

System of Particles

  • One part in which the system is treated as though it were a single particle (with the total mass of the system) located at a special point called the centre of mass. This part is often called the CM motion.
  • Another part was describing the internal motion of the system, as seen by an observer located at (and moving with) the centre of mass.

Center Of Mass

We consider a system composed of N point particles, each labelled by a value of the index I which runs from 1 to N. Each particle has its mass mi and (at a particular time) is located at its particular place ri.

The centre of mass (CM) of the system is defined by the following position vector

Center of mass
\(r_{CM}=\frac{1}{M}\sum_{i}m_{i}x_{i}\)

Where M is the total mass of all the particles.

This vector locates a point in space — which may or may not be the position of any of the particles. It is the mass-weighted average position of the particles, being nearer to the more massive particles.

As a simple example, consider a system of only two particles, of masses m and 2m, separated by a distance. Choose the coordinate system so that the less massive particle is at the origin and the other is at x =  as shown in the drawing. Then we have m1 = m, m2 = 2m, x1 = 0 , x2 = .

System of Particles

We find from the definition \(x_{cm}=\frac{1}{3m}\left ( m.0+2m \right )= \frac{2}{3}l\)

The location is indicated on the drawing above.

As time goes on the position vectors of the particles ri generally change with time, so in general, the CM moves. The velocity of its motion is the time derivative of its position:

\(v_{CM}=\frac{1}{M}\sum_{i=1}^{N}m_{1}v_{1}\)

But the sum on the right side is just the total (linear) momentum of all the particles. Solving for this, we find an important result:

Total momentum of a system
\(P_{tot}=Mv_{CM}\)

This is just like the formula for the momentum of a single particle, so we see that: The total momentum of a system is the same as if all the particles were located together at the CM and moving with its velocity.


Practise This Question

S1: The differential equation of parabolas having their vertices at the origin and foci on the x-axis is an equation whose variables are separable
S2: The differential equation of the straight lines which are at a fixed distance p from the origin is an equation of degree 2
S3: The differential equation of all conics whose both axes coincide with the axes of coordinates is an equation of order 2