20.3 Law of motion in angular form for a system of particles

This module is an extension of previous module in the sense that we move from consideration of a "particle" to that of a "collection of particles". If we have understood the derivation of angular momentum and second law of motion for a particle, then it would be near replica of reasoning for a system of particles here in this module. However, there are certain characteristic features about system of particles as different from a single particle. These characteristic features will be pointed out (emphasized) in the appropriate context.

Newton's second law for a system of particles in general motion

The concept of angular momentum can easily be extended to include particles, constituting a system. Each of the particles can be associated with certain velocity. In angular parlance, we can assign angular momentum to each of them, provided they are moving. They may be subjected to internal as well as external forces. This is very important difference between a "particle" and a "system of particles". When we consider a single particle, the force can only be external. A particle occupying a point of zero dimension can not be associated with internal force.

For visualization, we can again consider the example of particles or billiard balls that we used in the context of linear momentum. The situation, here, is same with one exception that we shall refer to a point (origin "O" as shown) for calculating angular momentum as against calculation of linear momentum that does not require any such reference point.

The angular momentum of a particle in three dimensional space is defined by the vector relation :

$$\begin{array}{l}\mathbf{\ell }=m(\mathbf{r}\mathbf{x}\mathbf{v})\end{array}$$

where “ r " and “ v ” denotes the position and velocity vectors respectively. We can combine angular momentum of one particle provided angular momentums are measured about a common point. Though, we can calculate angular momentum about different points, but then no physical meaning can be assigned to such calculation. This requirement is actually the reason that angular momentum, in general, is defined about a point - not about an axis. It would have not been possible to associate motion of particles with a common axis.

Since angular momentum is a vector quantity, the angular momentum of all the particles is equal to the vector sum of all the individual angular momentums. It is imperative that the summation would require application of vector addition rules to get the sum of the angular momentums.

The angular momentum of the system of particles is denoted by capital “ L ”. The particles may change their velocities subsequent to collisions among themselves (due to internal forces) or because of external forces. Consequently, angular moment of the system may change with time. The first time derivative of the angular momentum of a particle is equal to the torque on it :