14.3 Laws of motion and system of particles

The point of application of a single external force can not be associated with more than one particle. In this sense, the consideration to extend law of motion to many particles system may appear to be meaningless. However, we find that the dynamics of a bunch of even unassociated particles can be studied collectively with startling accuracy and relevance.

Let us begin with a collection of six billiard balls, positioned arbitrarily on a smooth billiard table. Imagine we have initiated one of the ball with a constant velocity.

Since other balls are placed unconnected, they are not affected by the motion of one of the six billiard balls. It may be surprising, but we will see that "center of mass (COM)" is, as a matter of fact, governed by Newton's second law of motion! As no external force is acting on the collection of balls (neglect forces like weight and normal force, which are perpendicular to the motion), their COM is moving with a constant velocity in accordance with Newton's second law.

$$\begin{array}{l}{F}_{\mathrm{Ext.}}=0;a_{\mathrm{COM}}=0;v_{\mathrm{COM}}=\mathrm{a\; constant}.\end{array}$$

Not only this, if the moving ball collides with other ball, then also the motion of COM of the system of particles remains same. The magnitude and direction of velocity of COM remain what they were before collision. The contact forces, being the internal forces, do not affect motion of the COM. Whatever be the sequence of subsequent hits among the balls, COM moves with constant velocity. This underlines the connection between Newton's second law and the COM of the system of particles.

What if the external force is not zero. Let us take another example where a ball is projected at an angle in air. The ball is split in mid air, say, due to implanted cleavage (plane of weakness). The two fragments move down towards the surface after the split. The ball is acted upon by the gravity (force of gravitation) before being split and follows a parabolic path. Since splitting along the cleavage had been an internal process, the system of two parts is acted by the same external force of gravity. Therefore, the COM of split parts also follows exactly the same parabolic path (neglecting air resistance) as the ball would have transversed, if it were not split! Here,

$$\begin{array}{l}{F}_{\mathrm{Ext.}}=mg;a_{\mathrm{COM}}=g;v_{\mathrm{COM}}=\mathrm{a\; variable}.\end{array}$$

The external force on the system of two parts is :

$$\begin{array}{l}{\mathbf{F}}_{\mathrm{Ext.}}=M{\mathbf{a}}_{\mathrm{COM}}\end{array}$$

where "M" is the total mass of the ball and " ${\mathbf{a}}_{\mathrm{COM}}$ " is acceleration of the COM.

In the nutshell, we conclude :

• COM represents an effective point, whose motion is governed by Newton's law of motion.
• The law of motion for system of particles is stated for the acceleration of COM - not for the individual particle. This equation does not give any information about the acceleration of individual particles.
• The law of motion for system of particles is stated assuming that all the mass of the system is concentrated at COM.
• The law of motion for system of particles is stated for the external forces acting on the particles of the system - not the internal forces.
• The law of motion for system of particles is stated for a closed system in which mass is constant i.e. there is no exchange of material "from" or "to" the system of particles.