8.3 Newton’s third law of motion  (Page 2/2)

It should be emphasized that though vector sum of two forces is zero, but this condition does not indicate a state of equilibrium. This is so because two forces, often called as action and reaction pair, are acting on different bodies. Equilibrium of a body, on the other hand, involves consideration of external forces on the particular body.

Deduction of third law from newton’s second law

We have pointed out that “action” and “reaction” forces are external forces on the individual bodies. However, if we consider two bodies forming a “system of two bodies”, then action and reaction pairs are internal to the system of two bodies. The forces on the system of bodies are :

If no external forces act on the system of bodies, then :

$$\sum F_{\text{ext}}=0$$

From second law of motion, we know that only external force causes acceleration to the body system under consideration. As such, acceleration of the “system of two bodies” due to net internal forces should be zero. Hence,

$$\sum F_{\text{int}}=0$$

This is possible when internal forces are pair forces of equal magnitude, which are directed in opposite directions.

The internal forces are incapable to produce acceleration of the system of bodies. The term “system of bodies” is important (we shall discuss the concept of system of bodies and their motion in separate module). The acceleration of the system of bodies is identified with a point known as center of mass . When we say that no acceleration is caused by the pair of third law forces, we mean that the “center of mass” has no acceleration. Even though individual body of the system is accelerated, but “center of mass” is not accelerated and hence, we say that "system of bodies" is not accelerated.

Referring to two charged body system that we referred earlier, we can consider forces $F_{12}$ and $F_{21}$ being the internal forces with respect to the system of two charged bodies. As such, applying Newton’s second law,

$$\sum F_{int}=F_{12}+F_{21}=0$$

$$\Rightarrow F_{12}=-F_{21}$$

The “action” and “reaction” forces are, therefore, equal in magnitude, but opposite in direction. Clearly, third law is deducible from second law of motion.

The important point to realize here is that “action” and “reaction” forces are external forces, when considered in relation to individual bodies. Each of the two forces is capable to produce acceleration in individual bodies. The same forces constitute a pair of equal and opposite internal forces, when considered in relation to the system of bodies. In this consideration, the center of mass of the body system has no acceleration and net internal force is zero.

Summary

Here, we summarize the interpretation of third law of motion as discussed above :

• This law does not address the issue of force and acceleration as applied to a body like first two laws.
• The law characterizes the nature of force irrespective of its class and genesis that they exist in pair of two equal but opposite forces. The existence of a single force is impossible.
• The pair of forces acts on two different bodies.
• The two forces do not neutralize each other, because they operate on different bodies. Each of the bodies will accelerate, if free to do so.