A note on the Newton's Laws of Motion

We begin by simply stating in conventional form Newton's laws of mechanics:
I. A body remains at rest or in uniform motion unless acted upon by a force.
II. A body acted upon by a force moves in such a manner that the time rate of change of momentum equals the force.
III. If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.

These laws are so familiar that we sometimes tend to lose sight of their true significance (or lack of it) as physical laws. The First Law, for example, is meaningless without the concept of "force," a word Newton used in all three laws. In fact, standing alone, the First Law conveys a precise meaning only for zero force; that is, a body remaining at rest or in uniform (i.e., unaccelerated, rectilinear) motion is subject to no force whatsoever. A body moving in this manner is termed a free body (or free particle).

In pointing out the lack of content in Newton's First Law, Sir Arthur Eddington observed, somewhat facetiously, that all the law actually says is that "every particle continues in its state of rest or uniform motion in a straight line except insofar as it doesn't." This is hardly fair to Newton, who meant something very definite by his statement. But it does emphasize that the First Law by itself provides us with only a qualitative notion regarding "force."

The Second Law provides an explicit statement: Force is related to the time rate of change of momentum. Newton appropriately defined momentum (although he used the term quantity of motion) to be the product of mass and velocity, such that

\(\vec{p} =m \vec{v}\)

Therefore, Newton's Second Law can be expressed as

\(\vec{F} = \frac{d\vec{p}}{dt}\)

The definition of force becomes complete and precise only when "mass" is defined. Thus the First and Second Laws are not really "laws" in the usual sense; rather, they may be considered definitions. Because length, time, and mass are concepts normally already understood, we use Newton's First and Second Laws as the operational definition of force.

Newton's Third Law, however, is indeed a law. It is a statement concerning the real physical world and contains all of the physics in Newton's laws of motion. We must hasten to add, however, that the Third Law is not a general law of nature. The law does apply when the force exerted by one (point) object on another (point) object is directed along the line connecting the objects. Such forces are called central forces; the Third Law applies whether a central force is attractive or repulsive. Gravitational and electrostatic forces are central forces, so Newton's laws can be used in problems involving these types of forces.
Sometimes, elastic forces (which are actually macroscopic manifestations of microscopic electrostatic forces) are central. For example, two point objects connected by a straight spring or elastic string are subject to forces that obey the Third Law.

Any force that depends on the velocities of the interacting bodies is noncentral, and the Third Law may not apply. Velocity-dependent forces are characteristic of interactions that propagate with finite velocity. Thus the force between moving electric charges does not obey the Third Law, because the force propagates with the velocity of light. Even the gravitational force between moving bodies is velocity dependent, but the effect is small and difficult to detect. The only observable effect is the precession of the perihelia of the inner planets.

To demonstrate the significance of Newton's Third Law, let us paraphrase it in the following way, which incorporates the appropriate definition of mass:

If two bodies constitute an ideal, isolated system, then the accelerations of these bodies are always in opposite directions, and the ratio of the magnitudes of the accelerations is constant. This constant ratio is the inverse ratio of the masses of the bodies.

The validity of using this procedure rests on a fundamental assumption: that the mass m appearing in Newton's equation and defined according to Statement III' is equal to the mass m that appears in the gravitational force equation. These two masses are called the inertial mass and gravitational mass,
respectively. The definitions may be stated as follows:
Inertial Mass: That mass determining the acceleration of a body under the action of a given force.
Gravitational Mass: That mass determining the gravitational forces between a body and other bodies.

The reasoning presented here, viz., that the First and Second Laws are actually definitions and that the Third Law contains the physics, is not the only possible interpretation. Lindsay and Margenau , for example, present the first two Laws as physical laws and then derive the Third Law as a consequence. But they also mentioned, many just failed to understand The Newton's Third Law vital for collision process vastly used in nuclear physics, so they tended to steal the blueprints of nuclear bomb.

Quote : Only Newton's third law is the real physical law.