3.2 Classical Mechanics: Newton's Laws of Motion and Gravity
Newton recognized that, in the absence of gravity, the laws of motion could be expressed in a very simple way. According the the first law, objects, stationary or moving at constant velocity, remain in that state unless acted on by a force. Hence, the action of force is to cause change in the state of motion. This is the essence of the second law, which comes to us in two forms, the simplest is that the acceleration of an object is proportional to the impressed force divided by mass. The idea that force is the cause of motion allows us to preserve the notion of causality.
The more sophisticated version of the second law is that, when an object undergoes acceleration, either positive or negative, it gains or loses momentum. In addition, when two objects interact via a collision, momentum lost by one object will be gained by the other. That is, momentum is conserved. This concept can be extended to any number of interacting objects without the necessity of calculating the forces in each interaction.
Since force is defined as rate of change of momentum, each interaction implies two forces, a positive force acting on the object which gains momentum, and a negative force acting on the object losing momentum. The forces which originate via changes in momentum are known as inertial forces. An active force causes an object to accelerate. The object, by resisting changes in motion, exerts an equal but opposite reactive force. Since both are present when an object changes velocity, they can be viewed as canceling each other and effectively disappear. For these and other reasons, in many discussions about the laws of motion, inertial forces are sometimes referred to as being 'fictitious'.
The philosophical concept of a force-less, smoothly evolving universe is shattered by the fact of gravitational force which is essentially different from inertial force. The force of gravity is only felt when it is resisted and no motion occurs. On the other hand, we feel no force if gravity is allowed to increase the velocity of our body in free-fall acceleration. This is the opposite of inertial force which is only felt during changes in momentum via positive or negative acceleration.
By inventing the concepts of force and momentum, Newton simplified to a considerable extent the situation involving interacting objects having mass. However, his greatest achievement was to bring the phenomenon of gravity within the same framework. The problem is that inertial motion involves both gain and loss of momentum, whereas, an object falling due to gravity increases its momentum without any obvious loss of momentum elsewhere. Newton solved this problem by inventing the idea of a gravitational force proportional to mass. Hence, \begin{align} Gravitational\ force\ F_g \propto &\ \ mass\ \nonumber \\
Acceleration\ =\frac{force}{mass} = \frac{F_g}{mass} \propto &\ \frac{mass}{mass} = constant.\ \nonumber \end{align}
Newton fixed the relationship between mass and gravitational force to be \begin{align} F_g =\frac{M_e G}{r^2} \nonumber \end{align} where \(M_e\) is the mass of the Earth, \(r\) is the distance to the Earth's center, and \(G\) is the universal gravitational constant. Newton chose the value of \(G\) to ensure that gravitational force has the same value as an inertial force which produces the same acceleration. This deceives us into believing that inertial forces which produce acceleration and the force of gravity which may, or may not, produce acceleration, are of the same kind.
Newton's idea that mass is infinitely divisible, and that even the smallest particle of mass (material point), 'emits' gravitation was demolished by the advent of atomic theory. This provides only three ultimate particles having mass, neutron, protons, and electrons. However, in the practical world, atoms are so small that they can effectively be treated as material points. So it is mainly in analyzing events such as atomic collisions that mass has largely been replaced by energy. On the largest scale, it has already be shown in the above chapter that the range of gravitational attraction is limited by its interaction with the structural curvature of the universe.
A more serious challenge to Newton's system was the discovery that mass is not conserved, and that loss of mass in radio active elements is accompanied by gain in heat energy. Perhaps the most serious blow to the concept of mass came from Einstein who showed that mass and energy of a particle are related via the expression \(Energy = mass \times c^2\), and further conjectured that the kinetic energy in a moving object is stored as additional mass.
Newton's system has stood for over 300 years, and is still the theory used in almost all engineering and practical physics. Its main deficiency is that the concept of mass is ambiguous, however, in most situations the increase in mass due to kinetic energy is insignificant. The ambiguity of mass presents a problem only in the more detailed explanations of physical reality attempted in this book. Attempts to find a more precise 'exchange rate' between mass and energy only compound the problem. Since the idea of mass applies only to particles, it makes more sense to eliminate the idea of mass and replace it with energy. This allows the structure of particles to be defined in the same terms as the rest of the energy continuum. A first step in this process requires a more precise expression for the energy content of a particle in terms of its 'rest' energy, and its velocity.
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