on interaction and inertial force
Anil Mitra, Copyright © December 20, 2019—December 20, 2019
on interaction and inertial force
The idea of inertial force can be understood intuitively as well as on a case by case basis for different inertial forces, e.g. linear and centrifugal.
However, the concept of inertial force is subtle enough that intuition is insufficient. For example other terms for inertial force are fictitious force and pseudo force (as well as d’Alembert force for the historical reason that d’Alembert used the concept of inertial force productively). But since such forces seem real, e.g. the centrifugal effect in a vehicle traveling a bend in the road, why would they be called fictitious? A further difficulty is that inertial forces can be introduced rigorously but without doing so the concept is confusing and its use cannot be guaranteed to be free of error (and will most likely result in error in all except the simplest of situations).
The plan, therefore, is to introduce inertial forces rigorously; once this is done, the intuition and physical nature of inertial forces can be made clear and analysis of actual situations solving can be made rigorous. We will find that it is best not to approach situations from an inertial force perspective but that inertial force concepts are useful in intuitive understanding of the physical phenomena.
It is essential to go to the foundation of Newtonian Mechanics.
The following is not formally systematic for the laws could also be written as axioms. It is however reasonably rigorous (in the sense that while it does not address all minor criticisms it could be extended to do so).
Definition. Every particle—i.e. point particle—has a quantity called its mass.
In modern terms, Newton defined mass as the product of density and volume but this definition is clearly circular. How is the problem to be resolved? In terms, as we will see, of its place or context within the net (axiomatic) system.
Statement. The mass of a system of particles is the sum of the masses.
Comment. We will see some motive for this and the vector addition for forces subsequently. However the true motivation is that (i) the system of definitions and axioms results in a consistent system that (ii) agrees with observation over a significant range of phenomena. That is, what is being employed is the hypothetico-deductive method of science. It is an interesting exercise to investigate a system of independent axioms.
Definition. A force is an interaction between particles.
That is all forces on a particle are applied forces.
According to the definition, ‘inertial forces’ are not forces. This, however, does not prevent introduction (definition) of ‘inertial forces’ and then demonstrating that there are situations in which they can be treated as forces.
Statement. Particles have no self or internal forces.
Statement. The net force on a particle is the vector sum of the forces of (all the particles in) the world on the given particle.
The net force can be zero if there is no force on the particle or if there are forces but they sum to zero.
In modern language, the first law says that a particle without any applied force has constant velocity (or constant momentum for momentum is defined as product of mass m and velocity v, written m v). Note—the velocity may be zero.
But note that if the particle has constant velocity in one framework, then it will not have constant velocity in a second framework that has acceleration relative to the first.
So the question arises—which framework shall we use?
Statement. There is a frame called an inertial frame in which if there is no net force on a particle, the particle will move with constant velocity (or momentum).
In a linear development, this axiom would have been placed before the first law but it is convenient to have motivated it before its statement.
Given an inertial frame (i) a second frame that has constant velocity but no rotation relative to it is also inertial, and (ii) the following frames are not inertial—one with linear acceleration relative to the first, one that rotates relative to the first (and of course one that has linear acceleration as well as rotation).
The second law of motion for a particle in an inertial frame is
F = m a
m is the mass of the particle
F is the net applied force—i.e. the sum of all the forces applied to the particle (by the world). That is, in the second law, force is by definition applied force or force of interaction.
a is the acceleration of the particle or body in an inertial system.
Note that the second law can be written
F = p', where p is the momentum and the prime ', denotes differentiation with respect to time.
Consider a system of particles. It is easy to show that the force equation results in
F = Fe + Fi = m a
F = net force = sum of all forces,
Fe = sum of external forces,
Fi = sum of internal forces,
m = sum of the masses, and
a = acceleration of the center of mass.
Now if the system is concentrated at a point a is just the acceleration of the point; but the above equation for acceleration is the equation for the center of mass if the body is extended; and may be taken to be the equation for small enough bodies.
If there are internal forces, it is conceivable that they do not sum to zero. That is a body without external forces could accelerate in an inertial framework. A consequence would be that conservation of momentum for isolated systems would not hold. The universe would be very unlike ours.
We therefore postulate—
Statement. (1) The action of one particle on another is reciprocated by an equal and opposite reaction; these forces are called action-reaction pairs or just the interaction. (2) The interaction lies the line joining the particles.
Theorem. The first part of the third law is necessary and sufficient for conservation of momentum of all systems (i.e. including the sub-systems of a given system).
Theorem. The third law is necessary and sufficient for conservation of angular momentum of all systems.
Comment. If particles had internal forces that did not sum to zero they would accelerate in an inertial frame without an external force. Momentum would not be conserved. Thus to preserve the Newtonian system, internal forces for a particle should sum to zero. However, zero sum internal forces for a particle have no significance in the Newtonian framework. This was why it was specified that particles have no internal forces.
Theorem. The motion of a composite body is given by
F = m a
F is the external or applied force
m the total mass
a is the acceleration of the center of mass or just the body itself if it is small enough to be regarded as a point particle.
Newton recognized (i) forces at a distance—gravitation, (ii) contact forces (‘impressed’), (iii) inertial forces.
However, his recognition of inertial forces was unnecessary within his system (he was using the vocabulary of his time and his intelligence generally enabled avoidance of mistakes).
We can combine the other two kinds of force under a single kind—just force (or applied force or force of interaction).
What Newton recognized as impressed are what we recognize as contact force—largely electrostatic and magnetostatic in nature.
Since Newton’s time further forces recognized in the classical framework are the electromagnetic (which subsume the static forces). The electromagnetic have deviations from the classical framework, e.g. action and reaction are not equal and opposite but the inconsistency in terms of the Newtonian framework can be patched up by recognizing that the electromagnetic field carries momentum.
Newton induced that gravitation on Earth and in the solar system are the same in kind and postulated an axiom of universal gravitation—
The gravitational attraction between any two particles is proportional to the product of their masses divided by the square of the distance between them.
Theorem. The gravitational field for a uniform spherical shell is (i) zero inside the shell and (ii) as if the entire mass is at the center for points outside the shell (and indeterminate at the shell).
Corollary. For a uniform sphere, at any point distant r from the center in or outside the sphere the field is as if the entire mass within a shell of radius r is at the center.
A fixed piece of matter may be defined as unit mass. A unit of force can be defined as (a) the force that has a certain reproducible effect such as the depression of a standard spring or (b) is required in order for the unit mass to have unit acceleration or (c) as the weight of a fixed mass (e.g. the unit mass).
All three systems are feasible and intertransleatable but (b) and (c) are fundamental and adapted to formulation of Einstein’s general theory of relativity.
Let aA be the acceleration of a body P in an inertial system A, and aB the acceleration in a non-inertial system B.
We are interested in the force law for B and therefore write
aB = aA + aI
F = m aA
Where F is the external force. We derive
F = m aA = m (aB + aI)
F – m aI = m aB .
If we now write
FI = – m aI (the inertial force) ,
FA = F (the applied force) .
FA + FI = m aB .
This may be read—In a non inertial system the second law holds if we add an inertial force FI to the applied force FA.
In terms of our axiomatic system, inertial force is not a force. From that point of view the question of its reality does not even arise.
Inertial force was used to advantage in the historical development of mechanics (under various names). However, none of these uses is necessary (the analogy between inertial force and applied force was useful) and the inertial force was but an artifact from the point of view of our axiomatic system.
Still, as noted earlier we can of course extend the concept of force by admitting the inertial forces as forces. This is fine provided (i) we recognize what we are doing, (ii) do not confuse the two kinds of force—i.e. inertial force is not an interaction. But the inertial force retains the character of an artifice and the question of reality is, after all, not that important in the Newtonian framework.
If one is in an accelerating car (either in linear line acceleration or in acceleration due to motion around a bend) one feels as if there is a force. What is that force? Is it an applied force associated with the motion? We know from the theory above that there are no such applied forces. So why do we feel it?
Let’s consider linear acceleration. If one is sitting upright without one’s back against the backrest then, in order for one’s upper body to accelerate in the forward direction, the torso below the upper part must apply a force to the upper part (and the upper part applies an equal and opposite force on the lower). That is the force one feels (if one’s back is against the backrest the rest provides the force and one’s back feels pressure). Thus while one feels force and it is associated with non inertial motion, it is an internal applied force that one feels and not an inertial force. But in this example, the magnitude of the inertial force and the applied force are the same and this may lead one to think that the inertial force is an applied force. It may also lead one to think of the applied and inertial forces as action-reaction pair—which they are not.
Because inertial and gravitational mass are numerically the same (to within experimental error) if one is in a space elevator one cannot detect the difference between smooth linear acceleration and a gravitational field (without looking outside).
This numerical equality led Einstein to postulate conceptual indistinguishability between gravitation and acceleration which, generalized via the tensor analysis to non uniform fields, led to the general theory of relativity.
Does this mean inertial forces that are fictitious relative to the concept of applied force in Newtonian mechanics are real in General Relativity (not that I do not say real in any unqualified sense)?
In a sense yes but not entirely for electromagnetism and the other forces cannot—have not yet—been brought under a common framework that shows equivalence of force and motion.
But even in that sense, a distinction between applied force and inertial force remains.
Still we ought to ask why some frames are inertial. Mach’s hypothesis, to which Einstein subscribed, is inertial frames are defined by the ‘fixed stars’. So, perhaps then, inertial forces are real and derive from acceleration relative to the universe—i.e. from some interaction with the universe as a whole. If so, then from the equivalence of gravitation and acceleration, gravitation may be a different kind of force than the other three ‘fundamental forces’.