Introduction
Inertial force, like centrifugal force, have come to mean both a real and a fictitious force. Science is here not particularly accurate. This confusion is likely, in addition to being indeed a subtle matter, the reason some people claim centrifugal force not to exist. In some ways it makes me think of that special group of people, not all that long ago, publising pamphlets and booklets arguing that the earth was flat. Those maintaining that centrifugal force to be a hoax appear to me to belong to this special kind of people. I will set out below to develop a mathematical model for inertial and for centrifugal force, simple and without any ambiguity. It is developed relying rigorously on mathematics but the emphasis is on appropriate model and graphical illustrations to appeal to our intuition.
Inertial forces
We have heard so often that the sum of forces is equal to mass times acceleration as expressed by Newton's 2nd law,
that we have forgotten that mass times acceleration not only has the dimension of force but also represents a real force, to be more precise, a real inertial force. It is hence equally true to say that the sum of external forces is equal to the inertial force of the particle, as it is 'resisting' the change in motion imposed by external forces. In this we follow the example of an illustrious scientist, d'Alembert .
In Fig1 two masses, m1 and m2, travel at constant speed towards each other with no external forces acting on them. When they collide they start exerting forces on each other. These forces are inertial type forces and are definitely real. Be aware that frequently one finds references to inertial forces as being fictitious, however they are associated with non-inertial reference frames. The mass m1 exerts an inertial force F1,2 on mass m2, whereas m2 exerts an inertial force F2,1 on mass m1. The forces F1,2 and F2,1 are of equal and opposite value. Again, it can't be said enough, these two forces don't act on the same agent, hence don't cancel each other!
Inertial and noninertial reference frames
Fig2a and Fig2b show some of the subtle features of inertial frames. In both cases the x'y'z' frame is linearly accelerating along the x-axis with an acceleration x''[t]. An observer scientist observes in case a) that the mass m is accelerating negatively along its primed x-axis with a magnitude of -x''[t], and concludes the existence of a fictitious force F'x operating on the particle m. In case b) the observer scientist observes that the mass m is not moving and the queer deflection of the cord from vertical. He notices and measures a tension T in the cord and derives F'x = -m g tan(β). These simple figures illustrate the many real life situations where, in non-inertial reference frames, there is first free motion and then subsequently containment. A typical case is the passenger first almost freely sliding before he crashes into the dash board during a head-on collision.Mass spring interaction (1)
M x''(t) + Mg = - k1 x(t) - k2x'(t)
Let's look at another simple example of a particle interacting this time with a spring/dashpot assembly as shown in Fig3. The mass M hits the assembly, consisting of massless spring and dashpot, with a velocity V. Simultaneously, the mass and the assembly start interacting through several forces, The mass compresses the spring/dashpot using its force of inertia and its weight. The spring/dashpot assembly , being compressed by the inertial force of the mass m, resists with two forces acting back on the mass M.. At all times the opposing forces are equal and opposing. However, the inertial force plus weight are acting on the spring/dashpot and the spring/dashpot forces are acting back onto the mass M. It is important to understand fully the significance of this last point - the two set of forces are equal and opposite but DON'T cancel each other. Many people readily seem to forget this simple truth as expressed by Newton's 3d law. The oscillating motion around equilibrium, as shown in Fig4, is primarily related to the inertial force of the mass M interacting with the spring/dashpot whilst decelerating. Once the mass M is at rest it is reduced to zero, as for the dashpot damping force, and the equilibrium value for x, xeq, is solely determined by the spring balancing the weight of the mass due to gravity force.
The mass M exerts on the spring/dashpot, in addition to its weight, an inertial force. It is transient but there is nothing fictitious about it. This real inertial force is battling with the forces produced by the spring/dashpot, being compressed. We will repeat the experiment but now with the mass M approaching the string/dashpot assembly with an angle of 90 deg as shown in Fig5. We will for convenience eliminate gravity. The assembly has a pivot at the lower end and the mass M is approaching with constant speed V and is aligned with a trap with catches the mass. The subsequent mathematical analysis of the governing DV-Equations below will clearly show the fascinating behaviour of a mass M rotating around a center in the presence of some form of retaining force.
Mass spring interaction (2)
We are now ready to attack the problem of finding out about the role/existence of centrifugal/centripetal force and explore/explain these entities for a particle rotating around a center. With reference to Fig5a, a mass M is traveling with constant speed V along a straight line until it is trapped at the extremity of a spring having some radial damping, further indicated, for convenience, by cord. Fig5b indicates the approximate initial trajectory of the mass m and shows the point of application of the action-reaction pair of forces acting respectively on mass and cord - Newton's 3rd law.
In this experiment we have inertia, not by altering the speed of a particle directly, but by trying to change the direction of its motion of the mass M. In the previous experiment, after a few transient oscillatory motions a stationery equilibrium was reached. Let's see now what happens in our second experiment. Even if the explanations are more addressed to our intuiton, the solution of the DV-Equations is rigorous and purely a mathematical affair.
The mass M wants to continue its straight line trajectory and moreover dislikes to be slowed down by the cord. It resists valiantly but starts slowly giving in to the persistent force exerted on it by the cord. The mass M starts increasingly slowing down whilst the cord continues to stretch and there comes a point where they the mass is running out of steam and the spring sees the opportunity to pull the mass towards its center. However by doing so the mass accelerates, increases its speed and tries again to escape. Due to the damping added into the system very soon there is an equilibrium situation reached between the mass trying to get away from the center and the spring trying to pull the mass in towards the center of rotation. The steady looking motion of the mass is in reality a continuous and stubborn dogfight going on between the spring pulling inwards and the mass wanting to escape. If nothing interferes this battle will last forever as it does for planets circling around their sun. This is clearly to be seen in Fig6 and Fig7. Fig6 shows the calculated vector field representation of the force exerted by the cord on the mass - the centripetal force. Fig7 shows similarly the calculated vector field representation of the inertial force exerted by the mass on the cord, effectively keeping it taut - the centrifugal force. As can be noticed these two forces are of equal and opposite value. I will repeat again that these forces don't cancel since they are not localized vectors, meaning that they don't have the same point of application.
Fig8 and Fig9 show the calculated vector field represetation of centrifugal and centrifugal forces when the initial transients have died out. It has become a 'stationery' looking equilibrium between the real force generated by the tension in the stretched cord and the real inertial centrifugal force caused by the mass resisting having his direction of motion continuously changed. I find it comical, re to JK et all, to state that centrifugal force not only exists but is in addition directly responsible for creating the centripetal force. It can can be seen that it is the inertia of the 'fleeing' mass (centrifugal force) which is directly responsible for the tension in the cord (centripetal force). They are like inseparable twins, but that should not come as a surprise for those who believe in the validity of Newton's 3rd law, action=reaction, real forces can't exist as a single entity, they always come in the form of pairs.
Conclusions - the circular motion of a particle around a center is directly explained by the interaction of two real forces. An inertial force exerted by the mass on the spring and a real force exerted by the spring on the mass, trying to pull the mass m towards the center. It is really a fascinating concept , very much like a snake trying to catch its tail and never able doing so, or perhaps trying to out-run your shadow. Nature really is quite fascinating. Most problems appear easy if clearly explained, however I can feel quite readily why it took so long before the concept of centripetal acceleration was finally developed by Newton, even with the science of astronomy being very alive and active for many centuries.
A different perspective
If we look at Figs 4 and 10, one will notice that the mass M and the spring in both cases, after some transient motion, reach an equilibrium situation. For Fig4 this is reached with a balance between two real forces - force due to deflection of spring and weight due to gravity. For Fig10 however it is between a real force (spring/centripetal) and an inertial force (centrifugal). We can also think of the inertial forces being setup by the rotation as the equivalent of a radial 'gravity-like' force field having a gravitational field strength of
r pushing mass particles radially away from the center with a force m
r. For a constant radial velocity this force field is proportional to r. This gives perhaps another intuitive way of looking at rotating systems. I like to point out that this is valid only if there is some force linkage with the rotating center, either through some form of pushing (retaining rotating surface), pulling (cord) or friction force.
Choice of reference frame
The analysis above used an inertial frame of reference. Using a noninertial frame, fixed to the spring mass assembly, has no particular advantage in this particular very simple case.
Definitions
I am sure that some are puzzled why it is that some so vehemently deny the existence of centrifugal force. I really don't know. It keeps surprising me. But then again there have been quite a few, not so long ago, who believed the earth to be flat. However, more seriously, a possible reason migh be that scientists themselves have created an ambiguity, using the expressions 'inertial force' and 'cenrtifugal force' to both have different meanings. This I have pointed out in various posts. Hence there is confusion, not so much for a scientist who is aware of this confusion, and relying on the context for proper interpretation. The problems start when people, not having a proper scientific back ground, start meddling in these matters and getting all mixed up. The solution for this ambiguity is very simple - define centrifugal force to be any force acting on an application line going through the center of rotation. Hence taking the adjective by its literal meaning - ‘center-fleeing’. This allows us to both have a real centrifugal force in a stationery reference frame as well as a fictitious centrifugal force in a rotating noninertial reference frame. Moreover the term 'fictitious' shoud be used solely to describe the type of inertial forces introduced by the acceleration/rotation of the noninertial reference frame itself and not be used as being equivalent to 'inertial'. Hence, as for centrifugal force, we can then define without ambiguity real inertial forces and fictitious inertial forces. In our familiar inertial reference frame inertial forces are very real and omnipresent all around us and seeing them as being fictious would perhaps push the paranoia of some so far that they would deny even with their last breath the reality of the inertial force of an high velocity impacting stray golf ball ending their existence on this earth.
mandrin