In this chapter we have learned how to find the details, and particularly, the time dependence of interacting systems: we use Newton’s second law. We know how motion is connected to forces. But as we see, things can get pretty “messy” very quickly, when we look at the details.
When we apply Newton’s laws to the vast majority of interesting questions, there is no simple solution to Newton’s 2nd law ( \(\Sigma F = ma\), or in its more general form, \(\Sigma F=dp/dt\) ). Except in a few special cases, the net force is a function of the positions of the systems that are interacting, and/or the net force can be time dependent itself. But with high speed digital computers, we can calculate what the positions and momenta of the pieces making up the systems are as a function of time. That is, we can determine the exact motion of the pieces of the systems as they interact. Examples we have mentioned before are human bodies in car crashes of various kinds. Rather than continually crashing cars with dummies inside into brick walls, it is now possible for engineers to determine what happens to bodies under differing automobile designs, safety features, and crash scenarios, by appropriately modeling the interacting pieces and applying Newton’s laws to all of the pieces. It gets complicated (getting at the details is always complicated), but the basic approach is what you now know how to do. Another example is figuring out when and how to fire the rockets that get various planetary probes out through space and after long journeys (sometimes years) arriving at the intended planet and orbiting and/or landing on it. How is this done? Simply applying the tools and concepts we have been studying in this chapter.
In both of the examples mentioned in the previous paragraph, Newton’s 2nd law must be used to get the time dependence of the forces and the motion. But much of the information that is wanted, is obtained much more simply by using conservation of energy, momentum, and angular momentum; i.e., taking a before and after the interaction approach. Working together, these are powerful approaches indeed. You have seen in several examples in DL activities and in the homework problems how both approaches used together, make it possible to get many answers to physical situations of interest.
We previously mentioned some of the limitations of our models. As we go to atomic sized systems Newton’s laws totally break down. Energy and momentum become quantized, but still exist and the idea of conservation exists. However, at atomic sizes we have to get a totally new model to replace the Newtonian model. As we get to systems the size of atoms and molecules, a different fundamental law of nature takes over. The various aspects of this very different law of nature are collectively known as quantum mechanics. One way to represent what is going on in quantum mechanics is with what are called wave packets (probability waves). Particles cease to have meaning in terms of what we normally think of as a particle. An electron is modeled not as a particle circling the nucleus, but is represented by a probability wave.
So, should we replace Newton’s laws with wave mechanics (or any other representation of quantum mechanics)? No, not for large scale objects. The very peculiar aspects of quantum mechanics become totally insignificant as the size of the objects approach macroscopic size. In the limit of large scale objects, Newtonian mechanics is a particularly useful approximation (model) of how matter interacts with other matter. Is it a correct model? Yes, in the sense that on a macroscopic scale, this is how matter behaves when it interacts. Is it an approximation? Not really. On a macroscopic level, the very weird quantum effects that the individual objects experience are simply not large enough to be observed when we look at the behavior of the macroscopic objects themselves (at least most of the time). Remember when we addressed the issue of models at the beginning of this course. We emphasized that we shouldn’t think of models as being right or wrong, but rather as useful or not useful. This is probably a good way to think about Newton’s laws. They are a very useful model of the interactions of (most) macroscopic objects as a whole.
Why do we insert the parenthetical “most” in the previous sentence? Because, even on a macroscopic level, sometimes quantum mechanics takes charge. Consider the phenomenon of superconductivity. This is truly a quantum mechanical effect on a macroscopic scale. The electrons going around the thousands of turns of wire in a superconducting magnet act in some ways as if they were one giant electron. And they go around the wire as if there were no electrical resistance! The currents in superconducting magnets can last months or even years. The magnetic field surrounding the space in a MRI machine (magnetic resonance imaging) is created by supercurrents in the superconducting magnet around the outside. The electrons act sort of like the entire magnet is one giant atom. But this only happens if the thermal energy is sufficiently low. Thermal energy is the “enemy” of macroscopic quantum effects. At the present time, these superconducting magnets must be kept at “helium temperatures”, so called, because liquid helium is used as the refrigerant. At atmospheric pressure the boiling point of helium is 4.2 K, and this is the temperature of the magnet coils.
There is another limit to the applicability of Newton’s laws that has nothing to do with size, but rather with speed. When objects move relative to each other with speeds that are not significantly less than the speed of light, \(3 \times 10^8 m/s \), very strange behavior begins to be observed. Clocks run slow. Distances shrink. We enter the strange world of special relativity. One thing Einstein’s theory of special relativity, developed a hundred years ago, is telling us, is that our concepts of space and time are not “correct.” That is, there really isn’t a three dimensional space and an independent time. Rather, space and time get all tangled up. This, of course, plays havoc with Newton’s laws! Every time you watch television, you are a witness to an example of one of the consequences of special relativity: namely, that nature has an ultimate speed limit of \(3 \times 10^8 m/s \), the speed of light. The electrons that are “fired” out of the back end of the picture tube travel toward the phosphors on the screen at almost the speed of light. As energy is transferred to the electrons by an electric field, their energy increases, but pretty soon their velocity approaches the speed of light. Does this mean their kinetic energy can’t increase further. No, it is as if their velocity tops out at just under the speed of light, but their mass increases. Pretty weird and fascinating stuff.