Their applications, physics, and math. -- Peter Ceperley
There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.
Einstein's equating of mass and energy through his famous equation E = mc2 was later verified by the invention of the atomic bomb. The atomic bomb converts a small amount of mass into heat and various forms of electromagnetic radiation. It is also thought that the sun derives its energy from the continual conversion of large amounts of mass into heat and light. Many other nuclear reactions similarly convert mass into energy.
Most physicists today consider mass and energy to be attributes of the same property. That is to say, that mass and energy are always proportional to each other as given by the above equation. Thus a hot object will be slightly more massive than when it is cold, although the difference in mass is so slight as to be very hard to detect.
11. Relativistic kinetic energy of a particle
If the mass increases with velocity as discussed in the previous chapter, what is the relationship between kinetic energy and velocity?
Classical kinetic energy
In classical (low speed) physics we are taught that the kinetic energy of a particle is given by:
KE = ½mv2 . (11.1)
We can derive this as
where all the vectors are in the forward direction of motion (in the x direction) so we just show these as scalars. Finishing the derivation, we have:
Derivation of relativistic kinetic energy
In the relativistic case, we repeat the same steps as above except that now the mass is a function of velocity as given by:
We now repeat the steps in (11.2) and (11.3) with this new non-constant mass that varies with velocity v. This calculation will be more laborious because of this new addition and will take a few lines of equations:
where we have used integration by parts for the second step. Also, please excuse the monster integral signs. This is one of the few flaws in the equation editor I use, MathCast, which is otherwise extremely good, especially considering that it is free, very easy to install, and makes entering equations very fast.
where u is defined as:
Continuing with (11.7), we have:
Interpretation of relativitic energy
It is standard to call the left-most term of the final expression in (11.10) the "total relativistic energy" or Etotal and the right-most term the "relativistic rest mass energy" or E0:
Making this substitution, we write (11.10) as:
Interpreted this way, the total energy of a particle (or object or mass) has two components, the kinetic energy and the rest mass energy. Identifying E0 as an energy was quite a leap of faith for Einstein, because at the time there was no conceivable way to extract this energy for use or even for experimental verification that this term indeed was an "energy" i.e. an ability to do work. He made the leap primarily because it made a nice equation for the energy as shown in (11.11) above. The ensuing years revealed the annihilation of matter and antimatter, nuclear fission, and nuclear fusion, all of which converted this rest mass energy to the more normal forms of energy (light and heat) verifying his bold assumption.
Incorporating the rest mass into the energy also allowed the energy and the three momentum components (px, py, and pz) to make up a "four vector", a concept we shall briefly explain in Chapter 14.
We can use our new mass/energy relationship (11.11) to rewrite the momentum/mass transformation, equation (10.25), at the end of the previous chapter as:
where in this equation we represent the total energy of the object (rest energy plus kinetic energy) as E . We have also multiplied the bottom items in (10.25) on either side of the equal sign by c2 .
Waves, Berkeley Physics Course - vol. 3, Frank S. Crawford, Jr. McGraw-Hill 1965. This book is suitable for an add-on to an introductory course on college or university physics. It discusses all sorts of aspects of waves and has a multitude of home experiments. One could probably make a great science fair project from one of them. As to its math level, it mostly uses algebra, with some calculus in the mix.
Physics of waves, by Elmore and Heald, originally published by McGraw-Hill in 1969, but currently published by Dover. This book covers many different wave systems, such as waves on a string, on a membrane, in solids, in fluids, on a liquid surface, and electromagnetic waves. It also covers the many aspects of waves. It has an excellent chapter on diffraction.
The Feynman lectures on physics, Feynman, Leighton, and Sands, Addison-Wesley 1963. Three volumes. These cover many aspects of physics. They are perhaps best suited for someone who has made it through an introductory sequence in college or university physics, and wants to read about the subject from a more sophisticated point of view. They are not particularly math intensive, more just into discussing concepts with some math as required. These are books you read to understand a physicist's mind. Perhaps 10% to 20% of the chapters are about waves and resonances.
Electromagnetic books that I use:
Engineering Electromagnetics, Hayt (with Buck on more recent editions), McGraw-Hill. An easy to read, compact junior-level text for electrical engineering students.
Fields and waves in communication electronics, Ramo, Whinnery, Van Duzer, Wiley. A upper level/graduate level text for electrical engineering student. Covers practically every aspect of applied electromagnetic fields in some depth. Is not a book to sit down and read for philosophy, but rather to look up the rational behind certain devices or design methods.