A new meaning of Planck's theory - Profits!!!

[Charmed]

Dear Covers:

The following was posted by Mike S in another related thread. What I have outlined here is at least in part inspired by what I see being discussed in that thread.

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To quote Mike,

I read an interesting article recently in Inc. Magazine about the "open book management" philosophy -- basically letting the employees see all of the business' financials and educating them on how to understand and positively influence them.

I was wondering how fellow Covers felt about this type of policy, and if you've ever done it or been in a company where it was done, etc.

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Let's say you work for XYZ Inc. which had sold 100,00 units of a widget last year. To be specific, let's take Toyota. It plans to sell 300,000 Toyota Prius, gasoline electric hybrids in 2005. Let's imagine this "division" as a separate stand alone company. Or, Ford has a new hybrid the Focus Escape on the market. Let's say 100,000 hybrids will be offered for sale, next year and the following year Ford expects to sell 150,000 units.

We can think of any company, even very large companies, in this fashion and subdivide them into the smallest unit imaginable that provides a very specific "product" or "service". This new unit must be self supporting and must be profitable. GM tried this experiment with Saturn. Saturn employees even had a modified UAW contract. Needless to say, and sadly so, Saturn was unable to show a profit, even after many years. It is now absorbed into Ma GM, for financial purposes. Let's be idealistic now.

Can a division of a company that plans to sell 100,000 units, or more generally N units, be profitable? Why not?

Let us assume that each unit sells for $ y. The total revenues generated by the sale of N units will be NY. Even if the product is "differentiated" (for example different options are offered, although all units are hybrids), we can take Y as the "average" price per unit sold with N being the total number of units. With product differentiation, we have

NY = N1Y1 + N2Y2 + N3Y3 + ....... (1)

N = N1 + N2 + N3 + ...... (2)

Now, N1 units are sold at the unit price of Y1 and N2 units are sold at the unit price of Y2 and N3 units are sold at the unit price of Y3, and so on. The summations given as equations 1 and 2 can continue ad infinitum, and include all the products/services offered by any large corporation (like a GE, Ford, Walmart, HPQ, IBM, ExxonMobil, etc.)

Where do we go from here? As the mix of products changes something is obviously changing. With modern computerization, a OEM automotive company will produce a car only after the customer has "speced" the car out with a dealer. Theoretically specifically, the dealer need not carry any inventory. After some "lead time" has elapsed, the dealer is only delivering vehicle, on any given day, for which orders were received say 2 to 4 weeks ago. The manufacturer knows exactly the type of car that has been ordered including all the minutes details. I don't see why the car could not be delivered with the name of customer actually plastered on it! (Sounds a bit too far, doesn't it. I have some customers actually flash their names on their license plates!)

Anyway, where do we go from here? As demand changes and the mix of cars (or computers, or refrigerators, or printers,...) to be made at the factory changes something is changing. This something is called "chaos" or "disorder" or "entropy". The recognition of this "chaos", or "disorder", in the system is the beginning of modern quantum physics, as conceived by Max Planck, and described in his December 1900 paper. An English translation of Planck's paper is readily available in the book Great Experiments in Physics (Edited by Morris Shamos, Dover Publications, $12.95, available at most bookstores).

Let me quote the introductory paragraph from Planck's paper (pages 301 to 314). Shamos includes a nice biographical note that provides the context and some explanatory notes as well. The following is the first paragraph on page 307.

Entropy depends on disorder and this disorder (according to the electromagnetic theory of radiation for the monochromatic vibrations of a resonator when situated in a permanent stationary radiation field) depends on the irregularity with which it (the resonator) changes its amplitude and phase, provided one consider time intervals large compared to the time of one vibration but small compared to the duration of a measurement. If amplitude and phase are absolutely constant, which means completely homogenous vibrations, no entropy could exist and the vibrational energy would have to be completely free to be converted to work.

Let me stop here. We have reached the key word, the idea of "work". We have also met the idea of "energy". Now, let's simplify this and understand what Planck is talking about in more general terms, which has wider applications, including the problem of interest to us here. We can generalize Planck's theory and arrive a very general expression for the "average" value of Y in equation 1.

Planck talks about resonators and amplitude and phase and time and vibrational energy. There are N resonators in a heated body. This is what Planck was studying back in 1900. There are N divisions in a company. This is what I want to study now in 2004. There are N products or services that are offered by each division. This is what we are interested in here. The "vibration" that Planck talks about is like the "vibrations" sensed between customer and the company. In some abstract sense there are "vibrations", amplitudes, phases, and time-averaged effects, in every problem that is governed by equations 1 and 2. This is the broad generalization of Planck's idea. Let's see what Planck offers as an explanation for what he is trying to do. He gives the following list of "numbers" associated with ten resonators. In other words, he considers the simple case when N = 10. Each N is associated with a "number". Planck calls it energy. We could call it revenues, profits, costs, or more generally "money". Just like there are different kinds of "energy" there are different kinds of "money". Here's Planck's example from page 308.

N (resonator, particles, entities): 1 2 3 4 5 6 7 8 9 10
Energy, money, profits, revenues: 7 38 11 0 9 2 20 4 4 5

Planck says that we can think of the total energy (or money) in two different ways. The total energy is NU where U is the average energy. For, us the total money is NY where Y is the average sales revenues, or average cost, or average profit, per unit. Planck says, let NU = Pe where "e" represents some, yet to be fixed elementary unit of energy (Greek symbol epsilon is used by Planck). In this example, N = 10 and P = 100. Imagine dividing $100 between 10 people. Person number 1 gets $7, person 2 gets $38 and so on. Planck is thinking about energy. We can think about money instead.

Or even "quality", such as number of "defects" in each car being surveyed, or the "opportunities" for defects in each product, and so on. Quantum theory can be extended readily, if we use our imagination.

Planck tells us that many different arrangements of the ten numbers in the second row (many different energy configurations, or money configurations) can be conceived, for the same total energy of 100e units (or same total revenues, or profits, the goal set by a company). And, he says, even if the same numbers appear in the second row, but in a different order, something has changed. This something is what Planck says must be called entropy.

So, there is another property entropy. Let the average entropy be S and the total entropy is then equal to NS. Now, Planck defines a new quantity called temperature T. What is temperature? Very simple. T = NU/NS = U/S.

Is it too much of a stretch now to say that we have the same entropy lurking in our problem and that our company also exhibits something called temperature? After we use such terms figuratively when we talk about a hot player, a hot movie, a hot stock, and so on. May be we can quantify this concept with some "numbers" now if we learn to quantify the idea of entropy. Let's see what Planck does next with entropy, or chaos, or disorder in his system of N resonators, particles (or for us N units sold, or N products and services offered).

Planck invokes the following equation for entropy S. This, amazingly, has to do with statistics and elementary combinatorial analysis that everyone interested in "quality" and SPC/SQC/TQM/Six Sigma, etc. is interested in.

S = k ln (Omega) ........(3)

where Omega = (N + P - 1)! / P! (N - 1)! ........(4)

here k is a proportionality constant. When Omega = 1, S = 0 since the natural logarithm of unity is zero. To understand the meaning of Omega think of the simpler combinatorial formula,

Omega = N! / r! (N - r)! .........(5)

This is the number of combinations of N objects of which r are identical. Further explanations for equation 4 may be found in Longair's book Theoretical concepts in physics (page 218, Figure 10.1, Cambridge Univ. Press, 1994) , and in the college textbook by Halliday, Resnick and Walker (Fundamentals of Physics, 1997, pages 521 to 525).

In Planck's problem, N and P are very large whole numbers. After a few simple and straghtforward steps, Planck approximates the factorials uses exponents (N- 1)! + N^N, P! = P^P and (N + P - 1)! = (N + P)^ (N + P). He is thus able to arrive at the formula relating entropy S and the average U, or in our case the average Y. This is given below. Interested Covers can have some fun and derive this result. It is actually quite simple to do.

The average Y = e / [exp(e/kT) - 1] ........(6)

Here "exp" means the exponential function. The average value of any property (or entity) of interest, or the average revenue, cost, profits etc. depend on three quantities. The elementary unit called "e", the constant k and what is called the temperature T.

What is "e" that has not yet been clarified? In Planck's theory, he sets e = hf where h is another constant (we now call it Planck's constant) and f is the frequency of vibration of the resonator. Now, we can do the same thing with our elementary "e" which is an elementary "quantum of money" instead of an elementary "quantum of energy".

So, I say, let e = hx where h is a new constant and x is some unknown property, or variable that we can observe, measure, and quantify using numbers, that affects profits, revenues, costs, defects, etc.

Hence, the general law is Y = hx /[exp(hx/kT) - 1] ........(7)


Next, we come to Einstein's extension of Planck's theory and his idea of a work function. This takes us to the simpler equation y = hx + c = hx - W where W is the work function. The idea of the work function has been discussed in my post on Law relating Views and Replies.

So, here we have a generalization of Planck's law. This, I believe, can be used to "design" corporations to deliver a rated amount of profits, just like we design heat engines to deliver a certain rated horsepower. From the heat engine, we thus go to what I call a Profits Engine.

Now, enough theoretical nonsense. Prove it, you say. I hear you. We are now ready to discuss how companies like Google, Microsoft and Toyota behave. I have picked these companies since these are examples of companies that are obviously "excellent" in some sense. Google has become a houshold word since its founding in 1998, as are Microsoft and Toyota, soon to become, they say, the largest automotive company in the world, surpassing GM.

We can learn how quantum physics of Planck can be applied, even in the business world, by studying the profits-revenues data for these companies. Cheers! It is reeaaalllly very very

Charmed

P. S. For completeness, let me add that Entropy is also defined as the negative of information. According to Brillouin, "Entropy measures the lack of information about the exact state of a system." (Quote from page 270 in the text Heat and Thermodynamics, by Zemansky.) Hence, as entropy S increases, information I decreases. The equation relation entropy and information is a simple one and is written as S = S0 - I. When I = 0, the entropy S = S0 has its maximum value. This is the "reference value", just like we must use some kind of a reference level for measuring heights, energies, temperature, etc. As information I increases, S decreases.

 

[Murph095]

I would like to put some actual numbers to this to check its usefullness. While I will never claim to be an avid quantum physics student but I would like to understand the business aspect to it. Let me disect this post a little more and see if I can formulate something. i will post back.

Murph

 

[Charmed]

Numbers with Murph

Hi Murph:

Let me know how your numbers turn out.

Now, imagine this. There are 10 divisions in a large corporation, each producing revenues (or profits). This is like 10 particles in different energy states in Planck's problem. What is the "average" energy of the system of 10 (more generally N) particles? This is the problem Planck tried to solve more than 100 years ago. Now, I think, with some creative thinking, we can use his ideas to solve many other problems in the business world.

What is the "average" revenue (or profits) produced by the 10 divisions of the large corporation? Can we answer this question?

Now, we must also bear in mind an inviolable law of the business world, viz., Profits = (Revenues - Costs). Planck invokes a similar inviolable law, which is called the first law of thermodynamics, to solve his problem.

In Planck's case, the law is written as dW = (dQ - dU) where dQ is the energy added in the form of heat (similar to revenues), dU is the change in the internal energy (similar to costs) and dW is the useful work done (similar to profits). Planck looks at the idealized case when no work is done (dW = 0) and all of the heat added goes into raising the internal energy of the body (dQ = dU). Then he invokes one more law, the entropy law, or the second law of thermodynamics. The temperature of the system is related to entropy and the heat added by dQ = TdS.

Now, we too can attribute two new properties called entropy and temperature to the business or economic system of interest and the analogy is complete. You then get the same equations that Planck got with a new meaning to each mathematical symbol with exact analogies to the business, economic world. Today, entropy is often related to "information" or the extent of "chaos" or "disorder" in a system. This idea of entropy or chaos is likewise a useful one for the business/economic world. Likewise, we can extend the analogy to many other systems where now produce "numbers" to understand how a system is behaving. For example, we produce numbers for 10 salesperson, or 10 different vehicle types sold by a car dealership, and so on. This there "entropy" in these numbers? Is there "energy" in this system? Is there what we call a "temperature" in this system? Yes, all these properties also exist in the non-physical world. Remember also dQ = CdT where dT is the change in temperature and C is the heat capacity of the system. (Planck idea of an elementary energy quantum was later used by Einstein to explain how specific heat changes with temperature. The specific heat is the heat capacity per unit mass. Now, we must understand what mass is in the business world as well. What is mass in physics?**)

Again, thanks and let's take this somewhere beyond just this post.


** What is mass in physics? The following is meant for the bravest.

If you think about it, Newton was the first to define mass. He does so on page 1, paragraph 1, Definition 1 of his Principia. Newton talks about all types of bodies, even porous bodies like packed snow, to enunciate the idea of mass. He then appeals to the process of weighing to give a precise meaning to the term mass. To Newton, mass = density times volume. If the volume is doubled, mass doubles. If volume is quadrupled, mas is quadrupled. And so on. This is how Newton introduces the idea of mass in physics. Einstein changes the meaning of time, by appealing to light as the universal body with which to build a clock. If we assume that the speed of light is a universal constant, this is one of the two postulates of Einstein's theory of relativity, we can build a clock using the equation t = 2D/c where D is the distance between two mirrors and c is the speed of light. All such clocks, let's call them Einstein-clocks, are synchornized. Today, all cellphone clocks are synchornized - at least to the second - because we use the GPS and the clock in the Naval Observatory at Washington DC, as our reference clocks. Such a synchronization was not possible 100 years ago. Check the cellphone clocks with your friends and you will see all show the same time. This has never before been possible in all of human history!

Now, if every observer, regardless of his or her state of motion (imagine people in different moving cars, or different moving aircrafts, or in different moving cars and aircrafts, and ships) measures the same speed of light, what would happen? The velocity addition law when the observer is moving is v' = v - U where v is the velocity of a moving body and U is the velocity of the observer. If a car is moving at a speed v = 100 mph, and the observer (a cop) is moving at 90 mph, the moving observer, say a cop in a moving vehicle, measures a speed v' = 100 - 90 = 10 mph. A stationary observer (a cop on the roadside in a stationary vehicle) would measure a speed of v = 100 mph. We arrive at this law using v = x/t and v' = x'/t' with primes denoting corresponding quantities for the moving observer. We know that distances will change when we are moving. The moving cop will cover a distance Ut in a time t. Hence, for the moving cop, the distance to be used in the speed calculation becomes x' = (x - Ut). But, we tend to assume that the times shown by the clocks for all observers are the same and so t = t' for all observers. This is physics before Einsten. Now Einstein tells us - not true! We have simply assumed that t = t'. This must be reconsidered, since the speed of light c is a universal constant.

If the clocks carried by the stationary and moving cop show different times, because they are moving at different speeds, what would happen? Starting with the premise of a constant speed of light (or what is called a postulate of the theory of relativity), Einstein derives a new law relating v' and v which includes U and the speed of light c. This is the starting point of the theory of relativity and the revolutionary ideas that changed physics. Einstein shows that v' = x'/t' = (v - U)/[1 - (Uv/c*c)]. Obviously, Einstein was able to derive a new relation between t and t' for the two observers, given that both measure the same speed of light c.

When U is very small compared to c the speed of light, or when v is small compared to the speed of light, the denominator 1 - (Uv/c*c) is approximately equal to 1. Then we recover the old rule v' = (v - U), for the adding of velocities. Einstein new law applies when bodies are moving close to the speed of light. Otherwise, even for bodies like the space shuttle orbiting the earth, the difference is quite small. This new law can actually be derived using just a knowledge of high school algebra and math.

Now, Einstein tells us that because of this new rule for addition of velocities, we actually must also redefine the law for addition of accelerations. Or, more generally, we must redefine what is called the parallelogram law for addition of vectors (velocity and acceleration are vector quantities, i.e., they have both magnitude and direction). Force is also a vector quantity. Forces acting in the same direction add up like numbers do. But forces acting in different directions do not. Later in the same June 1905 paper, Einstein proposes a modification of Newton's definition for force.

Einstein defines the force acting on a charged body (like a moving electron), using the equation force = mass X acceleration, or F = m (dv/dt) = ma where a = dv/dt is the acceleration. To Newton, however, force is equal to the rate of change of momentum, or F = dp/dt where p = mv is the momentum of a body with m being the mass and v its velocity and t is time. Hence, to Newton, force F = m(dv/dt) + v(dm/dt).

After changing the meaning of time, Einstein now stumbles upon the concept of force and makes a subtle change - indeed very important change. After redefining time, which redefines velocities and accelerations, he must finally address the idea of a force. Why does Einstein throw out the second term v(dm/dt)? This was one of the burning issues in physics 100 years ago.

Actually, Max Planck who developed quantum physics, played an important role here. He introduced a "compromise" definition for force, starting with m = bm0, which is the relativistic definition of force. Here m0 is the mass when the body is at rest and m is the mass when the body is in motion. The two are related by "b" where "b" depends on the velocity of the body. Placnk then does some simple math (based on elementary calculus) to show that Einstein's definition of force and Newton's definition of force can be reconciled. But, Planck himself overlooks some other subtle points regarding fthe orces acting on the electron that Einstein himself cautions us about in the June 1905 paper. The forces acting on the electron in the direction of motion (say x-direction) and in the two transverse directions (y and z directions) are not the same. This is what Einstein tells us. The forces are different, since accelerations are different in the x direction and the y and z directions.

Einstein throws out the term v(dm/dt) since the mass m must be a universal constant, having the same value for all observers. This is a consequence of his first postulate - viz., the laws of physics must be the same for all observers. Einstein talks about his first postulate when he discusses the idea of a force acting on a charged body. For an electric charge force F = qE where q is the charge and E the electric field strength. The charge q must have the same value for all observers. Einstein talks about charged bodies, like fast moving electrons, to develop his theory. Then he is confronted with the idea of mass, which was never introduced in the theory. He compares his new approach to the theory of a moving electron which had been developed by the Nobel laureate Lorentz, who had already received the Nobel Prize in 1902 and Einstein was an unknown in 1905.

To Lorentz, who was also working on the theory of the moving electron from a different angle, the mass of the electron must increase with its speed. Lorentz was using the old rules t' = t and v' = (v - U) when he developed his theory of the electron. So, he was forced to conclude that mass itself must change and that it changes differently in the x, y, and z directions. A round electron will therefore become elongated in the direction of motion. Pretty strange indeed!

Is an electron round, or a hard sphere? We do not know, but that is what Lorentz assumed. The round body called the electron has a certain mass m contained within it and it also has a certain amount of charge q. The charge q is constant but the mass changes!

To Einstein, the mass m must also be a constant. Only the acceleration and velocity of the electron that must be recalculated since an observer moving with the electron and a stationary observer would both measure exactly the same speed of light, c. So, Einstein says force = mass times acceleration and acceleration = dv/dt. LIke charge q in his equations, the mass m must be a constant. This is what we mean by "inertial" mass in physics. Newton is talking about "gravitational" mass since he uses a pan balance, or the force of gravity, to define the meaning of mass m. Einstein's meaning of mass and Newton's meaning of mass are to be carefully distinguished. The stumbling block is then the idea of force and how this should be defined in relativity. One of the texts on this subject considers three possible definitions for force. Einstein, being the master he was, picks the simplest of three definitions. He leaves the "confusion" created by the moving electron to the force, or acceleration, and keeps its mass a constant. So, what is happening with an electron moving at 99% or 90% of the speed of light? Such particles have now been observed. All of physics must be reconsidered if we accept what Einstein is telling us - that the mass m is a universal constant. The electron, I think, must be losing its charge and becomes a chargeless body when it attains the speed of light. One can actually develop a new theory along these lines - but no one dared to yet.

The 100th anniversary of these momentous events in physics is approaching - in 2005. Methinks, however, there are some issues that are lurking in the background and one cannot overlook what Newton did when he gave a scientific meaning to the word "mass" in physics. Newton also appeals to the idea of weighing, or the common pan balance, that humans had been using long before Newton to show "mass" can be measure precisely. He then talks about a pendulum, with bobs of various masses and relates this to the idea of time.

Can we overlook what Newton does with his ideas of mass and the pendulums. Today we have atomic clocks that can measure time with an accuracy of a femtosecond (one millionth of a billionth of a second) or even an attosecond (one billionth of a billionth of a second). So, Einstein's idea can be put to a critical test, if we read the June 1905 paper once again, and very very carefully. Einstein talks about how time will change as we move from the equator to the poles, along the same longitude. (The time zones that we are familiar with are due to differences in time at the same latitude with changes in the longitude.)

But Einstein tells us in his June 1905 paper that we cannot use a pendulum clock to test his theory. This means we must build the Einstein-clock with two mirrors held at fixed distance D. No such clock has ever been built and used to test the theory of relativity, and confirm the postulate of constancy of the speed of light for all observers. Amazingly, Einstein shows that the parallelogram law for addition of vectors - discovered by Archimedes - must be rejected, if the postulate of the constant speed of light is to be accepted. Newton talks about the parallelogram law in his Principia, after he introduces the idea of forces. Now, Einstein tells us we must abandon the parallelogram law for addition of velocities. Yet, he retains the parallelogram law for addition of other vectors, like vectors dealing with magnetic and electric fields. Even some physicists are still pretty confused about these basics, IMHO. We must actually reconsider all the electron experiments, starting with J. J. Thomson's experiments, performed in the late 19th and early 20th century. There is a still a lot to be learned there, once we accept the idea of a constant mass m for the electron, like Einstein tells us. But wiser heads prevailed and the young Einstein was thinking about gravity for the rest of his life. I wish he had debated the electron experiments, with the same vigour that he did when he debated Niels Bohr to show that quantum physics is not acceptable. GOD DOES NOT PLAY DICE, said Einstein.)

 

[Murph095]

Dang buddy, that was longer than my last thesis paper anyways like I said I will disect and give my best guess at some numbers and postulate what I can, stay tuned.

Murph

 

[Charmed]

A slightly modified discussion

** What is mass in physics? The following is meant for the bravest.

This is a slightly revised discussion. A few additional points have been added here. Thanks Covers and have a Merry Christmas and a Happy Holiday season. Drive carefully and hang on to your keys in coffee shops!

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If you think about it, Newton was the first to give a scientific meaning to the term mass. He does so on Page 1, Paragraph 1, Definition 1 of his momentous book called The Principia. In this opening paragraph, Newton talks about all types of bodies, even porous bodies like packed snow, to enunciate the idea of mass. He then appeals to the process of weighing to give a precise “measure” for what we call mass. The mass of a body m = W/g where W is the weight of the body measured using “weights” like pounds and ounces that had been developed by humans long before Newton. The term “troy ounce” comes from the fact that merchants from the ancient city of Troy (not where I live, we are the Troy of the 21st century) were known to cheat and sold less than the “standard” ounce of gold (or whatever precious commodity it was) when they traded.

The small g in m = W/g is a constant, the acceleration produced by the earth’s gravitational force (which existed before Newton) when the mass is placed on one pan of the balance. If M is the standard or reference mass, placed on the other pan of the balance, W = Mg = mg when the balance beam is horizontal and the two pans do not move anymore. Thus, the unknown mass m is equal to the standard or known mass M. Hence, the process of weighing does not require a knowledge of the unknown g, the gravitational acceleration. Newton thought about all these things before he enunciated the concept of mass in physics.

The precise value of g can now be determined experimentally and varies slightly from one place to the other on the surface of the earth. However, this did not prevent humans from using a pan balance to compare different “masses” or bodies with different weights. Today, we use the kilogram as the unit of mass. Forces are measured in units called newtons, named after Newton, of course. Grams, milligrams, tons, etc. are multiplies or submultiples of the kilogram. Weight, which is simply the force of gravity acting on the mass, is measured in Newton. If mass m = 1 kg and it is subject to an acceleration of 1 meter per second squared, the force is one newton. So, if a mass of 1 kg is subject to the standard gravitational acceleration of g = 9.81 meters per second squared, its weight is 9.81 newtons. When we use a spring balance, we still rely on the force of gravity with one pan being replaced by a spring and a linear scale for the mass. The linear scale was developed using various standard masses and the old pan balance. Modern electronic balances, with digital readouts, seem to make the spring invisible. But Mother Nature and gravity are doing exactly what they did before Newton arrived and conceived the ideas of mass, force, and the inverse square law for the force of gravity.

Thus, to Newton, mass = density times volume. If the volume is doubled, mass doubles. If volume is quadrupled, mass is quadrupled. And so on. This is how Newton introduces the idea of mass in physics.

After introducing the concept of mass, Newton enunciates the concept of a force and three laws, known as Newton’s laws of motion, which describe how forces act on bodies to change their state of motion. Thus, from mass we go to the idea of a force.

The notions of distance (x), time (t), velocity (v) and acceleration (a) were well understood before Newton and indeed given a scientific meaning by Galileo in the generation before Newton. Galileo discovered the law v = at for bodies falling near the surface of the earth. When a body falls, it gains speed. Its velocity increases linearly with time t. The ratio v/t = a = acceleration is constant for all bodies and independent of the mass m. The magnitude of “a” depends on where the body is falling. (Bodies falling on the surface of the moon or Mars have a different “a” compared to bodies falling near the surface of the earth.)

Einstein, on the other hand, changes the very meaning of time, by appealing to light as the body, which moves with a fixed speed, for all observers. Einstein felt that light, rather than the pendulum, could be used as the universal reference body to build clocks. The pendulum clock uses the equation T = 2Pi[(L/g)^0.5] where T is the period of the pendulum and “g” is the constant determined experimentally. If “g” varies, the period T will vary slightly, from place on the earth to the other, even if the length L of the pendulum is fixed. Astronomers of Newton’s times had actually performed such experiments with pendulums. A pendulum of the same fixed length L took different time to complete the same fixed number of vibrations in the tropics than it did in London or in Paris. Indeed, Newton discusses the significance of these early 17th century observations in his Principia.

But, if we assume that the speed of light is a universal constant, this is one of the two postulates of Einstein's theory of relativity, we can build a clock using the equation t = 2D/c where D is the distance between two mirrors and c is the speed of light. The clock ticks each time a light ray bounces back and forth between the two mirrors. The speed c is assumed to be a known quantity. We build many such identical clocks. All such clocks, let's call them Einstein-clocks, are then synchronized. Observers moving with different speeds use such clocks to perform experiments, such as the experiment discussed below with a stationary and a moving cop. The only difference is that we are dealing with fast moving spaceships (Cybercops in the 21st or 22nd century).

Today, all cellphone clocks are synchronized - at least to the second - because we use the GPS and the clock in the Naval Observatory at Washington DC, as our reference clocks. Such synchronization was not possible 100 years ago. Check the cellphone clocks with your friends and you will see all show the same time. This has never before been possible in all of human history! (However, one cannot overlook the basis of synchronization. Somewhere along the way, the time shown by the sundial was used to synchronize pendulum clocks and so on to the atomic clocks of modern days.)

After enunciating the two postulates of his new theory, Einstein then discusses a number of experiments with charged bodies moving at very high speeds, approaching the speed of light. If every observer, regardless of his or her state of motion (imagine people in different moving cars, or different moving aircrafts, or in different moving cars and aircrafts, and ships) measures the same speed of light, what would happen? The velocity addition law when the observer is moving is v' = v - U where v is the velocity of a moving body and U is the velocity of the observer. If a car is moving at a speed v = 100 mph, and the observer (a cop) is moving at 90 mph, the moving observer, say a cop in a moving vehicle, measures a speed v' = 100 - 90 = 10 mph. A stationary observer (a cop on the roadside in a stationary vehicle) would measure a speed of v = 100 mph.

We arrive at this law using v = x/t and v' = x'/t' with primes denoting corresponding quantities for the moving observer. We know that distances will change when we are moving. The moving cop will cover a distance Ut in a time t. Hence, for the moving cop, the distance to be used in the speed calculation becomes x' = (x - Ut). But, we tend to assume that the times shown by the clocks for all observers are the same and so t = t' for all observers. This is physics before Einstein.

Einstein tells us this cannot be true! Why? We have simply assumed that t = t', which means v’ = x’/t’ = x’/t = (x – Ut)/t = (x/t) – U = (v – U). The velocity v = x/t = distance x divided by time t. The velocity addition law must be reconsidered, since the speed of light c is a universal constant. For light, and light alone, Einstein says that the speed v = c = x/t = x’/t’ since all observers will measure the same speed. Since c is fixed and has the same value for all observers and we all agree (even before Einstein came along) that the distance x’ = (x – Ut) for the moving observer, it follows that t’ must differ from t when U is nonzero, if c is fixed for all observers.

If the clocks carried by the stationary and moving cop show different times, because they are moving at different speeds, what would happen? Starting with the premise of a constant speed of light (or what is called a postulate of the theory of relativity), Einstein derives a new law relating v' and v which includes U and the speed of light c. This is the starting point of the theory of relativity and the revolutionary ideas that changed physics. Einstein shows that v' = x'/t' = (v - U)/[1 - (Uv/c*c)]. Obviously, Einstein was able to derive a new relation between t and t' for the two observers, given that both measure the same speed of light c.

When U is very small compared to c the speed of light, or when v is small compared to the speed of light, the denominator 1 - (Uv/c*c) is approximately equal to 1. Then we recover the old rule v' = (v - U), for the adding of velocities. Einstein new law applies when bodies are moving close to the speed of light. Otherwise, even for bodies like the space shuttle orbiting the earth, the difference is quite small. This new law can actually be derived using just a knowledge of high school algebra and math.

Now, Einstein tells us that because of this new rule for addition of velocities, we actually must also redefine the law for addition of accelerations. Or, more generally, we must redefine what is called the parallelogram law for addition of vectors (velocity and acceleration are vector quantities, i.e., they have both magnitude and direction). Force is also a vector quantity. Forces acting in the same direction add up like numbers do. But forces acting in different directions do not. Later in the same June 1905 paper, Einstein proposes a modification of Newton's definition for force.

Einstein defines the force acting on a charged body (like a moving electron), using the equation force = mass X acceleration, or F = m (dv/dt) = ma where a = dv/dt is the acceleration. To Newton, however, force is equal to the rate of change of momentum, or F = dp/dt where p = mv is the momentum of a body with m being the mass and v its velocity and t is time. Hence, to Newton, force F = m(dv/dt) + v(dm/dt).

After changing the meaning of time, Einstein must finally deal with the concept of a force and makes a subtle change - indeed very important change. After redefining time, which redefines velocities and accelerations, he must finally address the idea of a force. Why does Einstein throw out the second term v(dm/dt)? This was one of the burning issues in physics 100 years ago.

Einstein’s logic is then as follows. First we consider the meaning of time, using the speed of light as a universal constant. Then we derive the law relating velocity and accelerations for moving observers. Then he considers the motion of charged bodies, like electrons, moving at very high speeds. These bodies (or particles) obey Maxwell’s laws of electromagnetism. The speed of light is a universal constant, a conclusion Einstein actually draws from Maxwell’s theory and the equation Maxwell had derived for the speed of light. Maxwell had shown that the speed of light (in a perfect vacuum) depends only on two constants that tell us how electric and magnetic forces act on charged bodies.

Thus, Einstein arrives at the concept of a force (now acting on charged bodies, instead of electrically neutral bodies in Newton’s theory), which then leads to the conclusion mass = force/acceleration. The idea of mass cannot be introduced in any other manner since Einstein’s theory deals with bodies with charge q while Newton’s theory deals with bodies with mass m and zero charge (q = 0). Einstein knows that bodies like electrons with charge q also have a mass m. What is the mass of the electron? This is the question that Einstein tries to answer in the last three sections of his June 1905 paper, which should be studied very carefully by those who are interested in understand relativity.

Max Planck who conceived quantum physics, played an important role in these early 20th century developments. He introduced a "compromise" definition for force, starting with m = bm0, which is the relativistic definition of force. Here m0 is the mass when the body is at rest and m is the mass when the body is in motion. The two are related by "b" where "b" depends on the velocity of the body. Planck then does some simple math (based on elementary calculus) to show that Einstein's definition of force and Newton's definition of force can be reconciled. But, Planck himself overlooks some other subtle points regarding the forces acting on the electron that Einstein himself cautions us about in the June 1905 paper. The forces acting on the electron in the direction of motion (say x-direction) and in the two transverse directions (y and z directions) are not the same. This is what Einstein tells us. The forces are different, since accelerations are different in the x direction and the y and z directions.

Einstein throws out the term v(dm/dt) since the mass m must be a universal constant, having the same value for all observers. This is a consequence of his first postulate - viz., the laws of physics must be the same for all observers. Einstein talks about his first postulate when he discusses the idea of a force acting on a charged body. For an electric charge force F = qE where q is the charge and E the electric field strength. The charge q must have the same value for all observers. Einstein talks about charged bodies, like fast moving electrons, to develop his theory. Then he is confronted with the idea of mass, which was never introduced in the theory. He compares his new approach to the theory of a moving electron which had been developed by the Nobel laureate Lorentz, who had already received the Nobel Prize in 1902 and Einstein was an unknown in 1905.

To Lorentz, who was also working on the theory of the moving electron from a different angle, the mass of the electron must increase with its speed. Lorentz was using the old rules t' = t and v' = (v - U) when he developed his theory of the electron. So, he was forced to conclude that mass itself must change and that it changes differently in the x, y, and z directions. A round electron will therefore become elongated in the direction of motion. Pretty strange indeed!

Is an electron round, or a hard sphere? We do not know, but that is what Lorentz assumed. The round body called the electron has a certain mass m contained within it and it also has a certain amount of charge q. The charge q is constant but the mass changes!

To Einstein, the mass m must also be a constant. Only the acceleration and velocity of the electron that must be recalculated since an observer moving with the electron and a stationary observer would both measure exactly the same speed of light, c. So, Einstein says force = mass times acceleration and acceleration = dv/dt. Like charge q in his equations, the mass m must be a constant. This is what we mean by "inertial" mass in physics. Newton is talking about "gravitational" mass since he uses a pan balance, or the force of gravity, to define the meaning of mass m. Einstein's meaning of mass and Newton's meaning of mass are to be carefully distinguished. The stumbling block is then the idea of force and how this should be defined in relativity.

One of the texts on the theory of relativity discusses three possible definitions for force. Einstein, being the master he was, picks the simplest of three definitions. He leaves the "confusion" created by the moving electron to the force, or acceleration, and keeps its mass a constant. So, what is happening with an electron moving at 99% or 90% of the speed of light? Such particles have now been observed. All of physics must be reconsidered if we accept what Einstein is telling us - that the mass m is a universal constant. The electron, I think, must be losing its charge and becomes a chargeless body when it attains the speed of light. One can actually develop a new theory along these lines - but no one dared to yet.

The 100th anniversary of this momentous event in physics is approaching - in 2005. Methinks, however, there are some issues that are lurking in the background and one cannot overlook what Newton did when he gave a scientific meaning to the word "mass" in physics. Newton also appeals to the idea of weighing, or the common pan balance, that humans had been using long before Newton to show "mass" can be measure precisely. He then talks about a pendulum, with bobs of various masses and relates this to the idea of time.

Can we overlook what Newton does with his ideas of mass and the pendulums? Today we have atomic clocks that can measure time with an accuracy of a femtosecond (one millionth of a billionth of a second) or even an attosecond (one billionth of a billionth of a second). So, Einstein's idea can be put to a critical test, if we read the June 1905 paper once again, and very very carefully. Einstein talks about how time will change as we move from the equator to the poles, along the same longitude. (The time zones that we are familiar with are due to differences in time at the same latitude with changes in the longitude.)

But Einstein tells us in his June 1905 paper that we cannot use a pendulum clock to test his theory. This means we must build the Einstein-clock with two mirrors held at fixed distance D. No such clock has ever been built and used to test the theory of relativity, and confirm the postulate of constancy of the speed of light for all observers.

Amazingly, Einstein shows that the parallelogram law for addition of vectors - discovered by Archimedes - must be rejected, if the postulate of the constant speed of light is to be accepted. Newton talks about the parallelogram law in his Principia, after he introduces the idea of a force and the three law describing how forces act on bodies. But, Einstein tells us that we must abandon the parallelogram law for addition of velocities. The implications of this statement, which we find early on in Einstein’s June 1905 paper, have still not been fully grasped. Yet, he retains the parallelogram law for addition of other vectors, like vectors dealing with magnetic and electric fields. Even some physicists are still pretty confused about these basics, IMHO.

We must actually reconsider all the electron experiments, starting with J. J. Thomson's experiments, performed in the late 19th and early 20th century. There is a still a lot to be learned there, once we accept the idea of a constant mass m for the electron, like Einstein tells us. But wiser heads prevailed in the 20th century and the idea of mass increasing with velocity was universally accepted. The young Einstein started thinking about gravity, sometime after 1905, or even 1908, and it continued for the rest of his life. I wish Einstein had debated and critiqued the electron experiments (the 1908 experiments by Bucherer led to the acceptance of Einstein’s theory), with the same vigor that he did when he debated Niels Bohr to show that some of the fundamental assumptions of quantum physics are just not acceptable. GOD DOES NOT PLAY DICE, said Einstein.

P. S. Galileo discovered the law v = at which means time t = v/a with “a” being the acceleration of a body rolling down an inclined plane. For a body falling vertically, the acceleration a = g and has slightly different values at different points on the earth’s surface. Can we use this equation as the basis for building a new clock? If not, why not? The answer is obvious. Time t = v/a and we have introduced two new concepts velocity v and acceleration a. However, to measure “v” precisely we need a clock. To measure “a” precisely we need a clock. Now, apply the same logic and think about Maxwell’s equation for the speed of light, derived from the theory of electromagnetism, in exactly the same way. Can we really build Einstein clock and use the equation t = 2D/c to redefine our basis for the unit of time?

From Maxwell's theory the speed of an electromagnetic wave, c = (epislon*mu)^0.5 where epsilon and mu are two new constants that enter into the fundamental equations describing electric and magnetic forces. The inverse square law for the electrical force between two charged bodies introduces the proportionality constant "epsilon" while the magnetic force law (the force between the north and south pole of two bar magnets, as an example) introduces the proportionality constant "mu". When Maxwell substituted the then known values of epsilon and mu into his new equation, he found that c calculated in this fashion was very nearly (but not exactly) equal to the then known value of the speed of light. The speed of light had been determined using both astronomical and terrestrial methods. So Maxwell felt that light must be some kind of an electromagnetic wave. Einstein was born in 1879, the same year that Maxwell died, at the young age of 49. Maxwell's theory made a deep impression on the young Einstein. Indeed Einstein theory of gravitation was an attempt to do for gravity what Maxwell did for electricity and magnetism.

 

[Tim Folkerts]

Charmed,

Have you considered submitting your posts to a forum of physicists or to a physic publications or to a local university professor? There are a lot of bright people here, but few are qualified to comment on subtle points of history, relativity and quantum mechanics. I'm sure other sources could give a lot more intellegient feedback. Perhaps you could try sci.physics.research or alt.sci.physics.new-theories.

One of the texts on the theory of relativity discusses three possible definitions for force. Einstein, being the master he was, picks the simplest of three definitions. He leaves the "confusion" created by the moving electron to the force, or acceleration, and keeps its mass a constant. So, what is happening with an electron moving at 99% or 90% of the speed of light? Such particles have now been observed. All of physics must be reconsidered if we accept what Einstein is telling us - that the mass m is a universal constant. The electron, I think, must be losing its charge and becomes a chargeless body when it attains the speed of light. One can actually develop a new theory along these lines - but no one dared to yet.

This seems to be your boldest claim. So I challenge you to be that prophet. Develop and publish that theory so that 100 years from now people will discuss you alongside Einstein, Planck, and Maxwell.

Hey, you could be "a prophet of new meanings" instead of "a new meaning of profits".


Tim F


P.S. If you are going to discuss "What is mass?" you might want to at least think about two other ideas:
1) The role of the Higgs particle in creating mass in the first place.
2) The equivalence of inertial and gravitational mass.

 

[Charmed]

Faster than light and Thanks Tim

Dear Tim F:

Thanks for the encouragement. I studied relativity, mostly for my own pleasure, over the last few years, in my spare time. I am not a physicist and have no formal degrees in physics. My degrees are in engineering but ever since I was a kid I wanted to understand what Einstein meant by time and why time (the rate at which clocks tick) depends on the state of motion of the observer.

As some of you might already have noted, I find that the simple equation y = hx + c with the nonzero c has a great many implications. We often tend to overlook the significance of the nonzero c. We use simple y/x ratios and percentages to draw important many far reaching conclusions - including many every day conclusions in the business world. My main interest, since 1998, had always been Planck's theory and how it can be extended to the business world.

I got interested in relativity only after I realized one fine day that Einstein was also using two simple linear equations to derive his two key equations for space and time in the theory of relativity. These can be expressed as:

x' = alpha(x - Ut)

t' = (beta)t + (gamma)x

With the first equation, Einstein modifies the familiar x' = (x - Ut) by introducing the factor "alpha". In old theory alpha = 1. With the second equation, Einstein modifies the meaning of time t. He also "mixes up" the space coordinate x and the time coordinate t with this equation. In the old theory, gamma = 0 and beta = 1.

Starting with these two simple linear equations, Einstein fixes the values of the three unknowns alpha, beta, and gamma, by considering the motion of an expanding spherical light front. Two observers are placed at the origin. The light front spreads out at the time t = 0 and t' = 0. At exactly the times t = 0 and t' = 0, one of the two observers starts moving, along the positive x and x' direction, with a fixed speed U while the other is stationary. Now, what would happen? Einstein solves this problem and determines the values of the three unknowns.

Anyone with high school level knowledge of algebraic equations can learn to derive the key equations of relativity. I recommend Bergmann's treatise, Introduction to theory of relativity. Bergmann was a disciple of Einstein and worked closely with him. Einstein has written a nice foreword to Bergmann's book. I got this book in 1998, mainly because I found that Einstein had personally endorsed it. After I learned to derive the key equations, I have spent a lot of "time" thinking about the assumptions that are built into the derivation of the equations relating x, x', t, and t'. How can we solve for three unknowns with just two equations to start with? Obviously much more is going on here.

Normally, most students do not spend time thinking about assumptions made to derive mathematical equations. As students we are more fascinated by the fact that we have mastered the derivation. Assumptions made before ink meets the paper must also be fully understood. One must actually think seriously about these assumptions and how Einstein then defines mass and then takes us to E = m(c*c). It is a fascinating journey.

But, IMHO there are many as yet unresolved problems. The values of alpha, beta, and gamma, if at all, ONLY apply to light. Einstein makes a giant leap of logic by using the same values for all moving bodies. I find this extension (very humbly speaking) a great leap and I was shaking my hand when I first realized that. But, I went along with the great master, to understand how he arrived at E = m (c*c). Then I realized that some more assumptions have been made through out the whole logical exercise that Einstein takes us through in his June 1905 and the September 1905 papers.

Take for instance the relation between v and v' which is now v' = (v - U)//[1- (Uv/(c*c)]. What is U? It is the velocity of the moving observer. What is v? it is the velocity of the moving body. If one observer, who is taken as stationary, measures the velocity v, the other will measure the velocity v'. But, Einstein is using the values of alpha, beta, and gamma that he derived for a light front. Can this be used to study the motion of a body moving at less than the speed of light? Einstein now tells us that the speed of light is the upper limit of speeds that can be attained by ANY moving body. Is this a foregone conclusion? Sometimes we do not realize that the conclusions are actually built into the equations that we derive (especially now with all kinds of computer simulations). Einstein’s extension of the logic (using the values of alpha, beta, and gamma from light to all other bodies) keeps bothering me here. Must we accept the speed of light as the upper limit? Or, can we build a rocket that can travel at speeds greater than the speed of light? I think we can and there is nothing in the theory of relativity (t and t’ are not the same, this is the simplest meaning of relativity) that stops us from building rockets that can exceed the speed of light.

Anyway, Tim, if you study the June 1905 paper, you will find that Einstein repeatedly sets U = v in all of this equations. This is called the "rest frame" in the theory of relativity. The observer is moving with a velocity U, which is now exactly matched to the velocity v of the body whose motion is being studied. This is like the moving cop observing you in a vehicle that is moving at the same speed as your own vehicle.

Now, ask yourself this question. What would happen if U is not exactly equal to v? I did and can show that E = Lambda[m*c*c] where the factor lambda depends on the ratio U/v. In Einstein's theory, U = v and E = m(c*c) and Lambda = 1. We can do the same thing for the moving electron - the only body whose motion Einstein considers in great detail. Then you will dare to do what I noted in the previous post. I have studied all of the electron experiments and read the original papers, starting with J. J. Thomson's papers. Indeed, J. J. Thomson's Nobel lecture is worth reading. This again tells us some of the subtle assumptions made by Thomson himself when he first conceived the idea of "cathode rays" being electrically charged particles with a mass m and charge q. Thomson could only deduce the value of the ratio m/q using Newton's laws of motion. But, as I noted in the earlier post, Einstein tries to deduce the mass m for the electron using his new theory. The rest is fascinating.

Yes, I have also studied the Higgs particle issue because of this. The mass of the Higgs particle is calculated using E = m(c*c). Since there is an unknown lambda, which must be accounted for in Einstein's derivation (since U/v is not necessarily equal to 1), we must also reconsider what we are doing with the Higgs particle. Physics, IMHO, is the same state it was at the turn of the 20th century. Many physicists IMHO are not thinking about the basics and inventing new theories to fix problems that run very deep.

My desire has always to be a student and understand one of the most fascinating ideas of space and time and why we must agree that mass increases with the speed of the body. I have come to the conclusion that mass m does not increase (even Einstein told us that, we just ignored what Einstein told us in his June 1905 paper) with speed. The electron experiments tell us that m/q first decreases with increase velocity (this is what we can deduce from J. J. Thomson’s original data), reaches a plateau (which is what Bucherer found) and then increases again as v approaches c (as Kaufmann found).

What does the variation in the ratio m/q for the electron mean? If m is constant q must be changing with increasing velocity. Indeed, attempts have been made to see if q depends on the velocity. However, the experiments were limited to speeds no more than 50% of c.

Yes, we have to redo these experiments. Since I am not a university professor and have no experimental facilities, I just cannot do anything about this. I have communicated these ideas though to some of the most respected physicists whom I know and we have had some interesting email exchanges. No one has yet taken me up on giving me some experimental facility. J. J. Thomson's experiments can be repeated and are indeed being repeated in many labs. But, let's reanalyze the data all over again and plot m/q versus v/c. Then we will begin to see what I am talking about here in these posts. I would be happy to provide you with the graphs I have prepared using the original experiments of J. J. Thomson, Bucherer, and Kaufmann.

Have a great Christmas and New Year and let's enjoy the remaining 12 days.

Charmed

 

[Charmed]

Profits of a Car Dealership and Planck's Quantum Theory

Dear All:

The following is an example of how we might be able to extend Planck’s ideas to the business world. It also highlights, IMHO, many more steps that must be taken. The following data was obtained from a recent issue of Forbes magazine (June 21, 2004), see article by Fahey, The Lexux Nexus.



1 Lexus 1280 42.5 54.4
2 M. Benz 680 52.4 35.6
3 BMW 709 43.7 31.0
4 Toyota 1323 23.0 30.5
5 Honda 1171 21.7 25.4
6 Infiniti 717 34.9 25.0
7 Acura 653 31.5 20.6
8 Ford 747 23.1 17.2
9 Chevrolet 635 23.5 14.9
10 Jaguar 332 42.6 14.1
11 Cadillac 146 40.8 6.0
12 Lincoln 116 38.7 4.5
13 Chrysler 158 23.6 3.7

The numbers, going horizontally for each brand, are units sold, average selling price per unit (in $, 000s) and total sales revenues (in $, millions). Table 1 in the Micorsoft Word file reproduces the same data and may be easier to read. The ranking is based on the sales revenues.

Notice that the number of units sold varies erratically with rank. A more systematic relation can be developed by introducing the idea of a "quantum", in this case of a "quantum" of money instead of a quantum of energy.

The Microsoft Word file provides details of the analysis and some additional notes. Two figures are included in the Microsoft PowerPoint attachment. I look forward to your comments and feedback.

Charmed

P. S. To Murph095: I hope you find this example of some interest. This is my first attempt to deduce the Planck number P from the data and it certainly looks quite interesting (see Figure 2 in the attached PowerPoint file).

There are other implications of the theory, such as a fixed quantum of profits for companies in each sector of the economy, as well as a fixed quantum for the economy as a whole. This can be deduced by carefully analyzing the quarterly and the annual profits and revenues data reported by various business magazines (Business Week, Forbes, Fortune).

 

[Charmed]

In Planck's own words

Dear Covers:

The following is taken from Planck's Nobel lecture, delivered June 2, 1920. I have highlighted some of his remarks from the first and the last paragraph of his lecture. The full text may be found in the link below.

(broken link removed)

************

When I look back to the time, already twenty years ago, when the concept and magnitude of the physical quantum of action began, for the first time, to unfold from the mass of experimental facts, and again, to the long and ever tortuous path which led, finally, to its disclosure, the whole development seems to me to provide a fresh illustration of the long-since proved saying of Goethe's that man errs as long as he strives. And the whole strenuous intellectual work of an industrious research worker would appear, after all, in vain and hopeless, if he were not occasionally through some striking facts to find that he had, at the end of all his criss-cross journeys, at last accomplished at least one step which was conclusively nearer the truth. An indispensable hypothesis, even though still far from being a guarantee of success, is however the pursuit of a specific aim, whose lighted beacon, even by initial failures, is not betrayed.

.....the problem of the quantum of action will not cease to inspire research and fructify it, and the greater the difficulties which oppose its solution, the more significant it finally will show itself to be for the broadening and deepening of our whole knowledge in physics.

*********

Perhaps, economics and finance and our observations on the business world also belong in the same arena as physics, through a broadening of the meaning of entropy, energy, and temperature.

 

[Charmed]

Car dealership example: Back to Basics

Dear All (especially Murph095):

Going back to basics in the car dealership example of the previous post, I did a few simple calculations to illustrate the significance of the Planck number P. The units sold per dealership (N) vary erratically with the average selling price M. This is illustrated in Figure 3 of the attached Microsoft PowerPoint file. This is an updated version of the earlier file.

So we really have three types of dealerships - the dealership with the highest revenue (product MN), the dealership with the highest price per unit (M) and the dealership with the highest units sold (N). Ranking of dealerships should take into account all three factors.

Now let's see how sales revenue (product MN) changes as a function units sold N and the Planck number P. This is illustrated in Figures 4 and 5. We see an upward trend in the graph of Revenues versus Units Sold (Figure 4). However, there is some scatter in this graph. Mercedes Benz is actually doing much better, for the number of units sold. This is due to the higher price per unit. However the graph of revenues versus the Planck number P is perfectly linear (Figure 5). As N increases P increases if we hold the quantum "epsilon" constant. This also follows from Revenue MN = NM = P*(epsilon), the starting point of Planck's analysis, extended here to the dealership problem.

As discussed in the appendices to the previous post, entropy S, or the extent of chaos or disorder in the system is related to both N and P. The idea of temperature T follows from the idea of entropy and energy. In our problem here, money plays the role of energy. Thus, we can arrive at the ideas of both entropy and temperature as well. The constant k in the entropy relation must however be deduced. This requires some more data on the system, which is presently lacking. The data we need ultimately relates to how N varies with M for a given dealership and for the 13 dealerships taken together.

Charmed