Section 4–2 Gravitational potential energy

(Perpetual motion / Gravitational potential energy / Principle of virtual work)

Feynman derives a formula for gravitational potential energy near the surface of the earth by using a relatively simple reasoning. In this section, the three interesting concepts discussed are the perpetual motion, gravitational potential energy, and the principle of virtual work.

1. Perpetual motion:

“…there is no such thing as perpetual motion with these weight-lifting machines. (Feynman et al., 1963, section 4.2 Gravitational potential energy).”

It is remarkable that Feynman relates the law of conservation of energy to the impossibility of perpetual motion. Importantly, we must be careful in defining the concept of perpetual motion. Strictly speaking, the motions of celestial bodies such as planets only appear perpetual, but their kinetic energy is gradually decreased due to the presence of interstellar medium and solar wind. Similarly, the electric current in a superconductor is gradually decreased after some time due to the effect of “flux creep.” Thus, it is a fact rather than a hypothesis that there is no such thing as perpetual motion with weight-lifting machines. In daily lives, weight-lifting machines lift and lower weights with a result of more thermal energy generated due to the presence of frictional forces and air resistance.

Some physicists prefer to distinguish perpetual motion machines into two kinds. For example, in a textbook titled Heat and Thermodynamics, Zemansky and Dittman (1981) write that “[a] machine that creates its own energy and thus violates the first law is called perpetual motion machine of the first kind. A machine that utilizes the internal energy of only one heat reservoir, thus violating the second law, is called perpetual motion machine of the second kind (p. 147).” Feynman did not praise many textbooks, but in chapter 45 of The Feynman Lectures on Physics, he says that “There are also good equation reference books, such as Zemansky’s Heat and Thermodynamics, where one can learn more about the subject (Feynman et al., 1963, p. 45-1).”

In general, there are two classes of machines: (1) non-reversible machines which include all real machines; (2) reversible machines which are not attainable in practice by having all careful designs of bearings and levers. Feynman mentions that if Machine A is a reversible machine, it can reversibly lower one unit of weight by one unit of distance and lift a three-unit weight by a distance X. However, it is potentially confusing to first state Machine B as not necessarily reversible, and then later suggest that it could be really reversible. Perhaps Feynman could first state Machine B as a perpetual motion machine of the first kind because this machine creates its own energy and violates the law of conservation of energy. Next, he could include Machine C as a closely reversible machine that is the best machine that we can have.

2. Gravitation potential energy:

“…We call the sum of the weights times the heights gravitational potential energy—the energy which an object has because of its relationship in space, relative to the earth (Feynman et al., 1963, section 4.2 Gravitational potential energy).”

Feynman derives the formula for gravitational potential energy near the surface of the earth by using the concept of a perpetual motion machine. This is a reversible weight-lifting machine that can lift one weight and lowering another weight at the same time. It functions like a frictionless seesaw that can swing up and down perpetually. For instance, it can continue reversibly to lower an object of 1 kilogram by a certain distance d, and lift another object of m kg by the distance d/m. However, it is impossible to have perpetual motion because of the presence of friction and air resistance. Although this kind of ideal perpetual motion machine is impossible experimentally, we can deduce intuitively a numerical quantity that remains constant when a system of objects is moving up and down simultaneously.

The numerical quantity, that is a sum of the weights times the heights, is commonly known as gravitational potential energy. This formula of potential energy (weight × height) is only valid if objects are not too far away from the earth such that the weight due to a gravitational force is approximately constant. However, the gravitational force weakens as the objects are significantly higher above the ground. On the other hand, if a perpetual motion machine involves electrical forces, or we are “lifting” charge carriers away from other charged objects by using some imaginary levers, then this numerical quantity is called electrical potential energy. This is based on the general principle in which the change in energy is equal to a force times the distance moved: (energy change) = (force) × (distance moved).

3. Principle of virtual work:

“…This approach is called the principle of virtual work because in order to apply this argument we had to imagine that the structure moves a little—even though it is not really moving or even movable (Feynman et al., 1963, section 4.2 Gravitational potential energy).”

Feynman suggests that we can work out the law of “balance” (or law of the lever) with regard to the statics of complicated bridge arrangements. We can apply the principle of virtual work in which we imagine a system has moved arbitrarily in very small (infinitesimal) displacements even though the system is not really moving or movable. For example, we can initially explain that the weight W times 4 nanometers (very small displacement) down, plus 60 pounds times 2 nanometers up, plus 100 pounds times 1 nanometers add up to nothing. Next, we can multiply these very small displacements by a factor to achieve the following equation as derived by Feynman: −4W + (2)(60) + (1)(100) = 0. In short, we have applied the principle of conservation of energy by imagining very small motions.

Interestingly, Feynman modifies the triangle in the epitaph of Stevinus (see the figure below) to a right-angled triangle. In essence, the tensions of the chains due to the weights in contact with both (upper part) sides of the triangle must be the same (T₁ = T₂) at the highest point of the triangle. Note that the tensions are not the same throughout the chain because it is not massless. Additionally, the number of “circles” in the lower part of the chain does not matter and they are symmetrical.

According to Dugas, Stevinus clearly stated the principle of virtual work as “The distance traveled by the force acting is to the distance traveled by the resistance as the power of the resistance is to that of the force acting (Dugas, 1955/1988, p. 127).” This statement is applicable to systems of pulleys based on the assumption of the impossibility of perpetual motion. In a similar sense, the tensions through the chain of weights along the two sides of the triangle (as shown in the epitaph of Stevinus) must be in balance.

Questions for discussion:

1. Why there is no perpetual motion machine in the real world?

2. What is the limitation of the formula for gravitational potential energy?

3. How would you explain the epitaph of Stevinus by using the principle of virtual work?

The moral of the lesson: the formula for gravitational potential energy near the surface of the earth can be derived by using a perpetual motion machine that is not achievable in practice.

References:

1. Dugas, R. (1955/1988). A History of Mechanics (trans. by J. R. Maddox). New York: Central Book Company.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.
3. Zemansky, M. W., & Dittman, R. H. (1981). Heat and Thermodynamics. New York: McGraw-Hill.

Section 4–1 What is energy?

(Conserved quantity / Indestructible blocks / Abstract quantity)

In this chapter, Feynman says that we have no knowledge of what energy is and we do not understand the conservation of energy. In this section, he discusses the concepts of energy as a conserved quantity, indestructible blocks, and abstract quantity.

1. Conserved quantity:

“…The law is called the conservation of energy. It states that there is a certain quantity, which we call energy, that does not change in the manifold changes which nature undergoes (Feynman et al., 1963, section 4.1 What is energy?).”

According to Feynman, the law of conservation of energy governs all natural phenomena. Remarkably, there is no known exception to this law within experimental accuracy. Based on this law, the amount of energy does not change in the manifold changes which nature undergoes. In short, it states that there is a numerical quantity that remains the same even when something happens. However, the law does not provide a description of a mechanism behind the natural phenomena. It seems a strange fact that the total amount of energy of a system must always be the same whenever we calculate some number by using various formulae of energy.

Importantly, Feynman advocates strict conservation of energy instead of statistical conservation of energy. During a Messenger Lecture, Feynman elaborates that “[i]t might have been that the law of energy conservation was not right; in fact, it was proposed by Bohr for a while that perhaps the conservation law worked only statistically, on the average. But it turns out now that the other possibility is the correct one (Feynman, 1965, p. 75).” Simply put, Bohr was willing to give up the law of conservation of energy during a beta decay. On the contrary, a firm belief in the strict conservation of energy has led to the discovery of neutrinos (Dennis’ blocks under the rug).

Note: In a paper titled The quantum theory of radiation, Bohr, Kramers, and Slater (1924) write that “a statistical conservation of momentum is secured in a way quite analogous to the statistical conservation of energy in the phenomena of absorption of light (p. 174).”

2. Indestructible blocks:

“…Imagine a child, perhaps ‘Dennis the Menace,’ who has blocks which are absolutely indestructible, and cannot be divided into pieces (Feynman et al., 1963, section 4.1 What is energy?).”

Feynman distinguishes the energy in an isolated system and real system by using Dennis’ block and Bruce’s block respectively. Initially, he illustrates the law of conservation of energy by using Dennis’ indestructible blocks. Being curious, Dennis’ mother discovers the law of conservation of blocks in which the number of blocks remains the same though some could be hiding under a rug. As an exception, the increase in the number of blocks may be explained by the additional blocks that Bruce brings in. This problem can be resolved by defining an isolated system, for example, Dennis’ mother closes the window and does not let Bruce in anymore. Nevertheless, the most remarkable fact is that there are no blocks because energy is not a material substance, but it is given meaning in mathematical calculations.

Some physics teachers explain the concept of energy by using Feynman’s blocks and consider energy as a quasi-material or pseudo-substance. On the contrary, Wilhelm Ostwald regards energy as a form of substance. For instance, Ostwald (1910/2013) writes that “we may call energy a substance unqualifiedly, since in every instance of which we know the principle has been maintained that a quantity of any energy never disappears unless an equivalent quantity of another energy arises (p. 136).” He was awarded Nobel Prize in Chemistry in 1909 for his work on catalysis and for his investigations into the fundamental principles governing chemical equilibria and rates of reaction.

Note: The law of conservation of energy is only applicable to an isolated system instead of a real system. Thus, Planck (1945) clarifies that “[a] system which changes without being acted on by external agents is called a perfect or isolated system. Strictly speaking, no perfect system can be found in nature, since there is constant interaction between all material bodies of the universe, and the law of the conservation of energy cannot be rigorously applied to any real system (p. 46).” Interestingly, Feynman uses Bruce’s blocks to illustrate how the total energy in a real system could be increased.

3. Abstract quantity:

“…It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas (Feynman et al., 1963, section 4.1 What is energy?).”

When Feynman says that we have no knowledge of what energy is, it does not mean that we know absolutely nothing about the nature of energy. Essentially, he considers energy as an abstract quantity and prefers to be open minded in the understanding of the mechanism behind the various mathematical formulae for different forms of energy. Furthermore, Feynman does not visualize energy as little blobs of a definite amount. In the last section of this chapter, Feynman explains that the frequency of light can be anything, and thus, there is no law that says that energy must always be quantized as a certain definite amount or come in lumps.

Physicists commonly cite Feynman and claim that there is no good definition of energy. Historically, Poincare (1906/1952) writes that “[a]s we cannot give a general definition of energy, the principle of the conservation of energy simply signifies that there is a something which remains constant (p. 166).” Currently, physics teachers may argue that we should clearly define the concept of energy and it is inappropriate to say we have not knowledge of what energy is. However, a definition of energy as “the capacity (or ability) to perform work” mainly states the effect of energy as work. This general definition still does not tell us about the nature of energy.

Note: Sir James Jeans explains that energy is a mathematical abstraction as follows: “the attempt to regard the flow of energy as a concrete stream always defeats itself… The concept of energy flowing about through space is useful as a picture, but leads to absurdities and contradictions if we treat it as a reality. Professor Poynting gave a well-known formula which tells us how energy may be pictured as flowing in a certain way, but this picture is far too artificial to be treated as a reality; for instance, if an ordinary bar-magnet is electrified and left standing at rest, the formula pictures energy flowing endlessly round and round the magnet… The mathematician brings the whole problem back to reality by treating this flow of energy as a mere mathematical abstraction (James, 1932, p. 129).”

Questions for discussion:

1. What does the law of conservation of energy really mean?

2. What is the most remarkable aspect of energy that must be abstracted from Dennis’ blocks?

3. Is it true that there is no good definition of energy?

The moral of the lesson: energy is an abstract quantity that is conserved and indestructible.

References:

1. Bohr, N., Kramers, H. A., & Slater, J. C. (1924). The quantum theory of radiation. In B. L. Van Der Waerden (Ed.). Sources of Quantum Mechanics. New York: Dover.

2. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Jeans, J. H. (1932). The Mysterious Universe. New York: Macmillan.

5. Ostwald, W. (1910/2013). Natural Philosophy. Auckland: The Floating Press.

6. Planck, M. (1945). Treatise on Thermodynamics. New York: Dover.

7. Poincare, H. (1906/1952). Science and hypothesis. New York: Dover.

Sitemap

The Feynman Lectures on Physics, Volume I

Chapter 1 Atoms in Motion

1-1 Introduction (Learning physical laws / Guessing physical laws / Changing physical laws)

1-2 Matter is made of atoms (The three states of matter: solid, liquid, and gas.)

1-3 Atomic processes (Evaporation of water / Dissolution of salt / Equilibrium)

1-4 Chemical reactions (Simple molecules / Special molecules / Human molecules)

Chapter 2 Basic Physics

2-1 Introduction (Idealization / Exception / Approximation)

2-2 Physics before 1920 (Electrical force / Electric field / Electromagnetic waves)

2-3 Quantum physics (Uncertainty principle / Particle-wave duality / light-matter interactions)

2-4 Nuclei and particles (Nuclear force / Elementary particles / Elementary interactions)

Chapter 3 The Relation of Physics to Other Sciences

3-1 Introduction (Natural philosophy / Relation to other sciences / Love)

3-2 Chemistry (Theoretical chemistry / Inorganic chemistry / Organic chemistry)

3-3 Biology (Nerves / Enzymes of Krebs cycle / DNA molecules)

3-4 Astronomy (Stars / An astronomer / Nuclear energy)

3-5 Geology (The condition of air / Mountain-forming processes / Earth’s Interior)

3-6 Psychology (Psychoanalysis / Physiology of sensation / Computing machines)

3-7 How did it get that way? (Physicist’s language / Changing physical laws / Turbulence)

Chapter 4 Conservation of Energy

4-1 What is energy? (Conserved quantity / Indestructible blocks / Abstract quantity)

4-2 Gravitational potential energy (Pendulum / Kinetic energy / Relativistic correction)

4-3 Kinetic energy (Pendulum / Kinetic energy / Relativistic correction)

4-4 Other forms of energy (Forms of energy / Independence of time / Available energy)

Chapter 5 Time and Distance

5-1 Motion (Motion / Galileo’s clock / Galileo’s inclined track experiment)

5-2 Time (Definition of time / Measurement of time / Periodicity of time)

5-3 Short times (Mechanical pendulum / Electronic oscillator / Nuclear vibration)

5-4 Long times (Natural time / Radioactive time / Astronomical time)

5-5 Units and standards of time (Units of time / Standards of time / Atomic clock standards)

5-6 Large distances (Nearby stars / Distant stars / Sizes of galaxies and universe)

5-7 Short distances (Molecular sizes / Nuclear sizes / Uncertainty principle)

Section 3–7 How did it get that way?

(Physicist’s language / Changing laws of physics / A glass of wine)

This section provides a brilliant ending to the chapter by combining physics with biology, chemistry, geology, astronomy, and psychology. In this section, the three interesting ideas discussed are a physicist’s language, changing laws of physics, and a glass of wine.

1. Physicist’s language:

“… In order for physics to be useful to other sciences in a theoretical way, other than in the invention of instruments, the science in question must supply to the physicist a description of the object in a physicist’s language (Feynman et al., 1963, section 3.7 How did it get that way?).”

According to Feynman, physics are useful in a theoretical way, other than in the invention of instruments, if a description of the object can be expressed in a physicist’s language. For example, a physicist may not be able to answer the question “why does a frog jump?” unless the frog is well defined or the number of molecules is specified with additional information. In addition, a physical theory is useful when physicists have some knowledge of the locations of the atoms. Similarly, to understand chemistry, we must know exactly what atoms are present such that we can analyze them.

In general, physicists may idealize a frog or a dog as a point object. Thus, they can formulate the motion of the frog or dog by assigning mathematical quantities such as its mass, velocity, or position. Furthermore, during a Messenger Lecture, Feynman elaborates that “physicists delight themselves by using ordinary words for something else (p. 84).” That is, physicists redefine words, such as force, action, heat, energy, and symmetry, in order that they have technical or mathematical meanings.

2. Changing laws of physics:

“…We do not imagine, at the moment, that the laws of physics are somehow changing with time, that they were different in the past than they are at present (Feynman et al., 1963, section 3.7 How did it get that way?).”

Curiously, Feynman mentions that there is no historical question being studied in physics. He elaborates that we do not have a question such as “Here are the laws of physics, how did they get that way?” In other words, physicists do not imagine that the laws of physics are changing with time or how they were different in the past as compared to how they are at present. However, Feynman suggests that if we can understand how the laws of physics are changing with time, it will be wrapped up with the history of the universe, and then physicists will be talking about the same problems with astronomers, geologists, and biologists.

In a sense, Feynman contradicts himself when he later discusses whether Newton’s gravitational law is varying with time in chapter 7 of The Feynman Lectures on Physics. In his own words, “it has been proposed that the gravitational constant is related to the age of the universe. If that were the case, the gravitational constant would change with time, because as the universe got older the ratio of the age of the universe to the time which it takes for light to go across a proton would be gradually increasing. Is it possible that the gravitational constant is changing with time? Of course, the changes would be so small that it is quite difficult to be sure (Feynman et al., 1963, section 7.7 What is gravity?).” Simply put, the gravitational law would change with time if the gravitational constant is observed to be changing with time.

Note: For example, Dirac (1938) writes that “the ratio of the gravitational force to the electric force between electron and proton varying in inverse proportion to the epoch, and since, with our atomic units of time, distance, and mass, the electric force between electron and proton at a constant distance apart is constant, the gravitational force between them must be inversely proportional to the epoch. Thus the gravitational constant will be inversely proportional to the epoch (p. 206).” In other words, one may imagine how the gravitational constant is varying with time in relation to the age of the universe. This is also known as Dirac Large Number hypothesis.

3. A glass of wine:

“…A poet once said, “The whole universe is in a glass of wine.” … But it is true that if we look at a glass of wine closely enough we see the entire universe (Feynman et al., 1963, section 3.7 How did it get that way?).”

A glass of wine can be analyzed from the perspectives of physics, geology, astronomy, chemistry, biology, and psychology.

(1) Physics: the twisting liquid wine which evaporates is dependent on the wind and weather as well as the reflections in the glass.

(2) Geology and Astronomy: the glass is an extraction of the earth’s rocks, and its atomic composition is related to the age of the universe and the evolution of stars.

(3) Chemistry: the chemicals in the wine may be explained by the ferments, the enzymes, the substrates, the products, and chemical processes.

(4) Biology: all life is fermentation and Louis Pasteur published Études sur le Vin that is about the diseases of wine.

(5) Psychology: the consciousness that watches the wine.

Note: it should be easy for Feynman to give another brilliant explanation if the poet says “[t]he whole universe is in a cup of tea.”

Questions for discussion:

1. How does one describe an object using a physicist’s language?

2. Are the laws of physics changing with time?

3. How do we see the entire universe in a glass of wine or a cup of tea?

The moral of the lesson: we may find a glass of wine tastes better if we can divide this glass of wine, this universe, into parts—physics, biology, geology, astronomy, psychology, and so on.

References:

1. Dirac, P. A. (1938). A new basis for cosmology. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 165(921), 199-208.

2. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Section 3–6 Psychology

(Psychoanalysis / Physiology of sensation / Computing machines)

Feynman suggests that there are changes in brain cells if they are made out of atoms. In this section, the three interesting ideas discussed are psychoanalysis, physiology of sensation, and computing machines.

1. Psychoanalysis:

“… Psychoanalysis has not been checked carefully by experiment, and there is no way to find a list of the number of cases in which it works, the number of cases in which it does not work (Feynman et al., 1963, section 3.6 Psychology).”

According to Feynman, psychoanalysis is not really a science because it has not been checked carefully by experiment. That is, there is no way to specify a list of the number of cases that a psychological theory works and the number of cases that it does not work. Thus, he reasons that psychoanalysis is at best a medical process, but perhaps it is closer to witch-doctoring. However, there are an increasing number of physicists and physics teachers using psychological method or psychoanalysis in physics education research to improve physics students’ learning of physical concepts.

In one of his Messengers lectures, Feynman says that we are unable to determine the correct theory from two theories that are psychologically different. In his own words, “[s]uppose you have two theories, A and B, which look completely different psychologically, with different ideas in them and so on, but that all the consequences that are computed from each are exactly the same, and both agree with experiment. The two theories, although they sound different at the beginning, have all consequences the same, which is usually easy to prove mathematically by showing that the logic from A and B will always give corresponding consequences. Suppose we have two such theories, how are we going to decide which one is right? There is no way by science because they both agree with experiment to the same extent (Feynman, 1965, p. 168).”

Note: During the Manhattan project in Los Alamos, Feynman used a psychological method to open locks. In his autobiography Surely you’re Joking, Mr. Feynman!, he reveals that “when I thought of the safecracker books again: Next, try the psychology method. I said to myself, ‘Freddy de Hoffman is just the kind of guy to use a mathematical constant for a safe combination.’ I went back to the first filing cabinet and tried 27-18-28-- CLICK! It opened! (The mathematical constant second in importance to pi is the base of natural logarithms, e: 2.71828…) (Feynman, 1997, p. 148).”

2. Physiology of sensation:

“… The other branches of psychology, which involve things like the physiology of sensation—what happens in the eye, and what happens in the brain—are, if you wish, less interesting (Feynman et al., 1963, section 3.6 Psychology).”

Feynman mentions that the physiology of sensation (another branch of psychology) which involves things such as what happens in the eye, and what happens in the brain are less interesting, but a real progress has been made in studying them. More interesting is the central problem of the mind (or the nervous system) when an animal learns something: it can do something that is different from what it could do earlier, and there must be changes in brain cells. Curiously, Feynman opines that we do not know where to look, or what to look for, when something is memorized. However, there are studies of brains using functional magnetic resonance imaging to understand working memories and logical reasoning of human beings. We need not find fault with Feynman here because magnetic resonance imaging was accomplished in 1977.

Note: In Volume I, Chapter 36 of his lectures, Feynman discusses the sensation of color. Initially, he elaborates that physics and other sciences are very closely interrelated, such that the separation of science into different fields is merely a human convenience and it is an unnatural thing. In essence, nature is not interested in the separation of science, and a color is not simply about physics of the light, but it is also a sensation that is different in different circumstances. Historically, Helmholtz proposes that there are three different pigments in the eyes which receive light rays. These pigments have different absorption spectra, for example, one pigment may absorb red color light more strongly as compared to blue and green.

3. Computing machines:

“… There is an analog of this to computing machines and computing elements, in that they also have a lot of lines, and they have some kind of element, analogous, perhaps, to the synapse, or connection of one nerve to another (Feynman et al., 1963, section 3.6 Psychology).”

Feynman assumes the brain is similar to an enormous mass of interconnecting wires such that it cannot be analyzed in a straightforward manner. Alternatively, the brain is analogous to computing machines in that they also have a lot of lines (or wires) and they have some kind of element similar to the synapse, or connection of one nerve to another. However, there is a lack of time to further discuss the relationship between thinking and computing machines. To conclude, we even have a limited understanding of how dogs work and thus, it will be a long time before we can have a better understanding of human behavior that is more complex.

Note: in the fall of 1983, Feynman gave a course on the physics of computation that is listed in the Caltech record as “Potentialities and Limitations of Computing Machines.” In the subsequent years 1984/85 and 1985/86, the lectures on computation were taped and it was from those tapes and Feynman’s notebooks that the lectures were published. In the preface of Feynman lectures on computation, Feynman (1996) explains that “[c]omputer science also differs from physics in that it is not actually a science. It does not study natural objects (p. xiii).” In the first chapter, he adds that computers “can guide weapons to their targets. They can book you onto a plane between a guitar-strumming nun and a non-smoking physics professor (p. 1).”

Questions for discussion:

1. Why psychoanalysis is not a science?

2. In what way is our mind (or brain) different when something is memorized?

3. How is the brain analogous to computing machines or computer?

The moral of the lesson: when an animal learns something, it can do something that is different from before, and there must be changes in its brain cells because they are made of atoms.

References:

1. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.

2. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Feynman, R. P. (1996). Feynman lectures on computation. Reading, Massachusetts: Addison-Wesley.