Section 5–2 Time

(Definition of time / Measurement of time / Periodicity of time)

In this section, Dr. Sands discusses a definition of time, measurement of time, and periodicity of time.

1. Definition of time:

“…It would be nice if we could find a good definition of time. Webster defines ‘a time’ as ‘a period,’ and the latter as ‘a time,’ which doesn’t seem to be very useful (Feynman et al., 1963, section 5.2 Time).”

Dr. Sands opines that time is one of the things that we probably cannot define in the dictionary sense. For example, Webster defines time as “a period” and defines the period as “a time.” Thus, it does not provide us with useful information on any characteristics of time. This is similar to a dictionary that defines space as “an area,” and defines the area as “an amount of space.” Essentially, there is a problem of circularity in defining fundamental physical concepts such as space and time. However, physicists could provide a theoretical definition of time or an operational definition of time. For example, Einstein (1905) suggests an operational definition of time when he writes that “a time is to be defined exclusively for the place at which the watch is located (p. 125).”

In physics, time can be defined in terms of characteristics such as “arrows of time.” For example, Gleick (1992) writes that “[p]hysicists had learned to distinguish three arrows of time. Feynman described them: the thermodynamic or ‘accidents of life’ arrow; the radiation or ‘retarded or advanced arrow’; and the cosmological arrow. He suggested keeping in mind three physical pictures: a tank with blue water on one side and clear water on the other; an antenna with a charge moving toward it or away; and distant nebulas moving together or apart (p. 126).” Alternatively, a theoretical definition of time may include the following three characteristics: linear (sequential order), unidirectional (irreversible), and relative (frame-dependent). The concept of time is related to Newtonian mechanics, the second law of thermodynamics, and theory of relativity.

Note: Another possible characteristic of time may be specified as probabilistic (apparent periodicity) from a perspective of quantum mechanics.

2. Measurement of time:

“…One way of measuring time is to utilize something which happens over and over again in a regular fashion — something which is periodic (Feynman et al., 1963, section 5.2 Time).”

Dr. Sands explains that what really matters is not how we define time, but how we measure time. One of the ways to measure time is by utilizing a physical phenomenon (natural or artificial) which happens repeatedly in a regular fashion. For example, a day is related to the relative movement of the earth and the sun. Nevertheless, when Dr. Sands says that days in summer are longer than days in winter, he is referring to the colloquial meaning of day as the time between sunrise and sunset. This is different from the meaning of a day as 24 hours.

In general, a measurement is essentially a comparison process. In other words, every measurement of time is a comparison between a duration of time and a standard unit of time. On the contrary, it is inaccurate to use one’s pulse to measure a pendulum’s period and then to use the period of the pendulum to measure the pulse of another person. Interestingly, an astronomer may measure time in terms of “space” or the distance traveled by light rays from a distant star, and another astronomer may measure space in terms of “time” for light to travel between two points (e.g. light years). This is another kind of circularity where time is defined in terms of space, and space is defined in terms of time. More important, space and time are also defined in terms of the speed of light.

Note: Similarly, in chapter 17, Feynman explains that “nature is telling us that time and space are equivalent; time becomes space; they should be measured in the same units. What distance is a “second”? It is easy to figure out from (17.3) what it is. It is 3×10⁸ meters, the distance that light would go in one second. In other words, if we were to measure all distances and times in the same units, seconds, then our unit of distance would be 3×10⁸ meters, and the equations would be simpler. Or another way that we could make the units equal is to measure time in meters. What is a meter of time? A meter of time is the time it takes for light to go one meter, and is, therefore, 1/3×10⁻⁸ sec, or 3.3 billionths of a second! (Feynman et al., 1963, section 17.2 Space-time intervals).”

3. Periodicity of time:

“…We can just say that we base our definition of time on the repetition of some apparently periodic event (Feynman et al., 1963, section 5.2 Time).”

Ideally, we should base our definition of time on the repetitions of a regularly periodic event. However, the concept of astronomical time based on the rotational period of the earth is apparently periodic because there are changes in gravitational fields due to changes in the locations of celestial bodies and changes in the distributions of mass on earth. In a similar sense, the concept of atomic time based on an atomic clock is apparently periodic because it can be changed by slight variations in gravitational fields, electromagnetic fields, motion, temperature or other physical phenomena. Therefore, our definition of “hour” and “day” are based on the repetition of apparently periodic events.

Note: A solar day is a period of time it takes for the earth to rotate about its axis such that the sun appears in the same position in the sky periodically. This is different from a sidereal day that means a period of time it takes for the earth to rotate about its axis such that distant stars appear in the same position in the sky periodically. Furthermore, astronomers define the equation of time as the difference between true solar time (based on the sun’s position in the sky) and mean solar time (related to the time of your watch) (e.g. Sajina, 2008).

Questions for discussion:

1. How would you define time?

2. How should time be measured?

3. Do we have a clock that has regular periodicity?

The moral of the lesson: what really matter is how time can be measured by a clock that has apparent periodicity.

References:

1. Einstein, A. (1905). On the Electrodynamics of Moving Bodies. In J. Stachel (Ed.), Einstein’s Miraculous Year: Five Papers that Changed the Face of Physics (pp. 123-160). Princeton: Princeton University Press.

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. Gleick, J. (1992). Genius: The Life and Science of Richard Feynman. London: Little, Brown and Company.

4. Sajina, A. (2008). On ships, trains, and the equation of time. Physics Today, 61(11), 76-77.

Section 5–1 Motion

(Motion / Galileo’s clock / Galileo’s inclined track experiment)

This chapter was delivered by Matthew Sands because Feynman was out of town. In this section, the three interesting points discussed are the concept of motion, Galileo’s clock, and Galileo’s inclined track experiment. (Feynman elaborates on the concept of motion in chapter 8.)

1. Motion:

“… the study of motion had been a philosophical one based on arguments that could be thought up in one’s head (Feynman et al., 1963, section 5.1 Motion).”

According to Dr. Sands, the development of the physical sciences to their present form has depended to a large extent on the emphasis which has been placed on the making of quantitative observations. Based on these observations, physicists can deduce important quantitative relationships, which are the heart of physics. On the other hand, the study of motion had been mainly philosophical based on arguments or thought experiments. Most arguments presented by Aristotle and other Greek philosophers were accepted as “proofs.” Thus, Galileo was skeptical of Aristotle’s theory of motion and carried out experiments on motions.

It is controversial to identify Galileo as the first physicist. For example, one may prefer Aristotle because he is the first person that writes a book titled Physics and proposes that the motion of an object is influenced by a motive force and resistance of medium. It is also debatable whether Galileo should be recognized as the “first experimental physicist.” For instance, Giuseppe Moletti reported in 1576 that objects made of the same material but having different weights reach the ground at the same time (Martínez, 2011). Currently, it is commonly agreed among historians that Galileo conceptualizes a thought experiment of dropping two spheres instead. Galileo’s dropping of two spheres of different masses from the Leaning Tower of Pisa could be an apocryphal story (Martínez, 2011).

Note: In chapter 8, Feynman discusses Zeno’s paradox of motion and defines motion as “the apparent change in its position with time.” In addition, he briefly explains some difficulties of defining space and time precisely from the perspectives of the theory of relativity and quantum mechanics. Although the concepts of motion, space, and time involve deep philosophical questions, Feynman suggests that we should avoid the paralysis of thought in defining everything more precisely.

2. Galileo’s clock:

“…he later devised more satisfactory clocks (though not like the ones we know), Galileo’s first experiments on motion were done by using his pulse to count off equal intervals of time (Feynman et al., 1963, section 5.1 Motion).”

Dr. Sands opines that Galileo’s first experiments on motion were done by using his pulse to measure time. However, Galileo’s measurement of a pendulum’s period by using his pulse while he was watching a lamp swinging in Pisa cathedral may be another apocryphal story (Glennie & Thrift, 2009). More important, the use of one’s pulse rate to measure time or a pendulum’s period in an experiment would introduce inaccuracy or subjectivity. Thus, the use of pulse to compare time is just a short-term improvisation. On the contrary, Santorius (1561–1636) was using a pendulum (or pulsilogium) to measure the pulse rate of patients. This invention was based on Galileo’s observation that a pendulum’s period is inversely proportional to the square root of its length.

Note: The pulsilogium (or pulsimeter) was composed of a heavy lead bob, a silk cord and a ruler (Sanctorius, 1631). The period of this pendulum bob can be adjusted by varying the length of silk cord from one end of the ruler such that it synchronizes with a patient’s pulse beat. On the scale of the ruler, we can read the frequency of the patient’s pulse beat.

3. Galileo’s inclined track experiment:

“…He allowed a ball to roll down an inclined trough and observed the motion. He did not, however, just look; he measured how far the ball went in how long a time (Feynman et al., 1963, section 5.1 Motion).”

Dr. Sands explains that Galileo’s first experiments on motion were carried out by using Galileo’s pulse to count off equal intervals of time. However, in his book Dialogues Concerning Two New Sciences, Galileo describes the use of a water clock to carry out his experiments on motions. In his words, Galileo explains that “[f]or the measurement of time, we employed a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent (Galilei, 1638/1914, p. 179).” It is possible that Galileo used a water clock and music (or a song) to measure time during experiments (Drake, 1975).

Feynman might attack Galileo’s deduction of how an object would move in a free fall from the inclined plane experiments. In Surely, You are joking, Mr Feynman!, he criticizes a physics textbook and says that “[t]here are no experimental results mentioned anywhere in this book, except in one place where there is a ball, rolling down an inclined plane, in which it says how far the ball got after one second, two seconds, three seconds, and so on. The numbers have ‘errors’ in them--that is, if you look at them, you think you're looking at experimental results, because the numbers are a little above, or a little below, the theoretical values. The book even talks about having to correct the experimental errors--very fine. The trouble is, when you calculate the value of the acceleration constant from these values, you get the right answer. But a ball rolling down an inclined plane, if it is actually done, has an inertia to get it to turn, and will, if you do the experiment, produce five-sevenths of the right answer, because of the extra energy needed to go into the rotation of the ball. Therefore this single example of experimental ‘results’ is obtained from a fake experiment (Feynman, 1997, p. 217).”

Questions for discussion:

1. Who would you identify as the first physicist?

2. Which clocks devised by Galileo are accurate for an experiment?

3. How would you perform Galileo’s inclined track experiment?

The moral of the lesson: it is important to perform experiments to study the motion of an object quantitatively.

References:

1. Drake, S. (1975). The Role of Music in Galileo's Experiments. Scientific American, 232(6), 98-104.

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. Galilei, G. (1638/1914). Dialogues Concerning Two New Sciences. New York: Dover.

5. Glennie, P., & Thrift, N. (2009). Shaping the Day: A History of Timekeeping in England and Wales 1300-1800. Oxford: Oxford University Press.

6. Martínez, A. A. (2011). Science Secrets: The Truth about Darwin's Finches, Einstein's Wife, and Other Myths. Pittsburgh, Pa.: University of Pittsburgh Press.

7. Sanctorius, S. (1631). Methodi vitandorum errorum omnium qui in arte medica contingunt. Geneva: P. Aubertum.

Section 4–4 Other forms of energy

(Forms of energy / Independence of time / Available energy)

Feynman relates the law of conservation of energy to the symmetry of time-displacement that was commonly discussed during advanced graduate courses on physics instead of introductory courses. In this section, the three interesting concepts discussed are forms of energy, independence of time, and available energy.

Note: If you listen to the audio CD of this lecture, Feynman says that “we know that it is not electrical, not gravitational, and not purely kinetic (not chemical), but we do not know what it is.” Currently, we know that nuclear force is a residual effect of the strong force or gluon field.

1. Forms of energy:

“There are many other forms of energy, and of course, we cannot describe them in any more detail just now … (Feynman et al., 1963, section 4.4 Other forms of energy).”

In general, different forms of energy include gravitational potential energy, kinetic energy, elastic potential energy, thermal energy, electrical energy, light energy, chemical energy, nuclear energy and mass energy. For example, elastic potential energy can be illustrated by pulling down a spring that can later lift a weight. As the spring passes through the equilibrium point, its elastic potential energy is converted into kinetic energy. If the spring is mounted vertically, there is also a transfer of some gravitational potential energy either going in or out of the spring. There is no change in gravitational potential energy if this experiment is performed “sideways” or horizontally.

Feynman also clarifies that there are not really many different forms of energy. Firstly, heat energy is essentially kinetic energy of atoms in random motion. Secondly, chemical energy can be understood as due to the kinetic energy of the electrons inside the atoms as well as the electrical energy of interaction of the electrons and the protons. Next, elastic potential energy is similar to the chemical energy because these two forms of energy are mainly the electrical energy of the attraction of the atoms. However, the gravitational potential energy of an object can be explained as field energy of a system because it is stored in the gravitational field instead of the object. Moreover, mass energy is now understood as contributed by the Higgs field. In short, we may replace potential energy by field energy (with the support of field theories).

Note: From a perspective of pedagogy, some physics educators suggest to teach students that there are different kinds of transfer of energy instead of different forms of energy (Falk, Herrmann, & Schmid, 1983). Similarly, we do not speak about different forms of electric charge. In other words, it is confusing to introduce different forms of electric charge, such as “electronic charge,” “protonic charge,” “Cl-ionic charge,” according to the charge carrier involved during a transfer of charge.

2. Independence of time:

“As independence in space has to do with the conservation of momentum, independence of time has to do with the conservation of energy… (Feynman et al., 1963, section 4.4 Other forms of energy).”

Feynman explains that the conservation of energy is very closely related to an important property of the world, things do not depend on the absolute time. Simply phrased, when we perform an experiment at a given moment and carry out the same experiment at a later moment, the laws of physics remain unchanged. If we assume that this is true, and add the principles of quantum mechanics, then we can deduce the law of the conservation of energy. This is related to Noether’s theorem in which conservation laws are related to symmetry principles: for example, time translation symmetry gives conservation of energy, space translation symmetry gives conservation of momentum, and rotation symmetry gives conservation of angular momentum.

One may wonder why Feynman states that the law of conservation of energy is closely related to the symmetry of time in quantum mechanics. This could be related to his proof of conservation of energy by using a transformation of time-displacement in his Ph.D. thesis titled A New Approach to Quantum Theory. Interestingly, based on an interview with Feynman in Austin, Mehra (1994) writes that “Feynman’s approach could be used in the theories with advanced and retarded interactions, where Noether’s theorem does not work without proper modifications. Feynman ‘proved a thing called Noether’s theorem, not knowing that it was known’ (p. 132).”

3. Available energy:

“With regard to the conservation of energy, we should note that available energy is another matter… (Feynman et al., 1963, section 4.4 Other forms of energy).”

It is important to distinguish “conservation of energy” and “energy conservation.” Although there is a law of conservation of energy, there is a problem of “energy conservation” because the available energy for human utility is not conserved easily. The laws which govern available energy are called the laws of thermodynamics and they involve a concept called entropy. Thus, a pertinent question is how we can continue to get our supplies of energy every day. Interestingly, we may say that our supplies of energy are from the sun, rain, coal, uranium, and hydrogen, but it is the sun that makes the rain and coal such that they all come from the sun.

Feynman also predicts that energy can be obtained from water if it can be controlled in thermonuclear reactions. However, scientists are developing hydrogen fuel cell cars that can run by water, and a design involves electrolysis a splitting of water into hydrogen and oxygen. The main problem is still about energy conversion efficiency that is a ratio of energy output to energy input.

Note: When Feynman says that “she liberates a lot of energy from the sun, but only one part in two billion falls on the earth,” it is based on a quick approximate by calculating the square of ratio of the earth’s radius to the distance between the earth and sun: (r/R)². To be more accurate, one may use the ratio (pr²/4pR²). Note that the radius of the earth (r) is approximately 6,371 km, whereas the distance between the earth and the sun (R) is approximately 149, 597, 870 km.

Questions for discussion:

1. What are the different forms (or types) of energy?

2. Why does an independence of time have to do with the law of conservation of energy in quantum physics?

3. Why does the available energy for human utility is not conserved easily?

The moral of the lesson: the law of conservation of energy is related to the symmetry of time translation.

References:

1. 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.

2. Falk, G., Herrmann, F., & Schmid, G. B. (1983). Energy forms or energy carriers?. American Journal of Physics, 51(12), 1074-1077.

3. Mehra, J. (1994). The Beat of a Different Drum: The life and science of Richard Feynman. Oxford: Oxford University Press.

Section 4–3 Kinetic energy

(Pendulum / Kinetic energy / Relativistic correction)

This section on kinetic energy is very brief as compared to potential energy. The three interesting points discussed are a pendulum, kinetic energy, and relativistic correction for the equation of kinetic energy.

1. Pendulum:

“…To illustrate another type of energy we consider a pendulum. If we pull the mass aside and release it, it swings back and forth. In its motion, it loses height in going from either end to the center (Feynman et al., 1963, section 4.3 Kinetic energy).”

Interestingly, Feynman demonstrates the law of conservation of energy by using a pendulum and he is confident that the pendulum would not rise to a greater height to hit him. He initially brings the pendulum to a height as high as his chin and then releases it. While the pendulum is swinging to and fro, it is possible to have a continuous change of gravitational potential energy to kinetic energy, and vice versa. During its motion, it may lose height in going from either end to the center with a gain in kinetic energy and a loss in gravitational potential energy. When the pendulum climbs up towards our chin, there is a loss in kinetic energy and a gain in gravitational potential energy instead.

One may google “Walter Lewin Kinetic & Potential Energy” to have a look at a similar demonstration by using a pendulum. In the video, Lewin explains that “physics works” and he is still alive because the pendulum did not hit him. Strictly speaking, it is possible that the pendulum gains additional kinetic energy during an earthquake.

2. Kinetic energy:

“…The kinetic energy at the bottom equals the weight times the height that it could go, corresponding to its velocity: K.E. = WH. ... We shall soon find that we can write it this way: K.E. = WV²/2g (Feynman et al., 1963, section 4.3 Kinetic energy).”

By using arguments about reversible machines (or law of conservation of energy), we can see that a pendulum at the lowest point of its path must have a quantity of energy which permits it to rise a certain height. Feynman says that this has nothing to do with any machinery by which the pendulum comes up or the path by which it ascends. Importantly, the kinetic energy of the pendulum at the lowest height equals to the weight of the pendulum times the height that it could go (K.E. = W ´ H). Furthermore, the kinetic energy depends on the velocity and a formula for kinetic energy of an object can be derived as K.E. = WV²/2g. (Feynman shows that the formula for kinetic energy K.E. = ½ mV² is correct in chapter 13.)

Note: During a British Broadcasting Corporation interview, Feynman (1994) explains that “[w]hen a ball comes down and bounces, it shakes irregularly some of the atoms in the floor, and when it comes up again, it has left some of those floor atoms moving, jiggling. As it bounces, it is passing its extra energy, extra motions to little patches on the floor (p. 127).” In short, a percentage of the kinetic energy of a macroscopic object is converted into the kinetic energy of microscopic objects (or atoms) during a collision.

3. Relativistic correction:

“…Both (4.3) and (4.6) are approximate formulas, the first because it is incorrect when the heights are great, i.e., when the heights are so high that gravity is weakening; the second, because of the relativistic correction at high speeds (Feynman et al., 1963, section 4.3 Kinetic energy).”

The equation K.E. = WV²/2g is a general formula for kinetic energy of an object moving with velocity V. Curiously, the motion of an object has kinetic energy has nothing to do with the fact that we are in a gravitational field. This is because the gravitational field g can be eliminated by writing W as mg such that the formula is simplified as K.E. = ½ mV². However, the kinetic energy of pendulum can be transformed from the energy stored in the gravitational field (or gravitational potential energy). Moreover, the equation for kinetic energy is only approximately correct because there is a need for relativistic correction when the object is moving at very high speeds. Thus, the law of conservation of energy is strictly true when we apply the exact formula for all forms of energy.

Note: The reality (or objectivity) of kinetic energy was questioned because it can be assigned any value depending on an observer’s frame of reference. For example, Heaviside (1891) has questioned the reality of energy due to the relative motion: “[b]ut we need not go so far as to assume the objectivity of energy. This is an exceedingly difficult notion, and seems to be rendered inadmissible by the mere fact of the relativity of motion, on which kinetic energy depends (p.425).” As an example, a book on a table may appear at rest from the perspective of an observer on earth. However, one may explain that the object is moving at a high speed due to the rotation of the earth. In other words, the kinetic energy of the object is not an invariant.

Questions for discussion:

1. Is it absolutely safe to demonstrate the law of conservation of energy by using some heavy bricks as a pendulum?

2. Does the kinetic energy of an object depend on the gravitational field (K.E. = WV²/2g)?

3. What is the limitation of formula for kinetic energy, K.E. = ½ mV²?

The moral of the lesson: the kinetic energy of an object is dependent on its velocity but the formula needs a relativistic correction when the object is moving at very high speeds (especially approaching the speed of light).

References:

1. 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.

2. Feynman, R. P. (1994). No Ordinary Genius: The Illustrated Richard Feynman. New York: W. W. Norton & Company.

3. Heaviside, O. (1891). On the Forces, Stresses, and Fluxes of Energy in the Electromagnetic Field. Proceedings of the Royal Society of London, 50, 126–129.