Section 13–2 Work done by gravity

(One-dimensional case / Any closed path / A simple triangle)

In this section, Feynman discusses work done by gravity from the perspectives of a one-dimensional case, any closed path, and a simple triangle.

1. One-dimensional case:

“This one-dimensional case is easy to treat because we know that the change in the kinetic energy is equal to the integral (Feynman et al., 1963, section 13–2 Work done by gravity).”

In a simple example, an object starts from a point and moves radially toward the Sun or the Earth. By using a convention in which moving radially outward is positive, the gravitational force is −GMm/r² and thus, the object’s gravitational potential energy decreases (or kinetic energy increases) as the distance fallen increases. One may ask if it is possible to find another formula for gravitational potential energy that is different from mgh such that the law of conservation of energy is still true. In Tips on Physics, Feynman (2006) replies that “the potential energy between two particles is −Gm₁m₂/r. It’s hard for me to remember which way the potential energy goes. Let’s see: the particles lose energy when they come together, so that means when r is smaller, the potential energy should be less, so it’s negative-I hope that’s right! I have a great deal of difficulty with signs (p. 48).” However, the sign of a work done and change in potential energy is also related to how we define a system.

The word system refers to one or more objects that we mentally group together as a matter of convenience, but it may affect how we classify forces into internal or external. For example, the gravitational force is an “internal” force instead of an external force (e.g., your hand) that does the work in the object-Earth system. The work done by an internal force results in a decrease in potential energy and it is commonly expressed as W = -DU. On the contrary, our hand (external force) can lift an object such that there is an increase in potential energy. In general, it is imprecise to simply write “potential energy of an object”; the potential energy belongs to a system which may consist of “the object and the Earth” (Lindsey et al., 2012). Similarly, weight is not a property of an object, but it is a measure of the gravitational force between the object and the Earth (Serway & Faughn, 2003).

2. Any closed path:

“…there is no friction the object should come back with neither higher nor lower velocity—it should be able to keep going around and around any closed path (Feynman et al., 1963, section 13–2 Work done by gravity).”

Feynman asks whether it is possible to make perpetual motion in a gravitational field by using a fixed, frictionless track. Some students may guess that it does not seem possible because the gravitational force varies in different locations and has different strengths and in different directions. Feynman suggests that it is impossible to have perpetual motion, but we ought to find out if this is indeed the case. Recently, Wilczek (2012) proposes the concept of time crystal and explains that “a system with spontaneous breaking of time translation symmetry in its ground state must have some sort of motion in its ground state, and is therefore perilously close to fitting the definition of a perpetual motion machine (p. 1).” In short, it is possible to have “almost perpetual motion” just like the electric current in a superconductor.

On the question “is the work really zero?” Feynman answers that it is zero and then examines it mathematically by using a simple path as shown in Fig. 13–3. That is, a small object is carried from point 1 to point 2, and then it is made to go around a circle to point 3, back to 4, then to 5, 6, 7, and 8, and finally back to point 1. The total work done by a gravitational force in moving around this path or any closed path is zero. Physics teachers may elaborate that this is true for a conservative force (it is discussed later in chapter 14). Generally speaking, examples of conservative forces include gravitational forces and electrical forces, whereas examples of non-conservative forces include frictional forces and air resistance. Some physicists may add that the work done is definitely zero provided it satisfies the condition: curl of F = 0.

3. A small triangle:

“…we see that the work done in going along the sides of a small triangle is the same as that done going on a slant because scos θ is equal to x (Feynman et al., 1963, section 13–2 Work done by gravity).”

In this example, Feynman discretizes a curve into a series of sawtooth jiggles as shown in Fig. 13–4. Importantly, each triangle has to be very small such that the (variable) gravitational force is approximately constant over the entire triangle. Thus, the work done in going along the sides (horizontal and vertical) of a small triangle is numerically the same as the work done in going on a slant side because s ´ cos θ is equal to x. Notably, the “smooth” closed path can be approximated by a series of radial and circumferential steps. In a sense, Fig. 13.4 is misleading because some students may think that there are only horizontal and vertical steps instead of radial and circumferential steps.

There are several issues in Feynman’s example of a mass on a vertical spring. If we pull the mass downward, there is a change in the elastic potential energy as well as the gravitational potential energy. Thus, physics teachers should simplify the example by discussing a horizontal spring that slides on a frictionless table instead of a vertical spring. Furthermore, the work done can be distinguished as “work done on a spring” and “work done by a spring.” For instance, if your hand pulls a mass (or extends a spring), the work done on a spring by your hand (external force) is (+kx)(+x) and positive. At the same time, the work done by the spring or internal force is (–kx)(+x) and it is negative. As a suggestion, physics teachers may end the section by explaining that we have usually ignored the work done in compressing an (non-rigid) object while pushing it.

Questions for discussion:

1. How would you explain the work done and a change in potential energy in a one-dimensional case?

2. Why is the total work done by a gravitational force in moving any closed loop is zero?

3. Why is the work done in moving along two sides of a small triangle is the same as the work done in moving on a slant side?

The moral of the lesson: the total work done by a gravitational (conservative) force in moving any closed path is zero.

References:

1. Feynman, R. P., Gottlieb, & M. A., Leighton, R. (2006). Feynman’s tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.

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. Lindsey, B. A., Heron, P. R. L., & Shaffer, P. S. (2012). Student understanding of energy: Difficulties related to systems. American Journal of Physics, 80(2), 154-163.

4. Serway, R. A., & Faughn, J. S. (2003). College Physics (6th ed.). Pacific Grove, California: Brooks/Cole.

5. Wilczek, F. (2012). Quantum time crystals. Physical review letters, 109(16), 160401.

Section 13–1 Energy of a falling body

(Rate of change of potential energy / Power expended / Work done by gravity)

In this section, Feynman discusses the rate of change of potential energy, power expended, and work done by gravity.

1. Rate of change of potential energy:

“…the rate of change of the kinetic energy is −mg(dh/dt), which quantity, miracle of miracles, is minus the rate of change of something else! It is minus the time rate of change of mgh! (Feynman et al., 1963, section 13–1 Energy of a falling body).”

Feynman did not begin chapter 13 by discussing the work-kinetic energy theorem. Firstly, he derives how the time rate of change of kinetic energy is related to the time rate of change of potential energy. Next, he specifies that the gravitational force is constant and it is equal to −mg. This is an approximation because we can show that GMm/(R + h)² » mg if the height h is less than 100 km above the sea level. In a sense, this section (13.1) can be titled as “work done by constant gravitational force, whereas the next section (13.2) is about “work done by variable gravitational force.” Furthermore, we assume there is no air resistance (an idealization) such that there is a miracle of miracles (Feynman’s words) in which the rate of change of the kinetic energy is minus the rate of change of potential energy.

In Tips on Physics, Feynman (2006) elaborates that “the value of the potential energy can be zero any place you want. The way we’re going to use potential energy is to talk about its changes - and then, of course, it doesn’t make any difference if you add a constant (p. 47).” In other words, what matters is the change in gravitational potential energy rather than its absolute value. (There are locations, e.g., sea level or somewhere at infinity, where they can be set to zero as a matter of convenience.) In this section, Feynman proves that the work done by gravity results in a change of gravitational potential energy. This is in contrast to conventional physics textbook authors that prove the work done by a force results in a change of kinetic energy. Nevertheless, one should appreciate the beauty of how the time rate of change of kinetic energy, −mg(dh/dt), is the minus time rate of change of mgh (potential energy).

2. Power expended:

“We thus have a marvelous theorem: the rate of change of kinetic energy of an object is equal to the power expended by the forces acting on it (Feynman et al., 1963, section 13–1 Energy of a falling body).”

Using vector analysis, Feynman shows that dT/dt = F_xv_x + F_yv_y + F_zv_z. = F.v. He also expresses F.v as the power being delivered to an object: the dot product of a force acting on an object and the velocity of the object. Moreover, he describes a theorem: the rate of change of kinetic energy of an object is equal to the power expended by the forces acting on it. Weaker students may prefer Feynman adds one more step such that d(½mv²)/dt = d(½mv.v)/dt = ½m(v.dv/dt + dv/dt.v) = m(dv/dt).v = F.v. Conversely, some textbook authors simply state dT/dt = F.v. To be precise, physicists prefer using the term instantaneous power instead of power.

In daily lives, the word power may mean average power. It is worthwhile to distinguish average power and instantaneous power. Essentially, average power (ΔW/Δt) is the amount of work done (ΔW) by a force divided by a period of time (Δt). In general, if the force acting on an object is constant and the average velocity is v_ave, then the average power can be shown to be equal to F.v_ave. If the object’s initial velocity is zero, then it is possible that the average power is equal to F.v_max/2 in which v_max is the maximum velocity of the object. On the other hand, the instantaneous power is numerically equal to the average power as the time taken, Δt, approaches zero. Mathematically, the time rate of change of energy transferred is equal to the dot product of the force at an instant and the instantaneous velocity, F.v.

3. Work done by gravity:

“We also see that it is only a component of force in the direction of motion that contributes to the work done (Feynman et al., 1963, section 13–1 Energy of a falling body).”

The work done by a force on an object can be written as F.s or F_xdx + F_ydy + F_zdz. In a daily life example, the gravitational force can be vertical, and it has only a single component, say F_z, that is equal to –mg. (Based on the convention that upward is positive, the gravitational force is acting downward and thus, it is negative.) Using this example, Feynman shows that ∫²₁F.ds = −mg(z₂ − z₁). In short, this is equivalent to the work-potential energy theorem, DW = DP.E. (or DW = −DP.E. for gravity). For instance, if your hand lifts a book upward, the work done by your hand (external force) is (+mg)(+h) and it is positive. At the same time, the work done by the gravitational force is (–mg)(+h) and thus, negative. Because of Newton’s third law of motion, we have two different works that are equal in magnitude and opposite in sign.

In Tips on Physics, Feynman (2006) explains that “[i]n certain cases, that integral can be calculated easily ahead of time, because the force on the particle depends only on its position in a simple way. Under those circumstances, we can write that the work done on the particle is equal to the change in another quantity called its potential energy, or P.E. Such forces are said to be ‘conservative’: DW = DP.E. (p. 44).” Strictly speaking, we need to idealize the particle moves slower than a snail (its speed is infinitesimally close to zero) such that there is no gain in kinetic energy. Some may prefer a more realistic work-energy theorem that includes both potential energy and kinetic energy: DW = DP.E. + DK.E.

Questions for discussion:

1. How do you explain the time rate of change of kinetic energy, −mg(dh/dt), is the minus time rate of change of mgh (potential energy)?

2. How do you prove that the rate of change of kinetic energy of an object is equal to the power expended by the forces acting on it?

3. How do you show that the work done by a force is positive or negative?

The moral of the lesson: average power is equal to work done per second and it is only a component of force in the direction of motion that contributes to the work done.

References:

1. Feynman, R. P., Gottlieb, & M. A., Leighton, R. (2006). Feynman’s tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.

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.

Section 12–6 Nuclear forces

(Interaction range / Mathematical approximation / Fundamental machinery)

In this section, Feynman discusses nuclear forces from the perspectives of interaction range, mathematical approximation, and fundamental machinery.

1. Interaction range:

“These forces have a very tiny range which is just about the same as the size of the nucleus, perhaps 10⁻¹³ centimeter (Feynman et al., 1963, section 12–6 Nuclear forces).”

Feynman concludes this chapter with a brief discussion of nuclear forces. He mentioned that there was no known law of nuclear forces. Currently, it is still challenging to accurately calculate the force between two nuclei. For instance, Wilczek (2007) writes that “[i]n principle, the equations of QCD contain all the physics of strong internucleon forces. But in practice, it is extremely difficult to solve the equations and calculate those forces (p. 156).” The physical law of nuclear forces is now known as quantum chromodynamics (QCD) and it is still true that these forces have a very tiny range which is about 10⁻¹³ centimeter (or about the size of a nucleon). One may criticize the use of the term nucleus here because it is possible to be much larger than a nucleon.

Physicists, nowadays, use the term “strong force” instead of “nuclear force.” We may estimate the range of strong force to be about the size of a nucleon because the quarks enclosed are point-like objects. Wilczek (2015) explains that “[t]he use of ‘strong’ and ‘force’ together is potentially ambiguous because ‘strong force’ could be taken to mean a powerful source of acceleration. Thus when speaking of the gravitational influence of a neutron star, or a black hole, one might say that gravity exerts a strong force on a nearby planet. To avoid ambiguity, in such cases I use terms like ‘powerful force’ or ‘powerful interaction,’ avoiding ‘strong force’ and ‘strong interaction’ (p. 389).” Strictly speaking, the concept of force is also not suitable in QCD (or quantum theory), but the concept of energy is more natural.

2. Mathematical approximation:

“Any formula that can be written for nuclear forces is a rather crude approximation which omits many complications… (Feynman et al., 1963, section 12–6 Nuclear forces).”

Feynman mentions that any formula that can be written for nuclear forces is only a crude approximation which neglects many complications. He adds that nuclear forces disappear as soon as the particles are at a great distance apart, and states that they are very strong within the 10⁻¹³ cm range. Physicists currently explain that the nuclear force is a residual effect of the strong force. In short, the strong force is relatively stronger as the quarks separate from each other, and it is weaker if the quarks approach one another. The idea that strong force becomes weaker at short distances is known as asymptotic freedom (or “charge without charge”).

According to Feynman, nuclear forces do not vary inversely as the square of the distance, but they decrease exponentially over a distance r according to the equation, F = (1/r²)exp(−r/r₀), in which the distance r₀ is of the order of 10⁻¹³ centimeter (or 10⁻¹⁵ m). Mathematically, this is incorrect because it suggests that the nuclear forces approach infinity when the distance r approaches zero. Based on experiments of the deep inelastic scattering of electrons off nucleons, physicists deduce that quarks behave like free particles in the asymptotic limit of zero separation. Interestingly, Zee (2015) explains that “[q]uarks are like some lovers: When they are far apart, they want each other, but when they are close together, they barely acknowledge (p. 205).”

3. Fundamental machinery:

“…we do not understand them in any simple way, and the whole problem of analyzing the fundamental machinery behind nuclear forces is unsolved (Feynman et al., 1963, section 12–6 Nuclear forces).”

Feynman says that the laws of nuclear force are very complex and he does not have a simple way of understanding these laws. (He mentioned initially that there was no known law of nuclear forces.) The problem of analyzing the fundamental machinery of nuclear forces was unsolved when this lecture was delivered. George Zweig, Feynman’s doctoral student, developed a model and named the fundamental particles as “aces” in 1964. Gell-Mann also proposed a quark model within the same year. In a public lecture on QED, Feynman (1985) expresses that “Murray Gell-Mann nearly went crazy trying to figure out the rules by which all these particles behave, and in the early 1970s they came up with the quantum theory of strong interactions (or quantum chromodynamics), whose main actors are particles called ‘quarks’ (p. 132).”

Just like the electromagnetic force is mediated by photons, the strong force is mediated by gluons. During the public lecture on QED, Feynman (1985) explains that “[s]omething else has been invented to go back and forth and hold quarks together; something called ‘gluons’ (p. 134).” The gluons are labeled different colors (red, blue, and green) that behave like an electric charge. The color charge is conserved in the interactions of quarks such that the color of a quark is changed after absorption of a color gluon. However, Feynman (1985) responds that “[t]he idiot physicists, unable to come up with any wonderful Greek words anymore, call this type of polarization by the unfortunate name of ‘color,’ which has nothing to do with color in the normal sense (p. 136).”

Questions for discussion:

1. How would you explain the strong force is a short range force?

2. How would you justify the word “strong” as used in strong force?

3. How would you describe the fundamental machinery behind the strong force?

The moral of the lesson: we may understand the nature of strong force from the perspectives of its interaction range, relative strength, and the interactions of quarks.

References:

1. Feynman, R. P. (1985). QED: The strange theory of light and matter. 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. Wilczek, F. (2007). Particle physics: Hard-core revelations. Nature, 445(7124), 156-157.

4. Wilczek, F. (2015). A Beautiful Question: Finding Nature’s Deep Design. New York: Penguin Press.

5. Zee, A. (2015). Fearful Symmetry: The Search for Beauty in Modern Physics. Princeton, NJ: Princeton University Press.

Section 12–5 Pseudo forces

(Centrifugal force / Horizontal pseudo force / Gravity)

In this section, Feynman discusses centrifugal force, horizontal pseudo force, and gravity. A pseudo force is sometimes known as a fictitious force, inertial force, or d’Alembert force.

1. Centrifugal force:

“Another example of pseudo force is what is often called ‘centrifugal force’ (Feynman et al., 1963, section 12–5 Pseudo forces).”

Feynman gives an example of a pseudo force that is also known as “centrifugal force.” He adds that an observer in a rotating coordinate system, e.g., in a rotating box, will look for unknown forces that throwing things toward the walls (outwardly). According to Feynman, the centrifugal force is simply due to the fact that the observer does not have Newton’s coordinate system or the simplest coordinate system. In a sense, centrifugal force is a corrective term invented in a non-inertial frame of reference such that Newton’s second law of dynamics still holds. To be precise, one may explain that the centripetal force is a real force that is measurable by an observer in an inertial frame of reference, whereas the centrifugal force is a “corrective force” that is deducible by an observer in a rotating frame of reference.

In Principia, Newton describes an experiment of a rotating bucket of water and explains the parabolic shape of the water surface. He idealizes the water rotates with respect to absolute space. In 1966, Feynman declined a suggestion of the editor of The Physics Teacher to discuss a sprinkler problem that is related to a centrifugal force and objected to it being called “Feynman’s problem” (Feynman, 2005, p. 211). His reason was this problem is discussed in Mach’s (1883) “The Science of Mechanics.” Mach argues that the parabolic shape of the water surface in a rotating bucket should be explained in relation to the rest of the matter in the universe. Notably, Feynman discussed Mach’s principle with Wheeler before performing a sprinkler’s experiment that led Feynman banished from the lab (Wheeler, 1989).

2. Horizontal pseudo force:

“…because of the horizontal acceleration there is also a pseudo force acting horizontally and in a direction opposite to the acceleration (Feynman et al., 1963, section 12–5 Pseudo forces).”

Another example of a pseudo force can be illustrated by pushing a jar of water (e.g., to the right) along a table such that there is an acceleration. Feynman explains that the pseudo force is acting horizontally (to the left) and in a direction opposite to the acceleration. The resultant of the pseudo force and gravitational force makes an angle with the vertical, and the surface of the water is perpendicular to the resultant force during the acceleration (the water in the “backward” side of the jar moves upward.) When the jar decelerates because of a frictional force (e.g., to the left), the pseudo force is reversed (to the right) and the water in the “forward” side of the jar moves upward (Fig. 12–4). In short, the water surface is perpendicular to the effective gravity (as a result of the pseudo force).

The angle of inclination of the water surface is dependent on the horizontal push or frictional force; the angles need not be the same in both cases. We can have an alternative explanation on the angle of inclination of the water surface. In the case of acceleration (horizontal push), a water molecule on the water surface appears stationary because the pseudo force that is acting toward the left is balanced by the horizontal component of a normal reaction (to the right) on the water molecule. In the case of retardation (frictional force), a water molecule on the water surface appears stationary because the pseudo force that is acting toward the right is balanced by the horizontal component of a normal reaction (to the left) on the water molecule. The water molecule’s weight is balanced by the vertical component of the normal reaction.

3. Gravity:

“The possibility exists, therefore, that gravity itself is a pseudo force (Feynman et al., 1963, section 12–5 Pseudo forces).”

Feynman asks whether gravity is a pseudo force simply because we do not have the right coordinate system. For instance, a man in a stationary box on the earth finds himself held to the floor of the box with a gravitational force. Assuming there was no earth at all and an enclosed box moves with an acceleration g, then the man in the box might deduce a pseudo force (or gravity) which would pull him to the floor. In a sense, it is difficult to distinguish to what extent a given force is gravity and pseudo force, based on Einstein’s principle of equivalence. Some physicists may expect Feynman to clarify that the equivalence principle is, an approximation, within a small region of space. However, Feynman questions the principle by asking what happens to the people in Madagascar, on the other side of the earth—are they accelerating too?

There is no agreement among physicists whether gravity is a real force or pseudo force. Advocates that gravity is a real force may suggest one to drop an iPhone on his feet, and check whether it really hurts. On the other hand, some may argue that a pseudo force is real when one is in an accelerating car that experiences an injury or a real impact. However, in a graduate course on gravitation, Feynman clarifies that “[i]t is not possible to cancel out gravity effects entirely by uniform accelerations… It is true that we cannot imitate gravity with accelerations everywhere, that is, if we consider boxes of large dimensions (Feynman et al., 1995, pp. 91-92). In essence, it is possible to distinguish an acceleration from the gravitational force in a sufficiently large region (as compared to an infinitesimally small region).

Questions for discussion:

1. Is centrifugal force real or fictitious?

2. How would you explain the angle of inclination of the water surface in a jar while it is accelerating or decelerating on a table?

3. Is gravitational force real or fictitious?

The moral of the lesson: the centrifugal force is a pseudo force that is deducible by another observer in a rotating frame of reference, but it is difficult to define precisely to what extent an observed force is due to gravity or acceleration.

References:

1. Feynman, R. P. (2005). Perfectly reasonable deviations from the Beaten track: The letters of Richard P. Feynman (M. Feynman, ed.). New York: Basic Books.

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. Feynman, R. P., Morinigo, F. B., & Wagner, W. G. (1995). Feynman Lectures on gravitation (B. Hatfield, ed.). Reading, MA: Addison-Wesley.

4. Mach, E. (1883/1989). The Science of Mechanics. La Salle, IL: Open Court.

5. Wheeler, J. A. (1989). The Young Feynman. Physics Today, 42(2), 24-28.