Section 14–5 Potentials and fields

(Potential = (−field).(ds) / F_x = −∂U/∂x / Field = grad potential)

In this section, Feynman discusses the integral of (−field).(ds), partial derivative of force, and gradient of potential.

1. Potential = (−field).(ds):

“Since the potential energy, the integral of (−force).(ds) can be written as m times the integral of (−field).(ds), a mere change of scale… (Feynman et al., 1963, section 14–5 Potentials and fields).”

The potential energy function U(x, y, z) is equal to the integral of (−force).(ds) and it can be written as m times the integral of (−field).(ds). (When the mass of an object is equal to 1 kg, the force is numerically equal to the field and the potential energy is numerically equal to the potential.) Strictly speaking, whether the work done is positive or negative is dependent on the convention chosen. One explanation is that an external force on an object within a gravitational field results in positive work and store “more” potential energy. On the other hand, the work done by the gravitational field may release the stored energy and thus, the potential energy is reduced through the negative work. In other words, the work done by an internal (gravitational) force results in a decrease in the potential energy of an object-Earth system.

In the last chapter, Feynman has already explained that there is no work done in moving an object from one place to another where the potential energy is constant because there is no (conservative) force. In this section, he suggests the convenience of giving a scalar function Ψ instead of writing three complicated components of a vector function C. Some may criticize Feynman for the use of the symbol C instead of g, which may refer to gravitational field strength or acceleration vector that is due to gravity. However, one may guess the symbol C is chosen to represent a conservative field. More important, a definition of conservative field could be provided and specifically, it may refer to a gravitational field or electrostatic field.

2. F_x = −∂U/∂x:

“Therefore, we find that the force in the x-direction is minus the partial derivative of U with respect to x: F_x = −∂U/∂x (Feynman et al., 1963, section 14–5 Potentials and fields).”

In general, the derivative of a function U(x, y, z) with respect to x can be written as dU/dx if we ignore other variables such as y and z. Feynman explains that we can write ∂U/∂x, or include a line beside it with a subscript yz at the bottom (∂U/∂x|_yz), which means “partial derivative of U with respect to x, while keeping y and z constant.” In addition, he mentions that the mathematicians have invented a new symbol to remind us to be very careful when we are differentiating such a function, by considering only x varies, whereas y and z do not vary. Historically, in Memoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations, Adrien Marie Legendre (1786) first used the “curly d” in the form of ¶u/¶x.

Partial derivatives differ from ordinary derivatives in five ways (Roundy et al., 2015): (1) Symbolic: we need to specify varying quantities and “fixed” quantities for a ratio of small changes. (2) Graphical: a partial derivative becomes the slope of the tangent plane (or tangent space for higher dimensions) in a given direction at a specific point. (3) Algebraic procedure: the steps to obtain a partial derivative are identical to an ordinary derivative if the variables are independent of each other. (4) Verbal: we must mention the independent variables, e.g., the ratio of change of volume as pressure is changed and temperature is held constant. (5) Experimental: The representation of derivatives in experiments must include measurable (dependent) variable, manipulated (independent) variables and controlled (constant) variables.

3. Field = grad potential:

“The x-component of this “grad” is ∂/∂x the y-component is ∂/∂y, and the z-component is ∂/∂z, and then we have the fun of writing our formulas this way… (Feynman et al., 1963, section 14–5 Potentials and fields).”

According to Feynman, the mathematicians have invented a glorious new symbol, ∇, called “grad,” which is an operator that makes a vector from a scalar function. In short, we can write ∇U instead of ∂U/∂x i + ∂U/∂y j + ∂U/∂z k. The symbol ∇ (pronounced as “del”) is not a specific operator, but rather a convenient mathematical notation that simplifies many equations. (It has a different meaning when it is written as ∇².) Historically, William Rowan Hamilton (1853) invented the ∇ symbol (or “nabla”) in his Lectures on Quaternions. Writing the symbol ∇ together with a scalar function U indicate a directional derivative that always points in the direction of greatest increase of U, and it has a magnitude equal to the maximum rate of increase at the point.

Feynman ends the section by saying it is easy to show that the force on a particle due to magnetic fields is always at right angles to its velocity. He concludes that no work is done by the magnetic field on a moving charge because the motion is at right angles to the force. However, this does not mean that the magnetic field is always a conservative field. In general, if a magnetic field is changing, then it generates an electric field according to Faraday’s law of electromagnetic induction. Notably, the induced electric field can form closed paths that are non-conservative. As a result, some energy can be dissipated as internal energy in resistive materials or radiated as light energy (electromagnetic waves).

Questions for discussion:

1. How would you explain the gravitational potential energy is equal to the integral of (−force).(ds)?

2. Why does the force in the x-direction can be more precisely written as the negative partial derivative of U with respect to x: F_x = −∂U/∂x instead of F_x = −dU/dx?

3. Are magnetic forces always conservative?

The moral of the lesson: the potential energy and the integral of (−force).(ds) can be written as m times the integral of (−field).(ds), or equivalently, force is the (negative) rate of change of potential energy and field is the negative rate of change of potential.

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. Hamilton, W. R. (1853). Lectures on Quaternions. Hodges and Smith, Dublin.

3. Legendre, A. M. (1786). Mémoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations. Mémoires de l’Acad. roy. des Sciences, 1788, 7–37.

4. Roundy, D., Weber, E., Dray, T., Bajracharya, R. R., Dorko, A., Smith, E. M., & Manogue, C. A. (2015). Experts’ understanding of partial derivatives using the partial derivative machine. Physical Review Special Topics-Physics Education Research, 11(2), 020126.

Section 14–4 Nonconservative forces

(Globular star cluster / Atomic level / Energy of light)

In this section, Feynman discusses non-conservative forces from the perspectives of a globular star cluster, atomic level, and energy of light. Physics teachers may change the sequence of teaching by discussing frictional forces at the atomic level (a concept that students are likely more familiar) followed by the globular star cluster.

1. Globular star cluster:

“…we analyze a system like that great globular star cluster that we saw a picture of, with the thousands of stars all interacting… (Feynman et al., 1963, section 14–4 Nonconservative forces).”

Feynman explains that friction is apparently a non-conservative force because it has been discovered that the fundamental forces between the particles are conservative. He suggests that we can analyze a system such as a star cluster that is drifting in space. The star cluster can be conceptualized as a single object if we do not have a powerful telescope to see the details. Gravitational forces are still responsible (or “wasted”) for increasing the kinetic or potential energies of the stars inside the star cluster. Strictly speaking, the total energy of the star cluster is still conserved, but it is our inability to see the detailed motions inside that suggest the energy of the star cluster is not conserved. However, Feynman could have explained the apparent loss of kinetic energy of the star cluster using the concept of friction.

In astrophysics, dynamical friction (or gravitational drag) causes a loss of kinetic energy of celestial bodies through gravitational forces with matters in space (Chandrasekhar, 1943). Similarly, a rocket may accelerate and gain kinetic energy through the slingshot effect (or gravity assist). Simply put, this is accomplished via gravitational interactions and it results in a loss of kinetic energy in a large celestial body such as a planet. Importantly, the total energy of the rocket and the celestial body remain constant. Using the law of conservation of energy, we can conclude that a loss of kinetic energy of a massive body is compensated by an increase in a less massive body. In short, a star cluster loses kinetic energy because of dynamical friction or Chandrasekhar friction.

2. Atomic level:

“When we study matter in the finest detail at the atomic level, it is not always easy to separate the total energy of a thing into two parts, kinetic energy, and potential energy… (Feynman et al., 1963, section 14–4 Nonconservative forces).”

According to Feynman, it is not always easy to separate the total energy of a thing into kinetic energy and potential energy at the atomic level. He elaborates that it is not strictly possible to treat heat and chemical energy as being pure kinetic energy or pure potential energy. In a sense, he was inconsistent in using the term heat that could mean “primarily kinetic energy” or “a mixture of kinetic and potential energy.” Physicists prefer to define heat as a process (Romer, 2001) and sometimes use the term internal energy instead of heat. In general, “internal energy of a real gas” can be expressed as a sum of the potential and kinetic energies of the molecules of a system. On the other hand, “internal energy of an ideal gas” does not include potential energy because we assume there are no interactions between molecules.

From a microscopic perspective, frictional forces (or electromagnetic forces) cause a loss of kinetic energy of a large object via electromagnetic forces between charges in the atoms. For example, an atom may accelerate and gain kinetic energy through a collision. This is accomplished through electromagnetic interactions and it results in a loss of kinetic energy in the large object. Importantly, the total energy of the large object and atoms (including electrons) remain constant. Using the law of conservation of energy, we can conclude that a loss of kinetic energy of the large object is compensated by an increase in kinetic energy of atoms. This is similar to how a star cluster loses kinetic energy because of friction as mentioned earlier.

3. Energy of light:

“Light, for example, would involve a new form of energy in the classical theory, but we can also, if we want to, imagine that the energy of light is the kinetic energy of a photon… (Feynman et al., 1963, section 14–4 Nonconservative forces).”

Feynman says that the law of conservation of energy will appear to be incorrect if we neglect to take interactions into account. (In biology, the law of conservation of energy may appear incorrect because biology students study open systems instead of isolated systems). Generally speaking, the kinetic energy of an object is not really lost because the atoms inside are jiggling with a greater amount of kinetic energy. In addition, he clarifies that the formula T + U = constant (14.2) would still be right if we conceptualize the energy of light as the kinetic energy of photons. However, there is also a continuous loss of light energy from the object that is in the form of infrared radiation. Physicists may add that the object would emit even more photons if its temperature is higher.

The emission of light energy can be related to the oscillations of charged particles. In Chapter 28, Feynman explains that light was recognized as electromagnetic influences extending over long distances and generated by an almost incredibly rapid oscillation of the electrons in the atoms (light has both particle and wave properties). Essentially, the fundamental mechanism for the emission of photons is due to the acceleration of charges (e.g., electrons). Furthermore, Feynman adds that “the charge moves around, the unit vector wiggles, and the acceleration of that unit vector is what we are looking for. That is all. Thus E = (−q/4πϵ₀c²)(d²e_r′/dt²) is a statement of the laws of radiation…(Feynman et al., 1963, section 28–2 Radiation).” However, there is also an apparent loss of light energy in the universe that is known as redshift.

Questions for discussion:

1. How would a globular star cluster lose kinetic energy as a result of friction?

2. Is it always easy to separate the total energy of a thing into two parts: kinetic energy and potential energy (at the atomic level)?

3. How does the energy of light help to maintain the law of conservation of energy?

The moral of the lesson: Non-conservative forces (e.g., frictional forces) appear only because we neglect microscopic complications.

References:

1. Chandrasekhar, S. (1943). Dynamical Friction. I. General Considerations: the Coefficient of Dynamical Friction. The Astrophysical Journal, 97, 255–262.

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. Romer, R. H. (2001). Heat is not a noun. American Journal of Physics, 69(2), 107–9.

Section 14–3 Conservative forces

(Path independence / Zero potential energy / Potential energy curves)

In this section, Feynman discusses conservation forces from the perspectives of path independence, zero potential energy, and potential energy curves.

1. Path independence:

“…in general the work depends upon the curve, but in special cases, it does not (Feynman et al., 1963, section 14–3 Conservative forces).”

In general, the work done by a force may be dependent on the path of an object. Specifically, the work done by a conservative force is independent on the path of the object, for example, whether it is curved or straight. Feynman mentions that the word “conservative” does not involve political ideas and it is misleading. In essence, the force is conservative (or “mechanical energy is conserved” during the work done) if the integral of the force times the distance in going from position 1 to position 2 is the same for any curve that connects this pair of points. Interestingly, Feynman explains in the next section that energy is still conserved even for non-conservative forces. However, some may criticize Feynman’s definition of conservative force.

Note: The terms non-dissipative force and dissipative force could be used instead of conservative force and non-conservative force respectively.

We should clarify that path independence is only a feature of a conservative force. A more comprehensive definition of conservative force may include the following: (1) The work done by a conservative force can be related to a difference between the initial and final value of a potential energy function: DW = -DU. (2) The work done is reversible and it is independent of the path of the object; the total work done is zero if the starting point and ending point are the same, DW = 0. (3) Conservative force is constant or time-independent. (4) The curl of a conservative force is zero, for example, the curl of an inverse square law force is zero. (5) The conservative force may be a spring force (F = -kr) or a central force (F = GMm/r² or F = Qq/4pe₀r²).

2. Zero potential energy:

“The constant has been chosen here so that the potential is zero at infinity. Of course, the same formula applies to electrical charges because it is the same law (Feynman et al., 1963, section 14–3 Conservative forces).”

According to Feynman, we can assume an object located at position P has zero potential energy, U(P) = 0. If we use any other point, say Q, instead of P, it would turn out that the potential energy of the object (at any point) to be changed only by the addition of a constant, C. Thus, the difference in potential energy between two points (1) and (2): [U(1) + C] – [U(2) + C] = U(1) − U(2) and this shows that it does not matter if we add a constant to the potential energy. Physics teachers may add that we are unable to determine the absolute potential energy (or absolute kinetic energy) by using any experiment. Essentially, physicists measure the change in potential energy (or its effect) and relate it to a conservative force.

We can choose a “standard” point P and set the potential energy at this point as zero. Although Feynman says that the point P can be arbitrarily chosen, he has demonstrated the usefulness of setting zero potential energy at infinity. Using a rocket problem, he assumes the kinetic energy of the rocket to be zero at a point somewhere far away and the potential energy is GmM divided by infinity, which is zero. In Tips on Physics, he writes that “K.E. + P.E. at a = K.E. + P.E. at ¥ (Feynman et al., 2006, p. 73).” Note that we can idealize an object that is significantly far from the Earth as located at infinity. Alternatively, physics teachers could explain that the potential energy –GMm/r is approximately zero where r >> R (R: radius of the Earth).

3. Potential energy curves:

“Therefore we find curves of potential energy in quantum mechanics books, but very rarely do we ever see a curve for the force between two molecules… (Feynman et al., 1963, section 14–3 Conservative forces).”

In a curve of the potential energy U(r) as shown in Fig. 14–3, the minimum of potential energy at r = d means that the work done (or the change in potential energy) is nearly zero if we move a short distance. There is only a little change in potential energy at the bottom of the curve, or equivalently, the force is approximately zero at the equilibrium point. One of Feynman’s reasons for discussing the potential energy curves is to bring out the idea that force (as compared to energy) is not particularly suitable for quantum mechanics. Physicists prefer to use the term interaction, but they still adopt phrases such as “strong force” and “weak force.” More important, we are unable to measure force by determining the acceleration of sub-microscopic particles; in a sense, the word force is used as a metaphor in quantum theory.

Feynman mentions that if several conservative forces are acting on an object at the same time, then the total potential energy of the object is the sum of the potential energies from each of the conservative forces. In addition, molecular forces are a complicated function of the positions of the atoms instead of simply a sum of terms from pairs of atoms. In Volume II, Feynman incorrectly reasons that “[f]or any radial force the work done is independent of the path, and there exists a potential. If you think about it, the entire argument we made above to show that the work integral was independent of the path depended only on the fact that the force from a single charge was radial and spherically symmetric (Feynman et al., 1964, section 4–4).” However, it can be shown that the curl of an inverse square force (1/r²) is zero, but the curl of a force that is inversely proportional to r is not definitely zero (Kleppner & Kolenkow, 2014).

Questions for discussion:

1. How would you criticize Feynman’s definition of conservative force?

2. Why do physicists define potential energy of an object to be zero when it is located at infinity?

3. Would you use the term conservative force in the context of quantum mechanics?

The moral of the lesson: in general, the work done by a force depends upon the path of an object, but we may say that the force is conservative if it does not depend on the path in special cases.

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. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

4. Kleppner, D., & Kolenkow, R. (2014). An Introduction to Mechanics (2nd ed.). Cambridge: Cambridge University Press.

Section 14–2 Constrained motion

(Frictionless constraints / Constraint forces / Analysis of constraint forces)

In this section, Feynman discusses frictionless constraints, constraint forces, and analysis of constraint forces.

1. Frictionless constraints:

“…suppose that we have a sloping or a curved track, and a particle that must move along the track, but without friction... These are examples of what we call fixed, frictionless constraints (Feynman et al., 1963, section 14–2 Constrained motion).”

Feynman gives two examples of fixed and frictionless constraints: (1) a particle moves along a sloping or a curved track that is frictionless; (2) a string constrains a weight to move circularly about a pivot point. He adds that the pivot point can be changed by having the string hit a peg such that the path of the weight is along two circles of different radii. Alternatively, Landau and Lifshitz (1976) explain that “[i]t is often necessary to deal with mechanical systems in which the interaction between different bodies (or particles) takes the form of constraints, i.e., restrictions on their relative position. In practice, constraints are effected by means of rods, strings, hinges and so on (p. 10).” Generally speaking, the word constraints can be classified as two categories: (i) constraints due to forces, and (ii) constraints due to equations (Waghmare, 1990). Physicists do not always conceptualize constraints as physical objects.

Some textbook authors describe at least four kinds of constraints: (1) Holonomic: the constraint on the positions of a system of particles, e.g., a particle is constrained to move on a plane: Ax + By + Cz = D. (2) Nonholonomic: constraints that restrict the velocities of particles, e.g., av₁ + bv₂ + cv₃ + … = N. (3) Scleronomous: constraints that do not depend on time explicitly, e.g., a pendulum of inextensible string can be described by the equation, x² + y² = l². (4) Rheonomous: constraints that specify time explicitly, e.g., a pendulum can be described by the equation, x² + y² = l²(t) where l(t) is the length of a string at time (t). Feynman would complain about these terms and possibly say velocity-independent (holonomic), velocity-dependent (nonholonomic), time-dependent (rheonomous), and time-independent (scleronomous).

2. Constraint forces:

“In motion with a fixed frictionless constraint, no work is done by the constraint because the forces of constraint are always at right angles to the motion (Feynman et al., 1963, section 14–2 Constrained motion).”

According to Feynman, “forces of constraint” mean forces which are applied to an object directly by a constraint itself. It may be a contact force with a track or a tension in a string. During an object’s motion with a fixed and frictionless constraint, there is no work done because the forces of constraint are always at right angles to the motion. However, some physicists may criticize Feynman’s definition of constraint forces. For example, we should not assume that frictional forces are definitely not forces of constraint. Note that static frictional forces can constrain an object to move circularly and the work done by static frictional forces is also zero.

We may find some of the following features in a more comprehensive definition of constraint forces: (1) It may be a contact force, tension, or static frictional force (non-sliding). (2) The constraint force on a particle could be exerted by a rod, string, or surface of a track. (3) It restricts the path of a particle or the motion of a system of particles. (4) It limits a particle’s freedom of motion under certain geometric or kinematic conditions. (5) The work done by a constraint force is zero because the force is perpendicular to the motion of the particle (W = 0). On the other hand, one may relate constraint forces to D’Alembert’s principle of virtual work.

3. Analysis of constraint forces:

“…the work done by the net force is equal to the sum of the works done by all the parts into which we have divided the force in making the analysis (Feynman et al., 1963, section 14–2 Constrained motion).”

Feynman explains that the work done by the resultant force can be analyzed as the sum of the works done by various “components” of forces. For instance, we can analyze the forces in terms of the x-component of all forces and the y-component of all forces, or using other co-ordinate systems. The analysis of constraint forces in this section appears simple because the constraints are idealized as fixed and frictionless. Physics teachers should add that the analysis involves mathematical skills that can be improved by practice. In some mechanical problems, it is either difficult or even impossible to determine the constraint forces on the objects.

There are at least two types of difficulties in solving mechanical problems that are related to constraints: “First, the co-ordinates r_i are no longer all independent, since they are connected by the equations of constraint: hence the equations of motion are not all independent. Second, the forces of constraint, e.g., the force that the wire exerts on the bead (or the wall on the gas particle), is not furnished a priori (Goldstein, Poole, and Safko, 2001, p. 13).” Simply put, some mechanical problems have many constraint forces and require as many independent equations of constraint that help to solve the equations of motion, provided it is solvable. In essence, the constraint forces are related to the motion of the particles in the system, and they can be calculated after the motion has been determined (or use F = -dV/dr for spherical co-ordinates).

Questions for discussion:

1. How would you classify the different types of constraints?

2. How would you criticize Feynman’s definition of constraint forces?

3. How would you determine the constraint motion or constraint forces?

The moral of the lesson: during the motion of a particle with a fixed and frictionless constraint, there is no work done by the constraint because the constraint forces are perpendicular to the motion.

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. Goldstein, H., Poole, C. P., & Safko, J. L. (2001). Classical Mechanics (3rd ed.). Massachusetts: Addison-Wesley.

3. Landau, L. D., & Lifshitz, E. M. (1976). Mechanics (3rd ed.). Oxford: Pergamon Press.

4. Waghmare, Y. R. (1990). Classical Mechanics. New Delhi: Prentice Hall.

Section 14–1 Work

(The word “work” / Physical work / Physiological work)

In this section, Feynman discusses the word “work” and definitions of physical work and physiological work.

1. The word “work”:

“The physical word ‘work’ is not the word in the ordinary sense of ‘Workers of the world unite!,’ but is a different idea (Feynman et al., 1963, section 14–1 Work).”

Feynman explains that the word “work” does not have the same meaning as it is being used in a political slogan, “Workers of the world, unite!” In one of his Messenger Lectures, Feynman (1965) mentions that “physicists delight themselves by using ordinary words for something else (p. 84).” In his public lecture on QED, he adds that “[p]hysicists often use ordinary words such as ‘work’ or ‘action’ or ‘energy’ or even, as you shall see, “light” for some technical purpose. Thus, when I talk about ‘work’ in physics, I don’t mean the same thing as when I talk about ‘work’ on the street (Feynman, 1985, p. 10). However, some physicists advocate banning the word “work” from physics. They prefer to use the phrase “force-displacement product” or “force-displacement integral” (e.g., Hilborn, 2000).

It is worthwhile to mention a historical definition of work and to know how it was previously defined. In Calculation of the Effect of Machines, or Considerations on the Use of Engines and their Evaluation, Coriolis (1829) writes that “[w]e propose the appellation dynamical work, or simply work, for the quantity òPds… This name will not be confused with any other mechanical denomination. It seems suitable to give the right idea of the thing, by maintaining its common meaning of physical work… this name is then suitable to designate the union of the concepts, displacement, and force (Capecchi, 2012, p. 368).” Students should realize that physicists need to redefine work for more complicated problems in physics.

2. Physical work:

“Physical work is expressed as ∫F.ds, called ‘the line integral of F dot ds,’… (Feynman et al., 1963, section 14–1 Work).”

In general, physical work can be expressed as ∫F.ds and it means that if a force is in one direction and an object is forced to move in another direction, then only the component of force in the direction of the displacement contributes to the work done. Although the rule is simply “force times distance,” it is only the component of force in the same direction as the movement of the object times Δs or, equivalently, the component of displacement in the direction of the force (F) times the force. In certain mechanical problems, it is imprecise to define Δs as “the displacement of the object” or simply “the displacement.” Importantly, physicists need a more specific definition of Δs depending on the nature of a mechanical problem.

Some textbook authors simply specify Ds as the displacement of the point of application of the force. To be precise, the displacement of an object is not necessarily the same as the displacement of the center-of-mass of the object for a deformable or rotating object (Jewett, 2008). In complicated problems, one may need to redefine work or introduce terms such as “pseudo-work” or “center-of-mass work.” Interestingly, Mallinckrodt and Leff (1992) classify seven types of work that can be done on a system of particles interacting internally or with its environment. In essence, a definition of work is related to the nature of force (external or internal) and conditions such as an inertial frame of reference (frame dependent or frame invariant).

3. Physiological work:

“It is a fact that when one holds a weight he has to do “physiological” work (Feynman et al., 1963, section 14–1 Work).”

According to Feynman, there are two kinds of muscles in the human body: (1) striated or skeletal muscle is the type of muscle we have in our arms which is under voluntary control; (2) smooth muscle is like the muscle in the intestines which works very slowly. He adds that we have to generate effort for striated muscles to hold up a weight due to a need for enormous volleys of nerve impulses coming into the muscles. Biology students may disagree with Feynman’s classifications by describing a third kind of muscles that are known as cardiac muscles. However, physics teachers may elaborate that most of the muscles in the human body work like a third class lever systems (first class lever systems have the most mechanical advantage).

Another example of physiological work can be illustrated by a person pushing against a fixed wall. There is also no external work because the displacement of the wall is zero. In this case, a considerable amount of energy is expended in a human body to keep the muscles balanced during the act of pushing. On the other hand, the efficiency of the muscles is about 20% during cycling (Davidovits, 2008). In other words, about one-fifth of the chemical energy in the muscle is converted into useful work. Specifically, one may measure physiological work in terms of oxygen consumption and heart rate using a respirometer and cardio-tachometer.

Questions for discussion:

1. Should we use the phrase “force-displacement product” or “force-displacement integral” to replace the word work?

2. Is there a more rigorous definition of work in physics?

3. How would you explain the physiological work of holding a weight?

The moral of the lesson: physicists use ordinary words such as work that means ∫F.ds and it is called “the line integral of F dot ds”; the term ds needs to be redefined depending on the nature of mechanical problems.

References:

1. Capecchi, D. (2012). History of Virtual Work Laws: A History of Mechanics Prospective. Milan: Springer-Verlag Italia.

2. Coriolis, G. (1829). Du Calcul de l'effet des Machines, ou Considérations sur l'emploi des Moteurs et sur Leur Evaluation. Paris: Carilian-Goeury, Libraire.

3. Davidovits, P. (2008). Physics in Biology and Medicine (3rd Edition). Burlington: Elsevier.

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

5. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

6. 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.

7. Hilborn, R. C. (2000). Let’s ban work from physics!. The Physics Teacher, 38(7), 447-447.

8. Jewett Jr, J. W. (2008). Energy and the confused student I: Work. The Physics Teacher, 46(1), 38-43.

9. Mallinckrodt, A. J., & Leff, H. S. (1992). All about work. American Journal of Physics, 60(4), 356-365.