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.

Section 13–4 Gravitational field of large objects

(Infinite plane sheet / Thin spherical shell / Inside a sphere)

In this section, Feynman discusses how a gravitational field or potential energy varies due to an infinite plane sheet, thin spherical shell, and inside a sphere.

1. Infinite plane sheet:

“The gravitational field C at a mass point produced by an infinite plane sheet of matter (Feynman et al., 1963, section 13–4 Gravitational field of large objects).”

Feynman illustrates the validity of Gauss’ law for gravitation by calculating the gravitational field produced by an infinite plane sheet of material. He explains that the gravitational force is independent of distance because if we are closer, most of the matter is pulling at an unfavorable angle; if we are farther away, even more matter is situated more favorably to exert a pull toward the plane. Specifically, the force is weaker by the inverse square if we are farther away, but within the same cone of solid angle, it would be compensated by more matter that is proportional to the square of the distance. In other words, the farther away an object is from the infinite plane, more mass will be included within a “circle of influence” (or field of view with a cone of solid angle). Physics teachers could emphasize that Gauss’ law is applicable here because the gravitational force follows the inverse square law.

Gauss’ law for gravitation can be stated as: the flux of gravitational field through any surface is proportional to the total gravitational charge (mass). Similarly, we can use Gauss’ law to solve an electrical problem involving a large electrically charged plate that has an amount σ of charge per unit area. The electric field at any point outside the large plate is approximately equal to σ/2ϵ₀ in the outward direction and the approximation gets better if the plate is larger. If the plate is infinitely large, any point at a distance away from the plate is equivalent to every other point away from the plate. In Feynman Lectures on Computation, he adds that “defining a charge density here is important — it would be meaningless to discuss the total charge for an infinite plate (Feynman, 1996, p. 232).” This is because the total charge would be infinity in the idealized infinite plate.

2. Thin spherical shell:

“Thus, for a thin spherical shell, the potential energy of a mass m′, external to the shell, is the same as though the mass of the shell were concentrated at its center (Feynman et al., 1963, section 13–4 Gravitational field of large objects).”

We have assumed the gravitational force produced by the Earth at a point on the surface or outside it, is the same as if all the mass of the Earth were located at its center. This assumption is related to Newton’s shell theorem in which we conceptualize a spherical body as thin uniform hollow shells. Feynman’s approach is to work with the potential energy instead of the gravitational force because we do not have to worry about angles, that is, we merely add the potential energies of all the particles. However, one may expect Feynman to take the derivative of the potential energy (or at least mention the result) to obtain the gravitational force. This step is not really complicated and it may help students to appreciate Newton’s shell theorem better.

It may not be clear to students how the work done results in the potential energy due to a ring of mass is −Gm′dm/r. In a sense, Feynman partially fills up this gap only in the next chapter. In Chapter 14, he adds that “[t]he potential energy of gravitation for point masses M and m, a distance r apart is U(r) = −GMm/r (14.5). The constant has been chosen here so that the potential is zero at infinity (Feynman et al., 1963, section 14–3 Conservative forces).” Importantly, the work done by gravity results in negative gravitational potential energy because the gravitational force (not external force) is attractive and brings objects closer. The decrease in potential energy also implies an increase in the kinetic energy of the objects.

Lastly, Feynman only mentions the Earth acts as if all the materials were at the center by imagining it as a series of spherical shells. It is not trivial to explain how the gravitational force between two spherical objects is equivalent to the case when the mass of both objects are located at their center of mass. Young and Freedman (2004) explain that “the forces the two bodies exert on each other are an action-reaction pair, and they obey Newton’s third law. So we have also proved that the force that m exerts on the sphere M is the same as though M were a point. But now if we replace m with a spherically symmetric mass distribution centered at m’s location, the resulting gravitational force on any part of M is the same as before, and so is the total force. This completes our proof.” Notably, they have assumed that the distribution of mass remains spherical (or undistorted) in the presence of a massive object.

3. Inside the sphere:

“If the potential energy is the same no matter where an object is placed inside the sphere, there can be no force on it (Feynman et al., 1963, section 13–4 Gravitational field of large objects).”

Inside a spherical shell, the work done on an object by gravity, W = −Gm′m/a, is independent of R and independent of position. Therefore, no force is needed to move the object anywhere within the hollow sphere and no work is done when we move it about inside. It is similar to no work is done in moving the object horizontally on a flat table. (In short, W = −Gm′m/a Þ same potential energy within a shell Þ gravitational force, F = 0 within a shell.) If a point mass m is positioned at a distance r from the center of the Earth, we can determine the gravitational force by ignoring the spherical shells whose radius are longer than r; we only need to focus on the spherical shells whose radius are shorter than r.

Note: Physics teachers could compare the potential energy inside and outside a shell as shown below.

(i) Outside the shell: r_max – r_min = (R + a) – (R – a) = 2a

 Þ W = −Gm′m(r_max – r_min)/2aR = −Gm′m/R

(ii) Inside the shell: r_max – r_min = (R + a) – (a – R) = 2R

 Þ W = −Gm′m(r_max – r_min)/2aR = −Gm′m/a (constant)

The validity of Gauss’ law depends on the inverse square law of Coulomb. In volume II, Feynman provides a better explanation: “[i]f the force law were not exactly the inverse square, it would not be true that the field inside a uniformly charged sphere would be exactly zero. For instance, if the force varied more rapidly, like, say, the inverse cube of r, that portion of the surface which is nearer to an interior point would produce a field which is larger than that which is farther away, resulting in a radial inward field for a positive surface charge (Feynman et al., 1964, section 5.8 Is the field of a point charge exactly 1/r²?).” Thus, the net electric field is also zero for any two symmetric or opposite cones (same solid angle) diverging from the opposite sides of a point P as the fields due to them are in opposite direction (See Feynman et al., 1964, Fig. 5–9). We can apply the same logic for gravitational fields.

Questions for discussion:

1. How would you explain the gravitational field produced by an infinite plane sheet of material is constant or independent of distance?

2. Why is the gravitational force due to the Earth at a point on the surface or outside it, is the same as if all the mass of the earth were located at its center?

3. Why is the gravitational force equal to zero for any point within a spherical shell of mass?

The moral of the lesson: the gravitational force due to the Earth at a point on its surface or beyond, is the same as if all the mass of the Earth were located at its center, can be shown using Newton’s shell theorem.

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

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

4. Young, H. D., & Freedman, R. A. (2004). Sears and Zemansky’s University Physics (11th ed.). California: Addison Wesley.

Section 13–3 Summation of energy

(Time derivative of energy / Energy between the pairs / Sum of the works)

In this section, Feynman discusses the time derivative of energy, gravitational potential energy between any pairs of particles, and the sum of works done that brought in the particles.

1. Time derivative of energy:

“… we see that Eqs. (13.16) and (13.15) are precisely the same, but of opposite sign, so that the time derivative of the kinetic plus potential energy is indeed zero (Feynman et al., 1963, section 13–3 Summation of energy).”

Feynman moves on to a more general consideration of what happens to the total energy of a large number of particles. He suggests that we can prove the constancy of summation of energies by differentiating the kinetic energies of all particles and gravitational potential energies of all pairs of particles in a system. This is not a conventional way to show the total energy is a constant. Alternatively, one may explain that the total energy of a closed (isolated) system is constant. On the other hand, the total energy of an open system is not constant because it allows energy transfer. The constancy of summation of energies is dependent on how we define a system of particles.

In chapter 4, Feynman states that the law of conservation of energy is closely related to the symmetry of time in quantum mechanics. In his Ph.D. thesis titled A New Approach to Quantum Theory, he writes that “in the theory of action at a distance, the kinetic energy of the particles is not conserved. To find a conserved quantity, one must add a term corresponding to the ‘energy in the field’ (Feynman, 1942, p. 11).” He proceeds to show the constancy of total energy of particles (or Noether’s theorem) using a transformation of time-displacement. Based on an interview with Feynman, Mehra (1994) writes that “Feynman proved a thing called Noether’s theorem, not knowing that it was known (p. 132).”

2. Energy between the pairs:

“…the potential energy is also simple, it being also just a sum of contributions, the energies between all the pairs (Feynman et al., 1963, section 13–3 Summation of energy).”

Feynman explains that the involvement of potential energy is simple because it is only about the gravitational potential energy between all pairs of particles. In addition, the symbol S means that given values of i and j occur only once and we can keep track of this if we let i range over all values 1, 2, 3, …, and for each i, we can restrict the range of j to be greater than i. Simply phrased, one may specify the condition i < j (or state it as a subscript) to prevent a possibility of ‘double counting’ in listing the terms. It could be clearer if Feynman also writes down all the terms, for example, -Gm₁m₂/r₁₂ -Gm₂m₃/r₂₃ -Gm₁m₃/r₁₃, for a system of three particles.

Physics teachers could explain that the gravitational potential energy is a property of a system of particles instead of an individual particle. In other words, we cannot divide the total gravitational potential energy by specifying the quantity of energy that belongs to every particle. Some textbook authors elaborate that “if M >> m, as is true for Earth (mass M) and a baseball (mass m), we often speak of ‘the potential energy of the baseball.’ We can get away with this because, when a baseball moves in the vicinity of Earth, changes in the potential energy of the baseball – Earth system appear almost entirely as changes in the kinetic energy of the baseball, since changes in the kinetic energy of Earth are too small to be measured (Halliday, Resnick, & Walker, 2005).” In essence, it is inappropriate to specify the gravitational potential energy of individual object if the objects in a system are comparable in mass.

3. Sum of the works:

“Therefore the work done is the sum of the works done by each because if F₃ can be resolved into the sum of two forces, F₃ = F₁₃ + F₂₃… (Feynman et al., 1963, section 13–3 Summation of energy).”

We can understand the sum of works done (or change in potential energy of particles) as follows: Firstly, when we bring in the first particle, there is no work done because no other particles are present to exert a force on it. Secondly, when we bring in the second particle, there is only work done between the two particles, W₁₂ = −Gm₁m₂/r₁₂. Thirdly, when we bring in the third particle, the work done is the sum of the works done against the first two particles because F₃ can be resolved as a sum of two forces, F₃ = F₁₃ + F₂₃. Mathematically, the net work done can be written as ½(−Gm₁m₂/r₁₂ −Gm₂m₃/r₂₃ −Gm₁m₃/r₁₃) and it is independent of the order of the particles that are brought to their final positions. However, one may explain that the potential energy of particles is stored in the fields instead of sharing among the particles.

In the case of electrostatic energy, it can be expressed as “U = ½[q₁ϕ(1)+q₂ϕ(2)]… That is why we need the factor ½ (Feynman et al., 1964, Chapter 8).” Using Feynman’s notations, the gravitational potential energy stored in three objects can also be written as U = ½[m₁Y(1) + m₂Y(2) + m₃Y(3)]. Initially, Feynman explains that the idea of locating the energy in the field is inconsistent with the assumption of the existence of point charges. To resolve this inconsistency, he suggests that “[o]ne way out of the difficulty would be to say that elementary charges, such as an electron, are not points but are really small distributions of charge… (Feynman et al., 1964, Chapter 8).” One may use the principle of mass-energy equivalence to argue that the gravitational field has energy, and thus it has mass.

Questions for discussion:

1. What is/are the implication(s) if the time derivative of the kinetic plus potential energy is zero?

2. How would you explain the gravitational potential energy is a property of a system of particles?

3. How would you explain the work done is the sum of the works done by each force?

The moral of the lesson: we can write the potential energy as a sum over each pair of particles because gravitational forces obey the principle of superposition of forces.

References:

1. Feynman, R. P. (1942/2005). Feynman’s thesis: A New Approach to Quantum Theory. Singapore: World Scientific.

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. Halliday, D., Resnick, R., & Walker, J. (2005). Fundamentals of Physics (7th ed.). New York: Wiley.

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