Section 9–4 What is the force?

(A program for forces / Gravitational force / Restoring force)

1. A program for forces:

“…we have to have some formula for the force; these laws say pay attention to the forces. Our program for the future of dynamics must be to find the laws for the force (Feynman et al., 1963, 9–4 What is the force?)”

The title of this section “What is the force?” is potentially misleading. Instead of explaining the concept of a force, it is about some formulae for the force to deduce the motion of objects. More important, Feynman mentions that it is about a program for the future of dynamics to determine the laws for the force. In essence, the section is an introduction to the use of numerical methods to deduce the motion of objects. In general, numerical methods may involve the use of algorithms to solve mathematical problems and it is built on the foundation of approximation theory. Furthermore, numerical analysis is revolutionized by the invention of computers as well as the development of mathematical theories and algorithms.

Physics teachers may elaborate the program for forces by using Wilczek (2005) words, “…the law F = ma, which is sometimes presented as the epitome of an algorithm describing nature, is actually not an algorithm that can be applied mechanically (pun intended). It is more like a language in which we can easily express important facts about the world. That’s not to imply it is without content. The content is supplied, first of all, by some powerful general statements in that language such as the zeroth law, the momentum conservation laws, the gravitational force law, the necessary association of forces with nearby sources and then by the way in which phenomenological observations, including many (though not all) of the laws of material science can be expressed in it easily… (p. 11).” According to Wilczek, the zeroth law may refer to the law of conservation of mass.

2. Gravitational force:

“Thus the law of gravity tells us that weight is proportional to mass; the force is in the vertical direction and is the mass times g (Feynman et al., 1963, 9–4 What is the force?).”

The gravitational (radial) force near the earth’s surface is proportional to the mass of the object and is nearly independent of height for heights that are small compared with the earth’s radius R. According to Feynman, Newton’s law of gravitation tells us that weight is proportional to mass and is equal to the mass times g: W = mg = GmM/R², in which g = GM/R² and it is known as the acceleration of gravity. Essentially, the free fall of an object under gravity, which leads to the equations v_x = v₀+gt and x = x₀+v₀t + ½gt², is related to a theoretical definition of weight, W = mg. However, there is no consensus on the definition of weight. For example, one may prefer an operational definition of weight that is based on a weighing scale.

The debate on the definition of weight started during the man-in-space project in the 1960s (King, 1962). Some physicists explain that an astronaut would feel weightless and “float in the air” in a satellite coasting around the Earth. Thus, physics educators review the definition of weight and debate whether “weight is a gravitational force” (gravitational weight) or “weight is measured by a weighing scale” should be adopted for science education. The issue is not simply between the “theoretical definition” of weight as a gravitational force and the “operational definition” of weight. The controversy is also about whether weight should be theoretically defined as a “gravitational force” or “force on the support” (alternatively, a support force).

3. Restoring force:

“…the force is greater, the more we pull it up, in exact proportion to the displacement from the balanced condition, and the force upward is similarly proportional to how far we pull down (Feynman et al., 1963, 9–4 What is the force?).”

Feynman gives an example of a gadget (or a spring) which exerts a force proportional to the distance and the force is directed in an opposite direction. Based on Newton’s second law, we can describe the motion using the equation m(dv_x/dt) = –kx. In short, it means that the velocity of an object (connected to the gadget) changes at a rate proportional to x and it is in the x-direction. For reasons of simplicity, we can choose k/m = 1 and solve the equation as dv_x/dt = −x. Feynman needs not say that nothing will be gained by retaining numerous constants or imagine there is an accident in the units. We can interpret this as a sense of humor and explain that physicists have the freedom to arbitrarily (instead of accidentally) define the units involved.

What seems missing here is an explanation of the equation involving Hooke’s law (F = –kx). This equation is explained later in chapter 12: “[t]his principle is known as Hooke’s law, or the law of elasticity, which says that the force in a body which tries to restore the body to its original condition when it is distorted is proportional to the distortion. This law, of course, holds true only if the distortion is relatively small (Feynman et al., 1963, section 12–3 Molecular forces).” In De potentia restitutiva, Robert Hooke (1678) succinctly phrases the law of elastic bodies as “Ut tensio, sic vis,” which means, “the extension is proportional to the force.” Interestingly, Hooke claims his discovery of the law by stating a Latin anagram “ceiiinosssttuu” (French, 1971) without clarifying the “elastic bodies.”

Questions for discussion:

1. Why do physicists need a program of forces?

2. Do you agree with Feynman’s definition of weight?

3. What does Feynman mean when he says that “nothing will be gained by retaining numerous constants” and “imagine there is an accident in the units”?

The moral of the lesson: physicists need a program for the future of dynamics to determine the laws for the force that involves the use of numerical methods.

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. French, A. (1971). Newtonian Mechanics. New York: W. W. Norton.

3. Hooke, R. (1678). De potentia restitutiva. In R. T. Gunther (1931). Early Science in Oxford, Vol. VIII: The Cutler Lectures of Robert Hooke. Oxford: Oxford University Press.

4. King, A. L. (1962). Weight and Weightlessness. American Journal of Physics, 30(4), 387.

5. Wilczek, F. (2005). Whence the force of F= ma? III: Cultural diversity. Physics Today, 58(7), 10-11.