Section 4–2 Gravitational potential energy

(Perpetual motion / Gravitational potential energy / Principle of virtual work)

Feynman derives a formula for gravitational potential energy near the surface of the earth by using a relatively simple reasoning. In this section, the three interesting concepts discussed are the perpetual motion, gravitational potential energy, and the principle of virtual work.

1. Perpetual motion:

“…there is no such thing as perpetual motion with these weight-lifting machines. (Feynman et al., 1963, section 4.2 Gravitational potential energy).”

It is remarkable that Feynman relates the law of conservation of energy to the impossibility of perpetual motion. Importantly, we must be careful in defining the concept of perpetual motion. Strictly speaking, the motions of celestial bodies such as planets only appear perpetual, but their kinetic energy is gradually decreased due to the presence of interstellar medium and solar wind. Similarly, the electric current in a superconductor is gradually decreased after some time due to the effect of “flux creep.” Thus, it is a fact rather than a hypothesis that there is no such thing as perpetual motion with weight-lifting machines. In daily lives, weight-lifting machines lift and lower weights with a result of more thermal energy generated due to the presence of frictional forces and air resistance.

Some physicists prefer to distinguish perpetual motion machines into two kinds. For example, in a textbook titled Heat and Thermodynamics, Zemansky and Dittman (1981) write that “[a] machine that creates its own energy and thus violates the first law is called perpetual motion machine of the first kind. A machine that utilizes the internal energy of only one heat reservoir, thus violating the second law, is called perpetual motion machine of the second kind (p. 147).” Feynman did not praise many textbooks, but in chapter 45 of The Feynman Lectures on Physics, he says that “There are also good equation reference books, such as Zemansky’s Heat and Thermodynamics, where one can learn more about the subject (Feynman et al., 1963, p. 45-1).”

In general, there are two classes of machines: (1) non-reversible machines which include all real machines; (2) reversible machines which are not attainable in practice by having all careful designs of bearings and levers. Feynman mentions that if Machine A is a reversible machine, it can reversibly lower one unit of weight by one unit of distance and lift a three-unit weight by a distance X. However, it is potentially confusing to first state Machine B as not necessarily reversible, and then later suggest that it could be really reversible. Perhaps Feynman could first state Machine B as a perpetual motion machine of the first kind because this machine creates its own energy and violates the law of conservation of energy. Next, he could include Machine C as a closely reversible machine that is the best machine that we can have.

2. Gravitation potential energy:

“…We call the sum of the weights times the heights gravitational potential energy—the energy which an object has because of its relationship in space, relative to the earth (Feynman et al., 1963, section 4.2 Gravitational potential energy).”

Feynman derives the formula for gravitational potential energy near the surface of the earth by using the concept of a perpetual motion machine. This is a reversible weight-lifting machine that can lift one weight and lowering another weight at the same time. It functions like a frictionless seesaw that can swing up and down perpetually. For instance, it can continue reversibly to lower an object of 1 kilogram by a certain distance d, and lift another object of m kg by the distance d/m. However, it is impossible to have perpetual motion because of the presence of friction and air resistance. Although this kind of ideal perpetual motion machine is impossible experimentally, we can deduce intuitively a numerical quantity that remains constant when a system of objects is moving up and down simultaneously.

The numerical quantity, that is a sum of the weights times the heights, is commonly known as gravitational potential energy. This formula of potential energy (weight × height) is only valid if objects are not too far away from the earth such that the weight due to a gravitational force is approximately constant. However, the gravitational force weakens as the objects are significantly higher above the ground. On the other hand, if a perpetual motion machine involves electrical forces, or we are “lifting” charge carriers away from other charged objects by using some imaginary levers, then this numerical quantity is called electrical potential energy. This is based on the general principle in which the change in energy is equal to a force times the distance moved: (energy change) = (force) × (distance moved).

3. Principle of virtual work:

“…This approach is called the principle of virtual work because in order to apply this argument we had to imagine that the structure moves a little—even though it is not really moving or even movable (Feynman et al., 1963, section 4.2 Gravitational potential energy).”

Feynman suggests that we can work out the law of “balance” (or law of the lever) with regard to the statics of complicated bridge arrangements. We can apply the principle of virtual work in which we imagine a system has moved arbitrarily in very small (infinitesimal) displacements even though the system is not really moving or movable. For example, we can initially explain that the weight W times 4 nanometers (very small displacement) down, plus 60 pounds times 2 nanometers up, plus 100 pounds times 1 nanometers add up to nothing. Next, we can multiply these very small displacements by a factor to achieve the following equation as derived by Feynman: −4W + (2)(60) + (1)(100) = 0. In short, we have applied the principle of conservation of energy by imagining very small motions.

Interestingly, Feynman modifies the triangle in the epitaph of Stevinus (see the figure below) to a right-angled triangle. In essence, the tensions of the chains due to the weights in contact with both (upper part) sides of the triangle must be the same (T₁ = T₂) at the highest point of the triangle. Note that the tensions are not the same throughout the chain because it is not massless. Additionally, the number of “circles” in the lower part of the chain does not matter and they are symmetrical.

According to Dugas, Stevinus clearly stated the principle of virtual work as “The distance traveled by the force acting is to the distance traveled by the resistance as the power of the resistance is to that of the force acting (Dugas, 1955/1988, p. 127).” This statement is applicable to systems of pulleys based on the assumption of the impossibility of perpetual motion. In a similar sense, the tensions through the chain of weights along the two sides of the triangle (as shown in the epitaph of Stevinus) must be in balance.

Questions for discussion:

1. Why there is no perpetual motion machine in the real world?

2. What is the limitation of the formula for gravitational potential energy?

3. How would you explain the epitaph of Stevinus by using the principle of virtual work?

The moral of the lesson: the formula for gravitational potential energy near the surface of the earth can be derived by using a perpetual motion machine that is not achievable in practice.

References:

1. Dugas, R. (1955/1988). A History of Mechanics (trans. by J. R. Maddox). New York: Central Book Company.

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. Zemansky, M. W., & Dittman, R. H. (1981). Heat and Thermodynamics. New York: McGraw-Hill.