(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.
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