Question: This question is about the rate of flow of water through a narrow tube that varies with the pressure difference above the tube. The pressure difference is proportional to the height (h) of water as shown in the diagram below. A student measures the height h in cm with a meter ruler. The flow rate of water (R) is obtained by measuring the volume of water collected in a measuring cylinder within a time of 100 sec. Students are expected to deduce the flow rate of water R for h = 0, which is about -1.6 cm3/s-1. Moreover, they have to explain why this value of R is not physically possible.
Mark scheme: (1) The flow rate is negative.
(2) This means that water is running uphill (or gaining potential energy) or words to that effect.
Comments:
In this question, physics teachers could let students try to make sense of negative flow rate of water. Currently, the mark scheme of this question does not accept negative flow rate because it implies water running uphill or gaining potential energy. However, students sometimes need to make sense of complex numbers or negative mathematical values. For example, it is possible to have negative acceleration, negative work, negative refractive index, negative energy levels, negative magnetic susceptibility, and even negative latent heat of solidification. Furthermore, in Feynman’s Tips on Physics, Feynman (2005) mentions that “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). Interestingly, David Gross and Frank Wilczek had made mistakes with regard to signs before completing their work that earns them a Nobel Prize.
In general, physics teachers and students should not simply dismiss answers that are negative in value. For instance, we could interpret that water does evaporate (at h = 0), and as a result, water molecules can move upward and gain gravitational potential energy. In fact, this happens daily and eventually results as rainfalls. Alternatively, the value of R may be explained to be negative because of experimental errors. In other words, the incorrect (extrapolated) flow rate could be due to the experimental setup or errors in measurement. Thus, it is possible that a physical quantity is negative because of theoretical and empirical reasons. However, one may still argue whether it is possible to have a flow rate of water as -1.6 cm3/s-1. Perhaps, students can design another experiment by using a larger apparatus in a higher humidity condition?
Note:
1. During Feynman’s Nobel Lecture titled The development of the space-time view of quantum electrodynamics, he explains that “one step of importance that was physically new was involved with the negative energy sea of Dirac, which caused me so much logical difficulty. I got so confused that I remembered Wheeler's old idea about the positron being, maybe, the electron going backward in time. Therefore, in the time-dependent perturbation theory that was usual for getting self-energy, I simply supposed that for a while we could go backward in the time, and looked at what terms I got by running the time variables backward. They were the same as the terms that other people got when they did the problem a more complicated way, using holes in the sea, except, possibly, for some signs (p. 26).”
2. In an article titled Simulating physics with computers, Feynman (1982) writes that “[t]he only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative, and that we do not know, as far as I know, how to simulate. Okay, that’s the fundamental problem (p. 480).”
3. In Feynman’s Tips on Physics, Feynman explains that “another question is, what happens if va exceeds the velocity of escape? Then vescape/va is less than 1, and b turns out negative - and that doesn’t mean anything; there is no real b (Feynman et al., 2006, p. 75).”
References:
(2) This means that water is running uphill (or gaining potential energy) or words to that effect.
Comments:
In this question, physics teachers could let students try to make sense of negative flow rate of water. Currently, the mark scheme of this question does not accept negative flow rate because it implies water running uphill or gaining potential energy. However, students sometimes need to make sense of complex numbers or negative mathematical values. For example, it is possible to have negative acceleration, negative work, negative refractive index, negative energy levels, negative magnetic susceptibility, and even negative latent heat of solidification. Furthermore, in Feynman’s Tips on Physics, Feynman (2005) mentions that “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). Interestingly, David Gross and Frank Wilczek had made mistakes with regard to signs before completing their work that earns them a Nobel Prize.
In general, physics teachers and students should not simply dismiss answers that are negative in value. For instance, we could interpret that water does evaporate (at h = 0), and as a result, water molecules can move upward and gain gravitational potential energy. In fact, this happens daily and eventually results as rainfalls. Alternatively, the value of R may be explained to be negative because of experimental errors. In other words, the incorrect (extrapolated) flow rate could be due to the experimental setup or errors in measurement. Thus, it is possible that a physical quantity is negative because of theoretical and empirical reasons. However, one may still argue whether it is possible to have a flow rate of water as -1.6 cm3/s-1. Perhaps, students can design another experiment by using a larger apparatus in a higher humidity condition?
Feynman’s insights?:
There are several opportunities to highlight Feynman’s insights pertaining to negative values in this question. Firstly, in an article titled The theory of positrons, Feynman (1949) suggests that “the ‘negative energy states’ appear in a form which may be pictured in space-time as waves traveling away from the external potential backward in time (p. 749).” For example, a positron could be visualized as an electron that is moving backward in time. Similarly, Dirac was awarded Nobel Prize (1933) because he established a connection between electrons in negative-energy states and positrons. In Dirac’s (1942) words, “[n]egative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative sum of money, since the equations which express the important properties of energies and probabilities can still be used when they are negative (p. 8).”
Furthermore, in an article titled Negative probability, Feynman (1987) writes that “[i]t is usual to suppose that, since the probabilities of events must be positive, a theory which gives negative numbers for such quantities must be absurd. I should show here how negative probabilities might be interpreted (p. 235).” That is, we need not simply reject the concept of negative probability and teach students that most negative physical quantities must be absurd. On the contrary, we should explain that an ability to interpret negative physical quantity may reflect a breakthrough in thinking.
Note:
1. During Feynman’s Nobel Lecture titled The development of the space-time view of quantum electrodynamics, he explains that “one step of importance that was physically new was involved with the negative energy sea of Dirac, which caused me so much logical difficulty. I got so confused that I remembered Wheeler's old idea about the positron being, maybe, the electron going backward in time. Therefore, in the time-dependent perturbation theory that was usual for getting self-energy, I simply supposed that for a while we could go backward in the time, and looked at what terms I got by running the time variables backward. They were the same as the terms that other people got when they did the problem a more complicated way, using holes in the sea, except, possibly, for some signs (p. 26).”
2. In an article titled Simulating physics with computers, Feynman (1982) writes that “[t]he only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative, and that we do not know, as far as I know, how to simulate. Okay, that’s the fundamental problem (p. 480).”
3. In Feynman’s Tips on Physics, Feynman explains that “another question is, what happens if va exceeds the velocity of escape? Then vescape/va is less than 1, and b turns out negative - and that doesn’t mean anything; there is no real b (Feynman et al., 2006, p. 75).”
References:
1. Dirac, P. A. (1942). Bakerian lecture: The physical interpretation of quantum mechanics. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 180(980), 1-40.
2. Feynman, R. P. (1949). The theory of positrons. Physical Review, 76(6), 749-759.
3. Feynman, R. P. (1965). The development of the space-time view of quantum electrodynamics. In L. M. Brown (ed.), Selected papers of Richard Feynman. Singapore: World Scientific.
4. Feynman, R. P. (1982). Simulating physics with computers. International journal of theoretical physics, 21(6/7), 467-488.
5. Feynman, R. P. (1987). Negative probability. In B. J. Hiley & F. D. Peat (eds.), Quantum implications: essays in honor of David Bohm (pp. 235-248). London: Routledge & Kegan Paul.
6. 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.
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