Suppose that in addition to the removal of y from the system at a rate –by, y enters the system at a constant rate a, independent of y and t. The net rate of change of y is given by
Then we go on to discuss how you can learn things about a differential equation without actually solving it.
It is often easier to write down a differential equation describing a problem than it is to solve it… However, a good deal can be learned about the solution by examining the equation itself. Suppose that y(0) = 0. Then the equation at t = 0 is dy/dt = a, and y initially grows at a constant rate a. As y builds up, the rate of growth decreases from this value because of the –by term. Finally when a – by = 0, dy/dt is zero and y stops growing. This is enough information to make the sketch in Fig. 2.13.
The equation is solved in Appendix F. The solution is
… The solution does have the properties sketched in Fig. 2.13, as you can see from Fig. 2.14.
Figure 2.13 looks similar to this figure
Sketch of the initial slope a and final value a/b of y when y(0) = 0. In this figure, a=b=1.
And Fig. 2.14 looks like this
A plot of y(t) using Eq. 2.26, with a=b=1.
However, Eq. 2.26 is not the only solution that is consistent with the sketch in Fig. 2.13. Today I want to present another function that is consistent with Fig. 2.13, but does not obey the differential equation in Eq. 2.25.
Let’s examine how this function behaves. When bt is much less than one, the function becomes y = at, so it’s initial growth rate is a. When bt is much greater than one, the function approaches a/b. The sketch in Fig. 2.13 is consistent with this behavior.
Below I show both Eqs. 2.26 and 2.26’ in the same plot.
A plot of y(t) using Eq. 2.26 (blue) and Eq. 2.26' (yellow), with a=b=1.
The function in Eq. 2.26 (blue) approaches its asymptotic value at large t more quickly than the function in Eq. 2.26’ (yellow).
The moral of the story is that you can learn a lot about the behavior of a solution by just inspecting the differential equation, but you can’t learn everything (or, at least, I can’t). To learn everything, you need to solve the differential equation.
By the way, if Eq. 2.26’ doesn’t solve the differential equation in Eq. 2.25, then what differential equation does it solve? The answer is
In Chapter 2 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss semilog plots, where the vertical axis is marked using a logarithmic scale. In this case, a constant distance along the vertical axis corresponds to a constant multiple in the numerical value. In other words, the distance between 1 and 2 is the same as the distance between 2 and 4, which is the same as the distance between 4 and 8, and so on. Looking at a semilog plot helps the reader get a better understanding of how logarithms and exponentials work. Yet, what would be a really useful learning tool is not something readers just look at, but something that they can hold in their hands, something they can manipulate, something they can touch.
Enter the slide rule. Sixty years ago, when electronic calculators did not yet exist, the slide rule is how scientists and engineers performed calculations. I didn’t use a slide rule in school. I’m from the first generation that had access to electronic calculators. They were expensive but not prohibitively so, and we all used them. But my dad used a slide rule. He gave me his, mainly as an artifact of a bygone era. I rarely use it but I have kept it in honor of him. It was made by the Keuffel & Esser Company in New York. It is a fairly fancy one and has a variety of different scales.
First, let’s look at the C and D scales. These are marked logarithmically, just like semilog paper. In fact, if you wanted to draw you own semilog graph paper, you could take out my dad’s slide rule, hold it vertical, and mark off the tick marks on your plot axis. On dad’s slide rule, C and D are both marked logarithmically, but they can move relative to each other. Suppose you wanted to prove that the distance between 1 and 2 is the same as the distance between 2 and 4. You could slide the C scale so that its 1 lined up with the 2 on the fixed D scale. If you do this, then the 2 on the C scale really does line up with the 4 on the D scale, and the 4 on the C scale matches the 8 on the D scale. The value on the D scale is always twice the value on the C scale. When you think about it, you have just invented a way to multiply any number by 2.
A slide rule showing how to multiply by 2.
This trick of doing multiplication isn’t just for multiplying by 2. Suppose you wanted to multiply 1.7 by 3.3. You could line the 1 on the C scale up with 1.7 on the D scale, and then look at what value on the D scale corresponds to 3.3 on the C scale. The slide rule has a handy little ruled glass window called the cursor that you can use to read the D scale accurately (if the cursor lands between two tick marks, don’t be afraid to estimate an extra significant figure based on where it is between ticks). I get 5.60. Use you calculator and you get 5.61. The slide rule is not exact (my answer was off by 0.2%) but you can get an excellent approximation using it. If my eyes weren’t so old, or if I had a more powerful set of reading glasses, I might have gotten a answer that was even closer. I bet with practice you young folks with good eyes and steady hands could routinely get 0.1% accuracy.
A slide rule showing how to multiply 1.7 by 3.3.
If you can do multiplication, then you can do its inverse: division. To calculate 8.2/4.5, move the cursor to 8.2 on the D scale, then slide the C scale until 4.5 aligns with the cursor. Then read the value on the D scale that aligns with 1 on the C scale. I get 1.828. My calculator says 1.822. When using the slide rule, you need to estimate your result to get the decimal place correct. How do you know the answer is 1.828 and not 18.28 or 0.1828? Well, the answer should be nearly 8/4 = 2, so 1.828 must be correct. Some would claim that the extra step of requiring such an order-of-magnitude estimate is a disadvantage of the slide rule. I say that even when using an electronic calculator you should make such estimates. It’s too easy to slip a decimal point somewhere in the calculation, and you always want to have a rough idea of what result you expect to avoid embarrassing mistakes. Think before you calculate!
A slide rule showing how to divide 8.2 by 4.5.
Suppose you have a number like 5.87 and you want to know its reciprocal. You could, of course, just calculate 1/5.87. But like most scientific calculators that have a special reciprocal key, dad’s slide rule has a special CI scale that performs the calculation quickly. The CI scale is merely the mirror image of the C scale; it is designed logarithmically, but from right to left rather than from left to right. Put the cursor at 5.87 on the CI scale, and then read the value of the C scale (no sliding required). I read 1.698. I estimate that 1/5 is about 0.2, so the result must really be 0.1698. My electronic calculator says 0.1704.
A slide rule showing how to calculate the reciprocal of 5.89.
One property of logarithms is that log(x2) = 2 log(x). To calculate squares quickly use the A scale (on my dad’s slide rule the A scale is on the flip side), which is like the C or D scales except that two decades are ruled over A whereas just one is over D. If you want 15.92, put 1.59 on the D scale and read 2.53 on the A scale (again, no sliding). You know that 162 is 256, so the answer is 253. My calculator says 252.81. Not bad.
A slide rule showing how to calculate the square of 15.9.
If you can do squares, you can do square roots. To calculate the square root of 3261, place the cursor at 3.261 on the A scale. There is some ambiguity here because the A scale has two decades so you don’t know which decade to use. For reasons I don’t really understand yet, use the rightmost decade in this case. Then use the cursor to read off 5.72 on the C scale. You know that the square root of 3600 is 60, so the answer is 57.2. My calculator says 57.105.
A slide rule showing how to calculate the square root of 3261.
There are additional scales to calculate other quantities. The L scale is ruled linearly and can be used with the C scale to compute logarithms to base 10. Other scales can be used for trig functions or powers.
I don’t recommend giving up your TI-30 for a slide rule. However, you might benefit by spending an idle hour playing around with an old slide rule, getting an intuitive feeling for logarithmic scaling. You’ll never look at a semilog plot in the same way again.
An episode of Meeting of Minds, with (l-r) Atilla the Hun, Emily Dickinson, host Steve Allen, Charles Darwin, and Galileo Galilei.
When I was a teenager, one of my favorite shows was Meeting of Minds. This television series, which aired on PBS from 1977 to 1981, featured historical figures interacting in a talk-show format. The host was the delightful comedian, musician, and television personality Steve Allen.
Below is an excerpt from when Darwin was discussing his voyage on the Beagle (in the video, this section starts at about 23:15).
Dickinson: How old were you when you made the trip?
Darwin: I was only 22. We set out from Plymouth on December the 27th, 1831.
Allen: A long or short voyage?
Darwin: Oh, it was five long years before I returned to England. It seems to me now that those five years constituted my real education.
This discussion took place when Darwin was discussing the controversy caused by his theory of evolution by natural selection (26:15)
Dickinson: Dr. Darwin, we know that your theories got you into the most dreadful trouble, that you were violently criticized, as was Signore Galilei. But what I cannot understand is how such common sense reasoning could have gotten you into such incredible difficulty.
Darwin: Oh my dear woman, the real trouble didn’t come about until after I presented to the world the idea that the common ancestry of all living things included man.
Galileo: Oh, how they must have howled for your blood when you said that!
I think the writers gave Galileo the best lines, like when Darwin and Galileo were discussing the dogmatism of Aristotle (38:45).
Darwin: It’s by no means a simple question as to what extent one should respect intellectual authority.
Galileo: There you are quite right. We must pay the most careful attention to what the great minds of earlier ages have discovered… But, we must never let our admiration for these great men blind us to the fact that they were only human, for their humanity means that they will inevitably fall into error at certain times.
Finally, a great soliloquy by Galileo about reason versus faith (45:50).
Galileo: There may be doubts and arguments as to the passages of scripture, problems of translation, et cetera. But there should be no room for doubt or argument about the evident facts of the physical world about us. Nature, unlike the sometimes confusingly worded scriptures, is inexorable and immutable, and does not care one jot whether her reasons and modes of operation are above or below the capacity of man’s understanding.
I like it best when the guests argue with each other (Dickinson gets so exasperated with Attila). For those of you whose interest is not science, you might enjoy listening to Dickinson read some of her poems or hear Atilla tell old war stories.
If I were Steven Allen and had to choose four guests to discuss physics in medicine and biology, who would I select? It’s tough to narrow it down to just four, but I would invite Marie Curie, Willem Einthoven, Alan Hodgkin, and Paul Lauterbur. Wouldn’t that be a fascinating discussion! Curie could tell us how she and her husband Pierre analyzed and purified tons of ore to isolate two new elements: radium and polonium. She could also recount her experience using medical x-rays during World War I. Einthoven could describe how he recorded the first electrocardiogram, and how he balanced his education in medicine with his interest in physics. Hodgkin could explain his research with Andrew Huxley that determined how nerves work. And Lauterbur could reflect on his invention of magnetic resonance imaging. Perhaps Curie might ask Lauterbur about who really invented MRI, him or Raymond Damadian. And Einthoven might probe into who deserves the credit for the voltage clamp, Hodgkin or Kenneth Cole. And they all could compare their experiences receiving the Nobel Prize (I can just hear Curie saying “but gentlemen, how many of you have two?”) What fun.
Exponential growth cannot go on forever. This fact is often
ignored by economists and politicians. Albert Bartlett has
written extensively on this subject. You can find several references
in The American Journal of Physics and The Physics
Teacher. See the summary in Bartlett (2004).
I started as an undergraduate in the fall of 1940 at Otterbein College in Westerville, Ohio. (It's a small, church-founded school, and my father was a Professor of Education there. Westerville back then was a separate town north of Columbus. Now it's been engulfed by Columbus.) In the spring of 1941, I was looking for some adventure, so I got a summer job washing dishes on an iron ore freighter on the Great Lakes. It took me several hitchhiking trips to Cleveland to get all the federal papers you need to work on the ships…
When September came along, I was having a good time, making good money, so I didn’t return to school. And that fall was Pearl Harbor. We were unloading the last cargo of iron ore for the season at the Wickwire-Spencer steel mill in Tonawanda, New York, on December 7. I’d finished cleaning up after lunch, then gone to my room and turned on the radio. I was the first person on board to hear the report...
Fortunately for all of us, Bartlett returned to college.
I said to myself, “I need to get back to college.” Writing from the boat, I applied to transfer to Colgate University in upstate New York. I was accepted, so, coming into Cleveland one day, I told the steward I was leaving to go back to college. I hitchhiked back to my home, changed from shipboard clothes to college clothes, hitchhiked up to Hamilton, New York, and started there in the fall semester of 1942.
Bartlett studied physics and math, and graduated from Colgate in 1944. Listen to how he ended up working on the Manhattan Project.
One of my professors heard that there was an address, “Box 1663,” in Santa Fe, New Mexico, where they were hiring physicists. Sounded like adventure to me, so I applied. Didn’t know what it was, and they didn't tell me and I was accepted.
My only instructions were to appear at 109 E. Palace in Santa Fe. So, after graduation, I hitchhiked home to Ohio, and to Springfield where a friend helped me get two new International Harvester trucks from the factory. I drove them to a dealer in Oklahoma City and then hitchhiked to Amarillo. All my life I’d wanted to ride a freight train, so I hopped a freight in Amarillo to Belen, south of Albuquerque. Then I hitchhiked to Santa Fe and reported to 109 E. Palace. The lady there in the office was Dorothy McKibben. She did some paperwork and had me get on an Army bus in the alley in back. It took me off through the desert and canyons up to Los Alamos. Quite an experience! I arrived on July 18, 1944 and worked for 25 months there.”
It wasn’t that I knew anything about mass spectrometry, but this was just at a time when the first plutonium was coming down from Hanford, and there was some indication that, in addition to the principal isotope 239Pu, there may be some 240Pu from an extra neutron capture in the Hanford reactors.
Now 240Pu has a high spontaneous fission probability, which would upset all their [bomb] calculations. So they wanted to know how much was there. They’d just requisitioned a mass spectrometer from a lab in Washington, DC, about the time I got there, and Bob Thompson, who’d been a PhD student with Al Nier at Minnesota, was setting it up. Bob took me on me sort of as an apprentice and took a real interest in me, telling me what I needed to study, giving me a stack of reprints to read, etc. He was very helpful. So we made the first measurements of the isotopic constitution of plutonium coming out of the reactors.
In 1946 he went to Harvard for graduate study in physics. His PhD project was to build
… a beta-ray [electron] spectrometer. I finished at Harvard and came out to Colorado in the late summer of 1950. I’ve been here ever since.
By Colorado, he meant the University of Colorado in Boulder. I interviewed at Boulder for graduate school around 1982. I can’t remember if I met Bartlett. I hope I did.
In 1978, I was national President of the American Association of Physics Teachers [AAPT]. I had four years in the AAPT presidential sequence vice-president, president-elect, president, and past president with duties in each of those offices. In ’78 I was the President, so I was quite involved with physics education, and I think that was why I was named an APS Fellow.
Bartlett is best known for his talk on exponential growth and the environment, which he gave over 1700 times (watch in on the YouTube videos below). Here is how it all got started.
Well, during all the uprising on the campus, there was a surge of student interest in the environment, and I had been slowly coming to the realization that students and other people didn’t have an understanding of the arithmetic of compound interest. So I started putting some notes together, and in September of 1969 the undergraduate pre-med honor society asked if I could speak on something or other at one of their meetings. I’d known many of these kids from the previous year when I was lecturing in the pre-med beginning physics course, so I put these notes together and on September 19, 1969 talked to them about growth and the problems of growth.
It didn’t have anything like the scope of today’s presentation. But, fortunately, I kept my notes and the next year I had two or three more invitations. It evolved as I became more interested. Finally in '78 or '79, I gave the talk 131 times. It was sort of like a Hubbert peak! It went down after that, but instead of going to zero, it’s leveled off at about 40 times a year.
With climate change becoming the major environmental issue of our time, we must all remember Bartlett’s famous quote:
The greatest shortcoming of the human race is our inability to understand the exponential function.
This inability is all the more reason to study Chapter 2 in IPMB, about exponential growth and decay. The survival of our species may depend on us somehow learning to understand exponential growth and its limitations.
A final reason to celebrate Bartlett is that this year, 2023, is the 100th anniversary of his birth: March 21, 1923. Happy 100th birthday, Albert Bartlett. We need you now more than ever!
The Most Important Video You Will Ever See (Part 1)
The Most Important Video You Will Ever See (Part 2)
The Most Important Video You Will Ever See (Part 3)
The Most Important Video You Will Ever See (Part 4)
The Most Important Video You Will Ever See (Part 5)
The Most Important Video You Will Ever See (Part 6)
The Most Important Video You Will Ever See (Part 7)
The Most Important Video You Will Ever See (Part 8)
Al Bartlett discusses population growth, climate change, energy, and consumption.
Two years ago, Japan developed a plan for the handling of Advanced Liquid Processing System (ALPS) treated water at the Fukushima Nuclear Power Station, which included a proposal to release the water into the ocean. Japan then asked the International Atomic Energy Agency (IAEA) to review the safety of their plan. The IAEA Director General established a task force to conduct this review, consisting of independent experts from all over the world (including Russia and South Korea). The task force recently published its report, whose purpose is “to present the IAEA’s final conclusions and findings of the technical review to assess whether the planned operation to discharge the ALPS treated water into the Pacific Ocean over the coming decades is consistent with relevant international safety standards.” The task force concluded that “the approach to the discharge of ALPS treated water into the sea… [is] consistent with relevant international safety standards” and that “the discharge of the ALPS treated water… will have a negligible radiological impact on people and the environment.”
One way to express risk is to compare medical doses to the natural background. We are continuously exposed to radiation from natural sources. These include cosmic radiation, which varies with altitude and latitude; rock, sand, brick, and concrete containing varying amounts of radioactive minerals; the naturally occurring radionuclides in our bodies such as 14C and 40K; and radioactive progeny from radon gas.
The effective dose of radiation is measured in sieverts, or more conveniently millisieverts (mSv). The typical effective dose from natural sources is about 3 mSv per year. What is the dose expected from releasing Japan’s radioactive water into the Pacific? According to the IAEA report, it is in the range from 0.000002 to 0.00004 mSv per year. So, we constantly are exposed to about 3 mSv/year of radiation and now we will experience 3.00004 mSv/year. The risk is negligible.
According to Table 16.6 in IPMB, flying in a plane at 40,000 feet—where cosmic ray exposure is increased—is 0.007 mSv per hour. That means the extra dose caused by the release of radioactive water is approximately equal to the extra dose received during 20 seconds of airplane flight.
The value of 0.00004 mSv/year assumes the water is slowly released as
planned. What if there is an accident? The IAEA report examined two
accident scenarios and concluded that the upper limit of exposure is
0.01 mSv per accident. In other words, if three of the holding tanks
accidentally dump all of their treated water into the Pacific at one
time, your dose would be less than one percent of your yearly dose from
natural sources.
Another way to look at it the risk is to analyze the amount of tritium released. The treatment of the water before release removes most of the radioactive contaminates except tritium, which is the radioactive isotope hydrogen-3. Tritium is usually found as part of a water molecule, so it is extraordinarily difficult to separate it form normal water. Tritium is constantly being created in our atmosphere by cosmic rays colliding with nitrogen. About 100,000 TBq is produced each year. A becquerel (Bq) is one nuclear decay and tera- (T) is the metric prefix for 1,000,000,000,000. How much tritium will be released each year from Japan’s wastewater? 22 TBq. In other words, the amount of tritium released is about 5000 times less than the amount naturally produced. Once the released water is diluted and mixed with ocean water, the increase in tritium concentration will be insignificant.
My conclusion is that releasing the treated water into the ocean is safe, with a large margin of safety. Russia and South Korea can relax. Lester Holt, I love ya, but you really gotta read the IAEA report more carefully.
There are plenty of things we all should worry about (just listen to the rest of the news report in the video below). The release of Japan’s wastewater into the ocean is not one of them.
NBC Nightly News with Lester Holt, August 24, 2023. See 14:30–16:40 for the report on Japan’s plan to release radioactive water into the ocean.
I am an emeritus professor of physics at Oakland University, and coauthor of the textbook Intermediate Physics for Medicine and Biology. The purpose of this blog is specifically to support and promote my textbook, and in general to illustrate applications of physics to medicine and biology.
