your weight is just your normal force!

Intro physics textbooks often jump over backwards to deal with the problem that astronauts (say) feel weightless, but in fact they are subject to a gravitational force that is only a few percent less than the gravitational force here on Earth. And then there is all the discussion of why they feel weightless when in fact they have nearly the same weights as we do.

I simply don't agree with this: In my view, your weight is your normal force against the floor when you are in static equilibrium in your local rest frame. Here are some arguments for my position:

the gravitational force is unobservable:
It is literally a constituent principle of modern physics that you can't tell a gravitational force from a non-gravitational force in an accelerated reference frame. So if we decide that “weight” is gravitational force, we have decided that weight is completely unobservable. So, presumably, everyone is wrong about their weight, and their weight is actually not a covariant property of anything.
everyone becomes right:
In the standard textbook view, astronauts are misguided about their weightlessness, as are passengers on the vomit comet. We have to say they “feel” weightless but aren't. Also, we have to say that people on a roller coaster who go over hills and valleys feel lighter and heavier, but when in fact (we have to say) actually nothing has changed. In my new view, the astronauts, passengers, and ride-goers are all correct: They really are weightless (in the space station and the comet), and they really are changing their weight (on the roller coaster) as they ride.
museum exhibits don't have to change:
It is still the case that you are heavier on Jupiter (if it had a surface) and lighter on Mars! Because the normal force you would feel would be higher and lower. Totally observable, totally true.
buoyancy gets taken care of naturally:
What does a helium balloon weigh? In the standard gravitational-force sense, something positive. But in the normal-force sense something negative! It has to be tied down to the floor. That seems sensible. Also, even humans have a buoyant force acting on them, it decreases their weight (in my view, but not in the standard view). Like should a doctor's office multiply everyone's weight measurement by (1+1/800) to account for buoyant force? They should if weight is weight is gravitational force, but not if weight is normal force. Again, this also connects to observability, and also the correctness of visceral feelings (like your feeling of weightlessness in a swimming pool). [Modification made later: Will Kinney (SUNY Buffalo) makes a great point: Your inner ear feels the normal force you would have with no abnormal buoyant force, whereas your feet on the floor feel a normal force that is modified if you are in a denser medium, so the buoyancy point here is complex to say the least.]
it disambiguates weight from mass better:
Mass is a gravitational charge, or an inertial constant. Weight is a force. If weight is going to be a force, it should be an observable, measurable force. Preferably the force you actually feel when you say “I feel heavy”. So make weight the observable force, and mass something to be inferred by inertial and gravitational arguments.

The funny thing about all these changes is that they change nothing in natural language or natural discussion of weight, and they greatly simplify physical discussions of weight. They also make it less true that physics is in Physics Land (tm) where all your intuitions are wrong! I hate Physics Land (tm) and this redefinition of the word weight tears down one of its (many, many) walls.


manipulating betting odds into probabilities and other odds

A colleague and I wanted to enter into a bet on a group-stage game of the World Cup of Football. We wanted our bet to have the following structure: He pays me something if Team A wins, I pay him something if Team B wins, and we push (that is, neither of us pays) if it is a draw between Team A and Team B. We are friends, so we wanted to give reasonable odds on this bet. How to calculate them?

Online, we saw the following odds posted on a reputable sportsbook (yes, such things exist) in European style:

Team A wins1.625
Team B wins6.710
tie 3.910

Is it possible to convert these data into what we want to know? Of course the answer is yes, but we need to make assumptions. Before we continue, I should note that these odds would be written in American style as follows:

Team A wins-160
Team B wins+571
tie +291

Because that may be confusing, let's just check in on what these odds mean. Looking at the European odds, what they mean is that if I bet $100 that Team A wins, I will be paid $162.50 if they do indeed win. That is, I will be paid back my $100 plus $62.50 in winnings. If I bet $100 that there will be a draw, then I will be paid $391.00 (my $100 plus $291 in winnings) if they do indeed draw. Looking at the American odds, what they mean is that if I bet $160 that Team A will win, I will be paid $100 in winnings plus my original bet back, or a total of $260, if they do indeed win. And they mean that if I bet $100 that Team B wins, I will be paid $571 plus my original bet back, or a total of $671, if they do indeed win. The +/- sign at the beginning of those odds makes a big difference! So in this sense, to a scientist, usually the European-style odds writing makes more sense.

Strictly, what's written in European-style odds is not the odds, but the odds-plus-one. If the book was not out to make money—If the book was just trying to break even—then these odds would be the book's approximation to the inverse of the probabilities of the outcomes. If you want to see why, you can think about placing a Dutch bet: Putting money on each outcome in proportion to the probability. If you do that, and the odds are just right, then you will get the same payout no matter what the outcome. That's an exercise to the reader! But if we interpret these European odds as inverse probabilities, then the implied probabilities of the various outcomes would be:

Team A wins0.615
Team B wins0.149
draw 0.256

Which you can get just by inverting the numbers in the first chart. The observant reader will notice that these three numbers don't add up to one! They add up to a bit more than one. Why? Because the book is making money, and so they pay out odds slightly lower than what's fair, which means that they have an edge over the bettors, on average.

Now if we make the assumption that the book is relatively capable (or that the betting public is pushing the book's odds to something sensible), then we can assume that these are close to the correct probabilities for the three outcomes. That's a great start! But remember that we want odds for a win bet, with the draw as a push. The idea is, we want to know: What's the probability that Team A wins given that there isn't a draw. For that we condition on there not being a draw and re-normalize that probabilities. Probabilities for these outcomes are:

Team A wins (conditioned on no draw)0.805
Team B wins (conditioned on no draw)0.195

I got these by re-normalizing the two outcomes my colleague and I care about to get unity. Now we can convert to European-style odds by inverting again, and the odds we get are:

Team A wins (conditioned on no draw)1.242
Team B wins (conditioned on no draw)5.128

That is I should bet $100, and my colleague $24.20, and whoever wins gets it all. If there is a draw, we take our money back. Or at larger stakes, I should bet $412.80, my colleague $100, and so on.


freshwater consumption

I'd like it to be a theme of my writing about teaching that I learn as much as any student in my class when I teach! It certainly is a fact. This semester has been huge for me because I'm teaching a subject I don't know much about (sustainability), and because I am teaching writing, which is a first (yes, in 25 years of teaching, I have never taught writing).

A few weeks ago I gave a (writing) assignment to estimate global freshwater needs by computing your own freshwater needs and extrapolating to the world. Then compare with estimates of global usage. My expectation: That a typical American college kid uses way more water than a typical person on Earth, and this would be a lesson about disparity. I wasn't wrong, but I also wasn't right either:

It turns out that global freshwater use is way larger than any estimate of your own personal water use, extrapolated to the globe! Isn't that odd? It's because global freshwater use is not dominated by household uses like laundry and toilets (which, by the way, don't need freshwater). It is dominated by agriculture. If you add in your own share of agricultural water use, a typical American uses way more than the average human! But it is hard to know or estimate this without significant engineering research.

There was one clue, however, in the essays I got: One student included the amount of water used to water a lemon tree in their dorm room. It was a trivial amount of water, so it was almost a joke. But then if you think about how much food we eat relative to the miniscule annual production of that tiny lemon tree, there is a little window into just how much water we must use for agriculture.


Is the greenhouse effect like a greenhouse?

I am teaching a class at NYU this Spring that I am loving: It is called (with grandiosity) The Art and Science of Approximate Reasoning: Physics, Sustainable Energy, and the Future of Humanity. We are computing what we can about sustainability, from a physics perspective and with order-of-magnitude reasoning. The strangest thing about the class (for me) is that it is a writing class. So I am grading writing assignments, something I am certainly not qualified to do.

I have learned a huge amount in this class. Here's one tiny, tiny example:

How does a greenhouse work, and is it anything like how the greenhouse effect on Earth works? The answer is extremely cool. In both cases (greenhouse and greenhouse effect) the system permits visible radiation to enter, but deters infrared radiation from escaping. In this sense, the two effects are very similar.

However, there are critical differences: In the case of the greenhouse, much of the effect is that the glass walls cut off convective cooling to the upper atmosphere. That is, they trap the heat near the heated ground, not letting it mix with the rest of the atmosphere. This effect probably dominates over the infrared-reflectance effect of normal glasses and plastics. The fundamental point is that (to first order) the Sun heats the Earth, which then heats the atmosphere. If part of the Earth is put inside a glass box, that part gets to hold its heat without sharing it entirely with the surrounding atmosphere. And indeed, greenhouses control their temperatures through ventilation management (management of convection to the outside).

In the case of the greenhouse effect, increasing the CO2 (and water and methane and so on) in the atmosphere increases the opacity at infrared wavelengths, so the Earth can't cool radiatively as efficiently as when the atmosphere is more infrared-transparent. That certainly is an effect in greenhouses, but not the only effect, and probably not the dominant effect.

The Earth's atmosphere can only cool (to space) radiatively, so the greenhouse effect is purely an adjustment of that radiative cooling. The human-built greenhouse can cool radiatively or convectively, and it uses both of those mechanisms for heat management.