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.

does the Earth really go around the Sun?

tl;dr: Executive summary: It is not fundamentally true that the Earth goes around the Sun; it is just easier to calculate things that way.

We like to say that the critical event that started the scientific revolution is the discovery that the Earth goes around the Sun, and not the other way around. This was incredibly important; the hypothesis by Copernicus led to the immensely important data-taking by Tycho Brahe and the quantitative, theoretical explanation of it by Kepler. Galileo's discovery of moons of Jupiter bolstered the case in important ways, and Newton's quantitative description of it all in terms of the inverse-square law solidified it all into an edifice of great importance, that is just as important and valuable today as it was then. It is also a great example of how a scientific discovery requires both observational and theoretical backing to become confidently adopted by the community.

In the 20th Century, Einstein brought us General Relativity, with the eponymous generality granting us immense coordinate freedom. That is, there are (infinitely) many ways we can make decisions about what is stationary and what is moving, and what we choose as reference points. In some choices, calculations are harder. In other choices, calculations are easier. In yet others, certain symmetries become more obvious or more valuable for making predictions. That is, GR delivers to us lots of choices about how to think about what's moving and how.

So the crazy insane thing is this: In GR, there is no answer to the question of whether the Earth goes around the Sun or whether the Sun goes around the Earth. There is literally no observational answer to the question, and no theoretical answer. All observations can be incorporated to an analysis from either perspective. The question of which goes around which is not a question you can ask in the theory.

That said, it really is far, far easier to do calculations in the Copernican frame. Indeed, absolutely all calculations of Solar System dynamics are done in this frame with post-Newtonian code. The way I see it (with modern eyes) is that Copernicus's hypothesis was based on parsimony or simplicity and was adopted for that reason. Brahe and Kepler confirmed that the data are consistent with Copernicus's simple model (though with the eccentricities added). After Brahe and Kepler it was still possible to understand the observations in an Earth-centered (or even stranger) coordinate system, but was far, far easier to do calculations in the heliocentric frame.

Even today, now that GR is our model of gravity, we still calculate the Solar System with Newtonian codes (with adjustments to approximate GR corrections). And even today, now that we have this amazingly accurate model of the Solar System, we still often calculate the positions of celestial bodies by looking at paths on the celestial sphere, as did Ptolemy. How we calculate something is incredibly context-dependent, and doesn't always respect our most fundamental ideas. And the truth of Copernicus's hypothesis really just represents the pragmatism of the present-day mathematical tools. All these thoughts bolster my rejection of scientific realism and play into questions of social construction and so on. It also bolsters my view that Ockham's Razor should be thought of as a statement about calculation, not truth.

Sure the Earth goes around the Sun! But let's remember that this is a statement about calculation and pragmatism, not the fact of the matter.

vectors and their derivatives

The time derivative of velocity is acceleration, both vectors of course. But I was reminded in office hours today of just how hard it is to get across the idea that the velocity vector and the acceleration vector can point in totally different directions. And some students have trouble seeing this when a ballistic stone is going upwards along some (parabolic) arc, some have trouble seeing it when it is going down, and some have trouble seeing it at the top. That is, different students have very different problems visualizing the differences of the vectors over time.

I said in lecture that this issue was deep but I didn't emphasize it enough. I feel like it is so big it almost needs its own week!

Apollo 11

Pete Mao (Caltech) sent me a note for Newton's birthday, pointing out that all sensible transfer orbits to and from the Moon ought to have half-periods (transfer times) of about 5 days. And yet, as he also pointed out, Apollo 11 came back in 2.5 days. What gives? Did NASA waste fuel to improve the filmic value of the mission, or does the enormous tidal effect of the Sun change the sensible set of transfer orbits for some reason I don't understand?

(His note on this subject also had a nice discussion about what a 5-year-old wants when he or she asks a scientific question, and why just answering it is the wrong response.)

everything is an approximation

One of the main things I emphasized in today's class (computing a trajectory in gravity near the surface of the Earth with no air forces) is that every calculation in physics is an approximation. The parabolic trajectory near the surface of the Earth is an approximation to the tip of a very eccentric ellipse, and the eccentric ellipse comes out only in the Newtonian approximation to GR, and even that only holds if there are no other forces acting (and there always are). There are also small adjustments for reduced mass, and if the object has non-trivial extension. Crazy! And in high school this is all taught like it is exact: Just plug numbers into the equations!

weight, gravity, and contact force

On the final exam, I asked the following:

Explain why the astronauts in the Space Shuttle are weightless.

I was lenient in grading. But my position is actually at odds with most of the textbooks. Here's why.

The standard textbook answer is something like Actually, the astronauts on the Shuttle still have weight, since there are still gravitational forces acting on them. However, they feel like they are weightless because they are in an accelerating reference frame that is accelerating at the acceleration that the gravitational force is providing. This will be followed with various things about equivalence and plummeting elevators and non-inertial forces and so on.

My explanation is that the gravitational force on an object is not the weight of the object, when the word weight is properly understood. The weight of an object is not the gravitational force but rather the contact force that holds the object up against gravity (and non-inertial forces). It is this contact force, after all, that a spring scale measures, because a spring scale does its job by providing a contact force. It is also this contact force, after all, that you perceive by having your feet pressed onto the floor or behind pressed into the seat of your chair. Indeed, gravitational forces can never be measured locally or internally (that's equivalence!), all you can measure is the stresses and strains required to oppose them in our non-inertial (by GR standards) frame.

My view makes the astronauts not misled but truly weightless. It also makes it true, not apparent, that one is lighter at the top of a hill and heavier at the bottom of a hill on a roller-coaster, and same for the related changes you experience in an elevator.

last class

I gave my last class today; still have to prepare a final. It has been a fun semester, and the students have performed well with a set of extremely difficult problems in lectures and recitations and problem sets, including ill-posed problems, approximate problems, and numerical problems. I ran out of time at the end, as I always do. Many subjects were not covered. Perhaps my biggest regret is not making it to the analysis of non-circular orbits using the effective one-dimensional radial potential. This analysis takes a long time to set up and then solve, but it is so damned beautiful.

blocks on planes

In class yesterday I compared two problems: the block on an inclined plane (sliding without friction) and the car sliding around an icy (frictionless), banked curve. In the latter, the challenge is to enter the banked turn at exactly the right speed that you make it out the other side without either sliding uphill or down. The nice thing about doing both these in one lecture is that they have nearly identical free-body diagrams, both of which have a massive particle acted upon by gravity and a normal force (at the same angle to the vertical if you set it all up correctly), and yet the physical situations are so different and the accelerations point in different directions. The comparison brings out a lot of conceptual material about contact forces and kinematic constraints.

The banked turn example is also hilarious, and has many nice details, such as that in general if you enter a conical banked curve and slide around to the other side without friction, you will come out with the car pointing in a strange (and non-trivial to calculate) direction.