Kaysville, UT homeowners show up in large numbers to oppose warming center

[Physics Guy]

This has certainly been a derail, but I am only doing that which has been done in other threads before now. That's a good excuse, right?

About inertia, maybe it helps to get more technical. For an object rising and falling under constant gravity, Newton's Second Law F = M a reads

h''(t) = - g

where h(t) is the height of the object (say, a ball), which is going to change over time, and g is the constant of gravitational acceleration, which over the Earth's surface is close to 9.8 meters per second per second, or 32 feet per second per second. The '' means the second derivative: the rate of change of the rate of change.

The general solution to this second-order differential equation is

h(t) = h0 + v t -(1/2) g t^2,

where h0 is the ball's initial height, v is its initial upward speed, and t is the elapsed time since the initial time. Newtonian dynamics tells us the differential equation, and the differential equation directly implies its general solution; Newtonian dynamics does not tell us either h0 or v. Newtonian dynamics is like an algorithm that prompts the user for inputs. In this case h0 and v are the inputs. Specify them, and the whole motion of the ball for all time is determined precisely, but until we specify h0 and v, we have no idea where the ball will be at any time.

The fact that h0 has to be supplied as an input is fairly intuitive. We think that a law of physics ought to be about explaining how things change, so we can maybe accept that it doesn't tell us how things initially are. It's a bit like logical reasoning, where logic does not supply an initial premise, but will let you deduce things from it. Physics should tell us how things will change, starting from wherever they are at some point. So we have to input h0.

The fact that we also have to input v is less intuitive. We really do have to say what v is, though. What v is can make a huge difference in how the ball moves. Unless v is very small, in fact, its effect will be decisive at first. The ball will keep rising at close to its initial speed. The initial trend will persist, at least roughly; the -(1/2) g t^2 part from gravity is always there, too, right from the start, but it may take some time to build up enough to be too big to ignore.

So we need to know v, and this might be disappointing: physics cannot in fact tell us how things will change, given how they are now. We also have to tell it how things are changing right now. That's what v tells us: how fast the ball is initially rising. But at least physics doesn't let us down too badly. Once we tell it that second number v, as well as h0, then physics tells us everything after, like a fortune teller who has to look at both your eyes and your palm but can then tell you your future. In effect, how things are initially changing has to be included in the full statement of "how things initially are".

Thinking of that initial speed v as part of "how things initially are" can be counter-intuitive, though. The ball's initial speed does not show up in a snapshot, the way its initial height would. The initial speed doesn't seem as though it is part of the initial situation. It seems instead to be part of the process of change—something that should be explained and not just taken as input.

The initial rate of change is an input, however—an initial condition, part of how things initially are. We cannot just present a snapshot of the initial situation, as if it were static, and ask physics to tell us what happens next, and explain why it happens. That's like asking the fortune teller to tell you your future without showing your palm. Her art just doesn't work that way. We also have to say how things are initially changing.

That initial rate of change, just like the initial snapshot of how things are, has a permanent effect on what happens. In that sense it persists.

This is what inertia means in Newtonian dynamics. Inertia is not a force like gravity. It isn't a thing that is otherwise defined and that explains why initial change rates persist. Inertia is simply a name for the fact, in Newtonian dynamics, that the initial rate of change does persist. And to say that some kind of change is occurring "because of inertia" is not to propose inertia as an entity which is the cause for the change, but rather to remind people that some component of how things are changing is always just due to the initial rate of change, and does not have any cause but that.

That's just how causality works, in Newtonian dynamics. The equations of motion are of second order. The initial rate of change is an input, with lasting consequences, just like the initial situation itself. Half of the necessary information to define the initial state of "how things are" is information about the initial rate of change. Half of the information about how things are is thus always hidden, in the sense that it does not show up in the snapshot. That's the principle of inertia.

I don't know whether inertia helps people understand Newtonian dynamics, exactly. It is necessary for understanding Newtonian dynamics, because it is an essential part of Newtonian dynamics. The word "inertia" is not important, but the second-order nature of the equations really is. Human intuition tends to think in terms of just two basic concepts: situation and change. Newton's revelation was that those two categories just aren't enough for reality. We need a three-level structure: initial configuration, initial rate of change, and acceleration. Only in those terms do cause and effect really work. That's important.

[bill4long]

By the way, for anyone who has had the experience of driving with children, I suggest a couple of variations on one of the usual songs. It's possible, and very simple but not easy for everyone, to sing Row, Row Your Boat, and Row Your Boat to the regular tune, ending up with, respectively, one and two beats at the end with no vocalization. Try it! I got a great feeling of accomplishment when I was finally able to sing all three versions, one after the other.

It's also fun to sing Take Me Out to the Ballgame by singing the first two words take me with two run up notes so that out starts with the note that is usually sung as take. Hilarity will ensue.

[malkie]

By the way, for anyone who has had the experience of driving with children, I suggest a couple of variations on one of the usual songs. It's possible, and very simple but not easy for everyone, to sing Row, Row Your Boat, and Row Your Boat to the regular tune, ending up with, respectively, one and two beats at the end with no vocalization. Try it! I got a great feeling of accomplishment when I was finally able to sing all three versions, one after the other.

It's also fun to sing Take Me Out to the Ballgame by singing the first two words take me with two run up notes so that out starts with the note that is usually sung as take. Hilarity will ensue.

Added to my list of mental strength trainers - thanks!

[bill4long]

It's also fun to sing Take Me Out to the Ballgame by singing the first two words take me with two run up notes so that out starts with the note that is usually sung as take. Hilarity will ensue.

Added to my list of mental strength trainers - thanks!

U bet.

[Chap]

This has certainly been a derail, but I am only doing that which has been done in other threads before now. That's a good excuse, right?

About inertia, [...]

I don't know whether inertia helps people understand Newtonian dynamics, exactly. It is necessary for understanding Newtonian dynamics, because it is an essential part of Newtonian dynamics. The word "inertia" is not important, but the second-order nature of the equations really is. Human intuition tends to think in terms of just two basic concepts: situation and change. Newton's revelation was that those two categories just aren't enough for reality. We need a three-level structure: initial configuration, initial rate of change, and acceleration. Only in those terms do cause and effect really work. That's important.

I'm sorry, and I don't want to sound rude or aggressive, but I remain unconvinced by the attempt your post makes to convince me that instead of relying on the application of mathematical analysis to well-defined physical quantities such as mass, length, time, momentum, energy, force, etc. I need to bring in this thing 'inertia' (which you agree is not a physical quantity) to understand Newtonian dynamics. I have never felt the need to do that, and I have studied mechanics to a fairly advanced level (think Hamiltonians, for example).

Assuming I had to explain (e.g.) the Newtonian dynamics of projectile motion to somebody, I am sure that trying to bring in 'inertia' would simply lead to unnecessary confusion.

[Physics Guy]

I'm not at all upset or offended by your persistence, Chap. I'm writing these long posts because to me this is an interesting point, and I'm trying to articulate it for myself. Perhaps as few as ten years ago I would still have been very much on your side, touting equations as the only real truth and dismissing vague words and "principles". I'm a bit surprised to find that, somewhere along the line, I began to think differently. So I'm glad to have you as a sounding board.

We can certainly do the mathematical analysis, for thrown balls and for the scale factor of the spacetime metric in cosmology. There really are often good reasons not to go through all that, however. In some audiences, for example, not everyone is familiar with differential equations.

In other audiences, on the other hand, everyone is familiar enough with differential equations that it saves time to refer to a relevant concept by its standard name, without explicitly unpacking all the details that everyone already knows well. We have to be able to refer to a book by its title, without having to recite the whole text, or we would never get to anything new.

It's not that expert discussions get away with using sloppy language. On the contrary, it's more precise to use a general term, when the relevant point to make is a general one. If one is discussing vehicles, it is more precise to say "cars" than to say "Honda Civics" when the point you are making is about cars in general and not about one particular make.

Higher-level technical language is used in all disciplines to refer compactly to general concepts. One can usually unpack the higher-level language down to concrete cases in particular examples, but this takes enough time that people don't do it more than they must. Devils lurk in the details, and one can easily get sucked down a rabbit hole once you get into detail. One might even discover that the general concept doesn't really make sense. All of this is common to most kinds of thinking.

Physics is probably unique, however, in the degree to which it is in fact possible to zoom in clearly, and translate the general concepts accurately into details so concrete that they can be measured precisely. It may take time, but with time, you can do it. People who don't yet know much physics probably do not expect this. Physics is unpackable to a bizarre degree, to the point where really understanding something in physics tends to raise one's standards for what it means to understand anything.

I guess that because this is probably the main novel thing about physics, it is probably the thing that education in physics most emphasises. "Don't just talk about things with words: show the thing itself by solving the equation!" In physics you can, at least sometimes. This is weird enough that it is hammered into all physics students. And so it probably should be.

Diving straight into solving equations is all that physics students need to learn physics—if by "learn physics" we mean, "pass the exam." When you actually start using physics in practice, however, after a while you have to learn to accept that those philosophers and historians kind of had a point after all. We do have to talk about things with words, because we have limited time—and because we generally don't know ahead of time what equations we even should be trying to solve. Figuring that out is usually the main job.

If I'm discussing a problem in a project with colleagues, we often sit in front of a whiteboard for hours. Maybe about half the time, someone is standing at the board actually solving an equation, or deriving a new equation to solve. The rest of the time we are discussing the problem in general terms, arguing about what issues are relevant, what simplifications could be instructive, and so on. If all we ever did was solve equations, we would spend years solving the wrong equations—equations that didn't answer any interesting questions—and never achieve anything. To be able to think through which equations we should be trying to solve, without taking all the time it would take to solve any equation, a higher-level language is indispensable.

So over the centuries a lot of general terms have been defined, even in physics, for example to refer quite specifically to certain common features of many dynamical systems, while precisely not saying anything about other features that the systems may also have. Inertia is one of these terms. It refers to a specific feature of dynamical systems which is so common that it is important in most systems, namely the fact that the dynamics is of second order, requiring two initial conditions per degree of freedom. In Hamiltonian terms: each degree of freedom requires two canonical coordinates, not just one. That is another way to express the principle of inertia.

If one didn't know how to solve an equation of motion, it certainly wouldn't be an adequate substitute just to know how to say that this rising motion is due to inertia. Not appreciating that everything implies quantitative solutions is not understanding physics at all, no matter how many words one can recite. It is also a limited understanding, however, to be unable to recognise and discuss the features of motion that are common in many systems, and embedded in our most basic theories.

The principle of inertia won't help you solve an equation, but it may help to relate the equation to other equations, and its solution to other solutions. That may help you judge how meaningful your equation was, and how important your solution is. That's also part of understanding the problem. So one should be able to illustrate inertia by solving the equations in a simple example, but one should also be able to look at a solution and recognize it as an example of inertia—because that's a pattern that shows up all over in physics. One has to understand that forests are composed of trees, but one should be able to see the forest as well.

And then maybe you're posting on a message board whose readers mostly aren't college-trained physicists. It's not going to be very helpful to post a differential equation. High-level language is all you can use. A term like "inertia" isn't going to convey all the details that the equation would convey to those who could read it. At best it's going to give people a notion that something is involved that's like that time when they shoved their car on a patch of ice and the car kept on sliding a bit, or something like that. But you know what? That's not bad. That actually is what the equations say. It isn't all that they say, but it's a big part of what they do say. And one should not let the limited nature of the high-level term blind one to its accuracy as the kind of term that it is.

It's worth being able to give people the gist of an equation that way, whether they're non-physicists reading from interest or colleagues helping you grope for a model. Words aren't so bad after all.

[Chap]

Try this for size:

Because of our experience of moving things about on the surface of the earth, we tend to assume that to move something from one place to another you need to push or pull it, and If you stop pulling or pushing, it stops.

However, experience out in space teaches us that if we have a moving body far away from friction with the ground, and with no drag from moving through the atmosphere, and away from a strong gravitational field like the one we have on earth, it will just keep on going in a straight line at constant speed for ever. (OK, it may hit something or it may get pulled into the gravitational field of a star or planet. But space is very big and very empty). If you want to change that situation so that the body speeds up, slows down or changes direction, that's when you have to exert a force on it for a while.

So if there was no atmosphere at the earth's surface, and if we switched off earth's gravitational field, then if you threw a ball upwards it would simply go up in a straight line at a steady speed indefinitely. The reason it doesn't is mainly because there is a downwards gravitational force acting on the ball, so that its upwards speed steadily decreases, then passes through zero and becomes an increasing downwards speed. (Of course atmospheric drag slows it down a bit too).

Nice clear explanation. Rigorously correct physics (no?). No maths. No mention of inertia.

So what's not to like?

[Physics Guy]

This statement doesn’t mention “inertia” for the same reason that Lincoln’s Gettysburg Address doesn’t mention “Gettysburg Address”.

… experience out in space teaches us that if we have a moving body far away from friction with the ground, and with no drag from moving through the atmosphere, and away from a strong gravitational field like the one we have on earth, it will just keep on going in a straight line at constant speed for ever.

The short name for this fact is “inertia”. That’s why Newton’s First Law, which states this fact, is called “the Law of Inertia”.

[bill4long]

This statement doesn’t mention “inertia” for the same reason that Lincoln’s Gettysburg Address doesn’t mention “Gettysburg Address”.

The short name for this fact is “inertia”. That’s why Newton’s First Law, which states this fact, is called “the Law of Inertia”.

That Chap thinks "staight lines" though spacetime are a thing is adorable.

Kindergarten was fun.

Meow!

[Chap]

That Chap thinks "staight lines" though spacetime are a thing is adorable.

We are talking Newtonian physics here. Or hadn't you noticed?