physics

Dimensional Analysis: A Secret Weapon in Physics

by

It’s not widely appreciated how often physicists can guess the answer to a problem before they even start calculating. By combining a basic consistency requirement with scientific reasoning, they can often use a heuristic approach to solving problems that allows them to derive most of a formula without doing any work at all. This week I want to introduce this to you, and show you some of its power.

The trick, called “dimensional analysis” or “unit analysis” or “dimensional reasoning”, involves requiring consistency among units, sometimes called “dimensions.” For instance, the distance from the Earth to the Sun is, obviously, a length. We can state the length in kilometers, or in miles, or in inches; each is a unit of length. But for today’s purposes, it’s irrelevant which one we use. What’s important is this: the Earth-Sun distance has to be expressed in some unit of length, because, well, it’s a length! Or in physics-speak, it has the “dimensions of length.”

For any equation in physics of the form X = Y, the two sides of the equation have to be consistent with one another. If X has dimensions of length, then Y must also have dimensions of length. If X has dimensions of mass, then Y must also. Just as you can’t meaningfully say “I weigh twelve meters” or “I am seventy kilograms old”, physics equations have to make sense, relating weights to weights, or lengths to lengths, or energies to energies. If you see an equation X=Y where X is in meters and Y is in Joules (a measure of energy), then you know there’s a typo or a conceptual mistake in the equation.

In fact, looking for this type of inconsistency is a powerful tool, used by students and professionals alike, in checking calculations for errors. I use it both in my own research and when trying to figure out, when grading, where a student went wrong.

That’s nice, but why is it useful beyond checking for mistakes?

Sometimes, when you have a problem to solve involving a few physical quantities, there might be only one consistent equation relating them — only one way to set an X equal to a Y. And you can guess that equation without doing any work.

Well, that’s pretty abstract; let’s see how it works in a couple of examples.

Read more

Dog Brains and Fishing Line: 2 Fun Articles

by

Nothing about quantum physics today, but … wait, everything is made using quantum physics… — Could you imagine getting a dog to sit absolutely still, while fully awake and listening to voices, for as much as 8 minutes? Researchers trained dogs to do it, then put them in an MRI [Magnetic Resonance Imaging] machine to … Read more

How Chinese Children Are Learning Physics

by

While we’re on the subject of China… The US has had space stations for decades, and people here now show limited interest, barely caring that the US currently has no rocket that can carry people to space.  Now China has its own rockets and space station, and, with plenty of excitement and national pride, is … Read more

Under Pressure

by

The scourge of “terrorism” — for today’s purposes, let’s take the word to mean attacks on civilians perpetrated by individuals or by small, stateless groups — is a part of human existence going back as far as you want to look. If a person has what he or she views as a grievance, then attacking people who are loosely connected to that grievance, in order to kill and maim some of them and frighten the rest, is obviously one of the options, immoral and hideous as it may be. There’s nothing modern about the strategy of terror.

What’s new about terrorism in the modern world is science. Science, via the technology that it makes possible, is a great multiplier. It allows an individual, or a small group, to exploit power inherent in nature, turning a task that no human could perform, or that would take a cast of thousands, into something that can be done with ease by a few people, or even just one. Of course this multiplied power has many benefits for us as individuals and for society as a whole; think of trains, tunnel-boring machines, skyscraper cranes, snow-blowers, pneumatic drills, aircraft engines, power plants, and on and on. But it also poses many risks and challenges that we have to face, as individuals and as a global civilization.

Read more

Another Storm Predicted

by

The greater New York region, having been broken into disconnected and damaged pieces by Hurricane/Nor’Easter Sandy, is still reassembling itself.  Every day sees improvements to electrical grids and mass transit and delivery of goods, though there have been many hard lessons about insufficient preparations.  Here’s an impressive challenge: over a million people and thousands of businesses lack electrical power; therefore many of them are running generators, to stay warm, keep food cold, and so forth; but the generators require fuel, typically diesel or gasoline; and so there is a greater need for fuel than usual; but a significant fraction of the petrol stations can’t pump fuel for their customers… because they lack electrical power and don’t have their own generators. These and other nasty surprises of post-storm recovery should be widely noted by policy makers and the public everywhere, especially in places that, like New York when I was a child, rarely experience disasters.

Unfortunately, another storm (a simple nor’easter) is now forecast for mid-week. While much weaker than the last, it is potentially still a dangerous situation for a region whose defenses are still being repaired.  As was the case with Sandy, the new storm was already signaled a week in advance by the ECMWF (European Center for Medium-range Weather Forecasting), the current European weather-forecasting computer program or “model”.  Confidence in the prediction has been growing, but still, predictions so far in advance do change.  Also one must keep in mind that a shift in the storm’s track of one or two hundred miles or so could very much change its impact, so the consequences of this storm, even if it occurs, are still very uncertain.  But again we are reminded, as we were last week, that weather forecasting has dramatically improved compared to thirty years ago; the possibility of a significant storm can now often be noted a week in advance.

What is this European ECMWF model? what is its competitor, the US-based GFS (Global Forecast System) model? And what about the other models that also get used?   All of these are computer programs for forecasting the weather; all of them use the same basic weather data as their starting point, and all have the same basic physics of weather built into their computer programs.  So what makes them different, and more or less reliable than one another?  I asked one of my commenters, Dan D., about this after my last post.  Here’s what he said, along with my best (and hopefully accurate) attempts at translation for less experienced readers:

Read more

Guest Post: Anand Gnanadesikan, Oceanographer

by

I know Anand Gnanadesikan, professor at Johns Hopkins University’s department of Earth and Planetary Sciences, from when we were both studying physics as undergraduates in college.  He wrote something today that speaks with more authority than I could in my post earlier this morning, and it is a pleasure to make it available to you. … Read more

North-East Winds A-Blowin’

by

The big storm of 2012 (at least, we hope it’s the biggest we’ll see this year) is approaching the New York City area, and though no one can predict in detail how bad it will be and for whom, there’s no question that with so much energy to play with, post-tropical quasi-hurricane quasi-nor’easter Sandy (also called “Frankenstorm” in honor of the Halloween holiday) is going to hit some of us very hard in the northeastern United States.  Not that it will be a disaster everywhere in the region.  With hurricane Irene last year, some areas just had a bit of wind and rain, while others had tremendous flooding that wiped out towns and roads and houses and history… and a few dozen lives, too. It will likely be the same this time.

How unusual is this storm?  Several weather forecasters have been quoted as saying that their supercomputer-based forecasting tools, which predicted Sandy to strengthen and become a monster in size, were doing things they’d never previously seen them do.  Right now, all you have to do is look at the weather map — the fact that there are tropical force winds extending over several hundreds of miles, and at the fact that the pressure of the atmosphere at the core of this storm is around 946 millibars and falling — to know there’s a lot of energy in this system that has to go somewhere, and is going to be taken out on somebody.   Although this is a Category 1 hurricane in terms of its fastest winds, 946 millibars is what one expects for a strong Category 3 hurricane; 1000 is average atmospheric pressure, and the mid-800s is about as low as it ever gets.  By comparison, the great blizzard of 1993 had a central pressure of about 960 millibars.  The Perfect Storm of 1991 (also a nor’easter-hurricane hybrid, like Sandy) had a central pressure of 972 millibars.  Anyone who thinks Sandy isn’t a dangerous storm hasn’t read enough history.

Read more

%d bloggers like this: