Claude Allègre and gravity

A corrected version: a multiplicative error of 1,000 fixed



Pétanque balls

When I posted about Claude Allègre, it had to be expected that some green mujahideens would instantly try to attack him personally. Indeed, one of them came to this blog and informed us about two episodes that are supposed to discredit him forever. These two memes that have been repeated by Wikipedia and many activist websites are related to asbestos and gravity:

1. In 1996, Allègre voted against the removal of asbestos from a campus
2. In 1999, Allègre said that there was a popular misconception that when a tennis ball and a much heavier pétanque ball fall from a tower, the pétanque ball will reach the ground before the tennis ball. However, as children should learn at school, all bodies accelerate in the same way.

Concerning the first issue, I am mostly convinced that asbestos is a carcinogen (and a commenter suggests that Allègre in principle believes it, too).

But that doesn't mean that I can endorse the precise counting of the deaths that were caused by asbestos on that campus: 22. And it doesn't mean I would necessarily vote to remove it. The removal is risky and expensive, too. And even those 22 people - however sad and huge a loss they may sound as - are much less than 1/1,000 of the people who have spent years on the campus since the war (and many more have died because of other reasons). There are 20,000 students living on the campus at each moment - and add the non-students and the "older generations" of the students etc.




Concerning the second issue, I support Allègre's viewpoint entirely. A hysteria against him began quickly. People would say that the equivalence principle only holds in the vacuum and the proposition won't be true in the air. Well, the equivalence principle holds everywhere but in the air, there is another force, air resistance or drag, that tries to slow down the ball.

However, the drag is something that everyone knows, unlike the equivalence principle, it has no deep physics in it, and moreover, it has a relatively small effect in this experiment, as we will see. People usually overestimate it. If a tennis ball and a pétanque ball fall from a 44-meter tower, comparable to a top floor of the Tower of Pisa, it takes 3 seconds for them to reach the ground. Indeed, the tennis ball will be slow. But how much slower?

Remember your guess and you may write it in the comments.

Now, what's the right calculable answer? How much important the air resistance actually is? We will be comparing the tennis ball with an "infinitely" dense object which falls along the standard parabola, unaffected by the drag. The real pétanque ball is somewhere in between the tennis ball and the superheavy object - but its trajectory will be much closer to the superheavy object.

The gravitational acceleration will be taken to be 9.81 m/s^2.

What about the drag? In the air, the Reynolds number is actually pretty high, many thousands for those speeds. So we can use Lord Rayleigh's formula for the drag deceleration,

a_drag = 1/2 rho v² C_d A / m

Here, the factor 1/2 is a number; "rho" is the density of the air, about 1.2 kg/m^3; "v" is the velocity; "C_d" is the drag coefficient, being 0.47 for a round ball; "A" is the reference area, namely the cross-sectional area "pi.r^2" where the radius of a tennis ball is 0.035 meters (three and a half centimeters); "m" is the mass, 0.057 kilograms (fifty-seven grams).

And what about the time delay? Just write a simple old-fashioned childish simulation in Mathematica (all quantities are in SI units):

dep = {0, 0};
vel = {0, 0};
dt = 0.001;
timemax = 3;

tenniscoef = 1/2*1.2*0.47*Pi*0.035^2;

For [i = 1, i <= timemax/dt, i++,
time = i*dt;
dep = dep + vel*dt;
vel = vel + 9.81*dt;
vel[[1]] = vel[[1]] - tenniscoef*vel[[1]]^2*dt/0.057
]

vel
dep
timedelay = (dep[[1]] - dep[[2]])/vel[[1]]

What are the results? Well, the final velocities after 3 seconds - near the ground - are 20 m/s and 29 m/s, the total accumulated depth after 3 seconds is 35 and 44 meters, and the time delay is 0.4 seconds (and probably less because the turbulent formula for the drag is an upper bound of a sort). So the delay is a sizable fraction of the total time but it is still a relatively small part.

Because the pétanque ball is comparably large and one order of magnitude heavier than the tennis ball, the effect of the drag on its acceleration would be smaller - by another order of magnitude - than for the tennis ball. So the pétanque ball will differ by 0.04 seconds from a superheavy object.

So from a sensible practical perspective, Allègre was right. The two balls reach the ground at nearly the same moment: less than half a second is close to the time that people, with their limited ability to react and sloppy hands, can distinguish. Of course, this difference can be measured by a careful experiment.

From a deeper, theoretical, pedagogical perspective, he was even more right because the equivalence principle - the universal gravitational acceleration for all objects - is much more important and sensible a starting point to study mechanics than the Aristotelian focus on friction.

People may accept that the acceleration is not quite proportional to the density but they still think that there is an increasing power law at game: this was the popular misconception that the minister would talk about. There's no such a power law, however. For heavy enough objects, the difference is undetectable. For the two balls, the difference in times was slightly above 10% even though one of them was roughly one order of magnitude heavier.

In some sense, this exercise is analogous to the greenhouse effect. There are many effects which exist but the environmentalists want you to think that one of the smaller, and not too theoretically deep, corrections - the greenhouse effect, when we talk about the climate - is the most important thing you should know. And you should forget and deny others.

Much like the drag acting on a tennis ball, it's not too significant. When you care about the overall qualitative behavior, won't hurt you in practice if you completely ignore it.

On the other hand, if you build your whole thinking around the greenhouse effect or the drag as the crucial effects, you will completely miss the point and arrive to qualitatively wrong answers to pretty much any question. And in theory, the drag is not the right principle to learn the rest of mechanics, either. It's one of the irrelevant technicalities that can be easily added when you develop the right formalism. Newton had to overcome Aristotle's wrong focus on the infinite-drag limit before he could actually build physics as we know it.

Of course, you can't explain such things to staunch religious believers such as the global warming advocates. Allègre is a heretic so he's being treated in this way. So the environmentalists hired a poor 85-year-old physics Nobel prize winner, Georges Charpak (who improved the Geiger counter). Charpak simply wrote that Allègre was wrong - and that settled the issue for the global warming enthusiasts. They have always cared about authorities - and, much more often, fake authorities - more than they care about the scientific truth.

However, it's turning, anyway. And the gravitational acceleration acting on all objects is equal. And for the two balls, the gravitational acceleration also explains most of the observations unless you use accurate enough gadgets to perform the experiments.

And that's the memo.

---

P.S. In the first version, I used the density "0.0013" instead of "1.2", ignoring that my input was in g/cm^3 rather than kg/m^3 I needed. Correspondingly, the differences came up 3 orders of magnitude smaller than the real ones.

Thanks to Roger (and Cyril) for having pointed the error out.

After the correction was made, I had to change some qualitative wording as well because the qualitative wording does depend on the numerical values which changed significantly. Note that when such numerical errors are found in the IPCC, the overall qualitative assessment never changes because they don't give a damn about the numbers. Their qualitative conclusions are pre-determined.

I do care about the numbers. So while I was originally surprised to see the delay being just in microseconds ;-) and it was one of the reasons why I decided that the surprising result deserved a posting, I trusted the calculation to some extent up to the moment when the actual mistake was found.