Lewin and Hawking flummoxed by air resistance!

 

Before reading this page, you might like to click on the link below to watch the Youtube video and see if you can resolve the paradox as to why the playing card wins the 80 cm sprint, but the balloon wins the 2 m race, as this page reveals all.

Physics Paradox

You may want to start by watching Lewin’s 8.01 lecture 12. He begins describing the calculations behind his ball/balloon dropping experiment at 33 mins and actually drops the balloon at 39 mins. If you find Lewin’s description hard to follow you may prefer to read my explanation first.

I thought Lewin’s calculations were absurdly wrong, because he seems to confuse the mass of the balloon with its net weight in air. He gives the weight of the balloon as 34 gram, and its diameter as 70cm. If we calculate the mass of the air inside a balloon of this diameter, we get a value of around 220 gram, giving a total mass for the air-filled balloon of 254 gram. I was thinking that the accelerating force due to gravity would just be the 34 gram times g (g is the symbol for the acceleration due to gravity at the surface of the earth, 10 m/s/s), as the air inside has no effective weight in air; but the mass to be accelerated must include the air inside the balloon and therefore be the full 254 gram. Therefore the balloon should not accelerate at 10 m/s/s but rather at 34/254 times 10 m/s/s = 1.3 m/s/s or about one seventh the normal acceleration due to gravity, g/7.

However Lewin writes the equation ma = mg – (Fres) on the blackboard ; so we can only assume that he is using 34 gram as the mass to be accelerated as well as the net mass that is subject to the force of gravity. This seemed to me to be totally absurd.

I could see no error in my logic, but then again was it likely that both Lewin and his graduate student Dave Pooley would make what might be considered such a basic error? And what about the hundreds of thousands of students and teachers who must have watched the lecture and apparently could not see a problem with the calculations? Somebody did make a comment about Archimedes Principle on Lewin’s YouTube channel about this lecture, though not specifically about the balloon experiment, but Lewin roundly dismissed it as irrelevant.

I checked the graph created by the graduate student Dave Pooley, which is shown in the lecture; and he clearly predicts that in the absence of air drag, the balloon would drop like a stone and accelerate to the earth at 10 m/s/s, not at 1/7 of that value as I thought. Was the graduate student completely crazy? If he saw a dead fish sinking to the bottom of a lake, would he say “if it was not for drag, that fish would sink like a stone”, when a fish is so beautifully slimy and streamlined that with one flick of its tail it can glide effortless at many metres per second? How could frictional drag possibly restrict the sinking of the fish to only a few metres a minute, if drag is somehow proportional to the speed? Its beyond ridiculous, how could somebody with so little understanding of the physical world possibly gain a degree in physics? Or perhaps it was me who was up the creek, maybe I had overlooked something? Dave Pooley is now an assistant professor at Trinity University in Texas, so is it really possible that both he and Lewin would get this so catastrophically wrong?

The predictions of Dave Pooley do agree fairly closely with the result of the experiment; the prediction was 1.8 second and the measured value 2.08 second, and Lewin did come up with a whole host excuses as to why the measured value should be slightly higher than the predicted value. Yet I am maintaining that the acceleration of the balloon is only a seventh of that predicted by Dave Pooley, how could my theory possibly agree with the experimental evidence? Time to take a closer look at Lewin’s experiment.

I assumed that the acceleration would be about 10/7 m/s/s, and decided to calculate the time the balloon would take to fall 3m if there was no air drag. Using the equation: distance = ½ at², I get 3m = ½ 10/7 t², so t² = 4.2, so t = 2.05 second. If my calculation was correct, it would imply that air drag has a negligible effect on the result of the experiment, so clearly there is something amiss. Is my logic flawed, or maybe there was something wrong with the way the experiment was performed?

So I replayed the experiment at ¼ speed on Youtube by clicking one of the icons at the bottom of the screen. It appears that Lewin does start the clock at about the same time as he releases the balloon. But if you stop the video when the balloon is a little bit less than ½ m above the ground, you will see that the clock has already stopped at 2.08 second. Assuming the terminal velocity of 1.8 m/s predicted by the graduate student, this means the clock was stopped about 0.25 second too early. So the real time looks like about 2.33 second, not 2.08. We can only assume that Lewin had done the experiment many times, and since he kept getting inconveniently high times, his finger was a little too eager on the trigger. Things are beginning to look somewhat bleak for the graduate student’s calculations; but is this extra quarter second really enough to salvage my claim that the balloon’s acceleration due to gravity is only 1/7 the rate employed by the Dave Pooley in his calculations?

So I decided to check the video to see how long it took for the balloon to fall the first 0.5m; the logic being that if my theory was correct then the balloon would fall so slowly to start with, that the drag over that distance would be negligible. I stopped the video and found that at 0.5m the timer read 0.7s. Ignoring drag, we can calculate an acceleration using the equation: distance = ½ at², so 0.5 = ½ a 0.7², so a = roughly 2 m/s/s. I assume that the weight of the balloon was accurately measured to be 34g, but to accelerate at only 1/5 g the mass of the balloon including air would have to be 170 gram. So the mass of the air would be 136 gram. Assuming a spherical balloon, and air density at 1.2kg per cubic metre, we get 4/3 Π r3 times 1.2 = 0.136. This gives us r = 0.3m. So I concluded that whilst the balloon might sometimes have a diameter of 70cm as Lewin claimed, that at the time of the experiment it was somewhat under-inflated and had a diameter nearer to 60 cm or 2 ft, which is perhaps how it looks in the video.

Then I thought, if the balloon had fallen for 0.7s at an acceleration of 2 m/s/s without drag as I had assumed, then its speed would have already reached 1.4 m/s, which is clearly too fast to ignore the effect of drag. So I tried to see how long the balloon took to fall the first 0.25m, and got a value of 0.49s, which again indicates an acceleration of 2 m/s/s, the same as the acceleration over the first 0.5m where the effect of air drag must have been greater on average. So I concluded that Lewin must have dropped the balloon before he started the timer. I managed to stop the video after the balloon had fallen a few centimetres, and indeed the timer still read 0. If the balloon had fallen for 0.1s at an acceleration of 2 m/s/s, then it would have gone 1cm, or if it had fallen for 0.2s it would have gone 4 cm. So I concluded that Lewin probably started the clock about 0.2s after releasing the balloon. I have not actually done any detailed calculations, but I would assume that this delay of 0.2s is sufficient to make the experimental evidence fit the assumption that the acceleration of the balloon is indeed somewhere around 2 m/s/s. Taking this extra 0.2s into account, the total time the balloon took to fall the 3m must have been close to 2.5s, and not “two point zero” as Lewin triumphantly proclaimed when he saw the clock had registered 2.08 second.

If you check the predictions of Dave Pooley’s graph with the results I extracted from the video, you will indeed see that at 0.25m and at 0.5m his times are wildly out. I also used the video in an attempt to measure the terminal speed, between 1.5m to 2.5m, and got a value of around 2 m/s, slightly higher than the graduate student’s estimate of 1.8 m/s. Did Dave Pooley manipulate his calculations to get a lower terminal speed so as to extend the time, in order to try to get it in line with the experimental result? Or maybe he was just given a value for the diameter of the balloon that was slightly too high, implying greater drag?

So, what do you think? Is my analysis correct? Or do you still think Lewin and Pooley were correct? Or maybe you do not care, and are just annoyed that I should criticise the great man?

What this episode tells us about Professor Lewin, is that for all his great intelligence, scientific training, knowledge of the history of physics and experimental bias, and years of studying and teaching, he was still able to make a catastrophic elementary error about a basic law of nature; and when the experimental evidence pointed to the fact that his assumptions were wrong, he was quite willing to disregard what nature was telling him. Essentially Lewin was subject to the same frailties as all other humans; so really the episode tells us not so much about Lewin himself, but more about human nature in general.

Lewin has made available hundreds of hours of his lectures for people to analyse, and catastrophic errors like this are extremely rare, so what happened in this case? In my experience, pretty much all sources treat a particular subject in physics in much the same way; whether they are online lectures, text books, TV programs, or Wikipedia articles. It is as though all physicists have been taught exactly the same syllabus, and because the lesson that ‘for bodies falling in air the effective gravitational mass is different from the effective inertial mass’ has not been specifically included in the syllabus, it means that many physicists have not fully grasped it.

In ‘The Brief History of Time’, Stephen Hawking wrote:

Of course a lead weight would fall faster than a feather, but that is only because a feather is slowed down by air resistance.”

Hawking does not do any calculations, and he uses the somewhat ambiguous term ‘air resistance’ rather than drag; but surely most readers would assume that Hawking is talking about the same ‘air resistance’ that they might experience on a bicycle; and one can only assume that that was what Hawking was talking about, because he specifically states ‘slowed down’. If we interpret ‘air resistance’ in the broadest possible manner, then we would have to say that a helium balloon is accelerated upwards by ‘air resistance’. All Hawking was really doing was parroting the old adage that in a vacuum, such as on the moon, all objects fall at the same rate because there is no ‘air resistance’; and he surely did not fully grasp the physics, otherwise why would he include a statement that was bound to confuse his readers?

The same equation applies to the helium balloon as to any other object: m(total)a = m(net)g, it is just that for the helium balloon m(net) is negative, since the helium balloon has a lesser mass than the air it displaces, so the helium balloon accelerates in the opposite direction. Of course once the balloon reaches its terminal velocity, the total mass is no longer needed and all we get is m(net)g = air drag. Hawking was talking about a feather not a balloon, but the same effect is important in the case of any object of low density whether its a balloon, a lump of expanded polystyrene, or a feather; though with feathers the situation is particularly complex, because a feather tends to trap air in its branches rather than actually containing a lot of it.

One of the things that makes Lewin’s lectures so entertaining, is the time and effort he put into preparing them; so we can only assume that he did the balloon dropping experiment many times, and kept getting times that did not agree with his calculated predictions. That is why he uses the excuse of turbulence prior to dropping the balloon, and is so relieved when he gets a reading of only 2.08 second in class. We could perhaps criticise Lewin for stubbornly refusing to accept the evidence that there was something seriously wrong with his calculations, but we should be grateful that he had the courage of his convictions; because if he had taken the coward’s way out and ditched the demonstration because he was afraid of being embarrassed, then we would have missed out on an episode that teaches us a bit about physics and a lot about human nature.

The lecture series was filmed in 1999 and made available online by MIT. MIT is certainly one of the world’s top universities, especially for physics as it has a number of Nobel physics Laureates on its staff. Over the years, hundreds of thousands of people must have viewed the lectures online, including some of the brightest students and teachers. I read that the lectures were also shown on a TV channel in the USA, and had millions of viewers. Can such a blatant howler really have escaped everybody’s notice? Probably most viewers accepted what they were told without question. Perhaps a few thought there was something a bit dodgy going on; but were not prepared to accept that Lewin could have made a error of basic physics, or were reassured by the apparent agreement of the calculations with the experiment. Possibly somebody did spot the error but did not want to hurt Lewin’s feelings. Or maybe physicists no longer expose each other’s howlers for fear of bringing the subject into disrepute and thus perhaps giving ammunition to the anti-science brigade?

I hope you enjoyed the story; and the moral is, that if your theory does not agree with experiment, change your theory rather than fiddling the experiment, or the evidence may later return to haunt you. Or else keep the details of the experiment secret, as with the Higgs boson, then nobody can ever prove you wrong.

 

© William Newtspeare, 2017. Unauthorized use and/or duplication of this material without express and written permission from this blog’s author and/or owner is strictly prohibited. Excerpts and links may be used, provided that full and clear credit is given to William Newtspeare, with appropriate and specific direction to the original content.