Thursday, January 21, 2010

Galileo's Genius: The Experiment That Never Was.

Many of us have this mental picture of Galileo standing on top of the Leaning Tower of Pisa and dropping a feather and stone to watch them fall on the ground simultaneously, thus upsetting the Aristotelian assertion that "heavier objects fall faster".

Of course, we know Galileo couldn't possibly have done that experiment, because if he actually had, the presence of air would have reaffirmed the Aristotelian world-view, and the feather would have fallen later.

So what exactly did Galileo do?

I haven't read his original work, but my PhD advisor - Ron Larson - told us this charming story during a class I took with him, and its stuck with me since.

Apparently Galileo did a thought experiment.

He used the idea of "reductio ad absurdum", which begins by assuming something that we wish to prove as false, and hitting a contradiction.

The proof is beautiful, and here is a rough outline.

Consider two objects, with the heavier object of mass M, and the lighter object of mass m. Let us assume that "heavier objects fall faster" - which is opposite of what we wish to prove.

Further consider a really tall tower. If you can think of an infinitely tall tower, that is even better.

Now we've assumed that mass M falls faster than mass m. Here's the beautiful construction: Consider tying the two masses with a piece of string - to create a new object of mass M + m, as shown below.

Let us drop this composite object from the top of our tall tower, and think about what may happen. Because of what we've assumed, the mass M is going to fall faster than m, and in a short while the composite object (that is still falling, remember) will have adopted a configuration that looks more like the following.

Let us now consider the tension inside the string (red line in my diagram). The big mass tugs on the small mass, while the small mass tugs back.

Thus, the small mass slows the big mass down.

Consequently, the composite object of mass (M+m) falls slower than the unfettered object of mass M. But the composite object is heavier, and should fall faster.

Reductio ad absurdum!



Galileo is reported to have dropped a ten-pound weight and a one-pound weight off the Leaning Tower of Pisa, and proved that both fall at the same speed. general principle was being demonstrated, the fact that objects of any weight fall at the same speed.He could not measure speeds with perfect accuracy, he could never prove that these two weights fell with the same speed. But, the alternative hypothesis, predicted by Aristotle, was that the ten-pound weight would fall ten times faster than the one-pound weight. Galileo clearly shattered Aristotle's hypothesis. And the experiment was more or less proof of Galileo's hypothesis. Aristotle was not a person to perform experiments. Galileo described the above experiment in his writings, reasoning much as I do above (as I was amused to find out). But he never claimed to have performed it. Had he actually performed the experiment, it is likely that he would have reported that. So, science historians feel that he never actually did the experiment.

Galileo's results (as opposed to Aristotle) became obvious from Newton's laws of motion, and his law of gravity, which were published several decades after Galileo's death.

It would seem that Einstein's General Relativity (a more accurate law of gravity) says that a heavier object actually does accelerate very slightly faster than does a lighter object, in a vacuum.

Incidentally, the Leaning Tower of Pisa began to lean before it was half built. The clue is that the tower is also bent. The builders obviously bent it to compensate for the leaning of the lower portions. So the lean is greater at the bottom than at the top. Efforts have been made to stop it from leaning further (as otherwise it will fall down eventually) including injecting concrete into the ground underneath. I have not heard if these efforts have been completely successful.

Constantin Bogdan Petrescu said...

Thanks for the interesting article!

Sachin Shanbhag said...

You are welcome, Constantin