Rotating Earth: the proofs or significance of Leon Foucault’s pendulum – Part 1

rotating-earth

In an earlier post, we had discussed proofs of the round shape of the Earth. This included some ancient and some modern proofs. There was, in general, a consensus that the shape of the Earth was spherical and not flat and the proofs were given since the time of ancient Greeks. Only in the middle ages, there seems to have been some doubt regarding the shape of the Earth. But amongst the learned people, there was never a doubt about the shape of the Earth. Counter-intuitive it may seem when you look at the near horizon, it is not that counter-intuitive. We can find direct proofs about it by looking around and observing keenly.

But the rotation of Earth proved to be a more difficult beast to tame and is highly counter-intuitive. Your daily experience does not tell you the Earth is rotating, rather intuition tells you that it is fixed and stationary. Though the idea of a moving Earth is not new, the general acceptance of the idea took a very long time. And even almost 350 years after Copernicus’ heliocentric model was accepted, a direct proof of Earth’s rotation was lacking. And this absence of definitive proof was not due to a lack of trying. Some of the greatest minds in science, mathematics and astronomy worked on this problem since Copernicus but were unable to solve it. This included likes of Galileo, Newton, Descartes, and host of incredibly talented mathematicians since the scientific revolution. Until Leon Foucaultin the mid-1800s provided not one but two direct proofs of the rotation of the Earth. In this series of posts, we will see how this happened.

When we say the movement of the Earth, we also have to distinguish between two motions that it has: first its motion about its orbit around the Sun, and second its rotational motion about its own axis. So what possible observational proofs or direct evidence will allow us to detect the two motions? In this post, we will explore how our ideas regarding these two motions of the Earth evolved over time and what type of proofs were given for and against it.

Even more, there was a simple geometrical fact directly opposed to the Earth’s annual motion around the Sun and there was nothing that could directly prove its diurnal rotation. (Mikhailov, 1975)

Let us consider the two components of Earth’s motion. The first is the movement around the Sun along the orbit. The simplest proof for this component of Earth’s motion is from the parallax that we can observe for distant stars. Parallax is the relative change in position of objects when they are viewed from different locations. The simplest example of this can be seen with our own eyes.

Straighten your hand, and hold your thumb out. Observe the thumb with both the eyes open. You will see your thumb at a specific location with respect to the background objects. Now close your left eye, and look at how the position of the thumb has changed with respect to the background objects. Now open the right eye, and close the left one. What we will see is a shift in the background of the thumb. This shift is related by simple geometry to the distance between our eyes, called the baseline in astronomical parlance. Thus even a distance of the order of a few centimetres causes parallax, then if it is assumed that Earth is moving around the Sun, it should definitely cause an observable parallax in the fixed stars. And this was precisely one of the major roadblock

Earth moving around an orbit raised mechanical objections that seemed even more serious in later ages; and it raised a great astronomical difficulty immediately. If the Earth moves in a vast orbit, the pattern of fixed stars should show parallax changes during the year. (Rogers, 1960)

The history of cosmic theories … may without exaggeration be called a history of collective obsessions and controlled schizophrenias.
– Arthur Koestler, The Sleepwalkers

Though it is widely believed that Copernicus was the first to suggest a moving Earth, it is not the case. One of the earliest proponents of the rotating Earth was a Greek philosopher named Aristarchus. One of the books by Heath on Aristarchus is indeed titled Copernicus of Antiquity (Aristarchus of Samos). A longer version of the book is Aristarchus of Samos: The Ancient Copernicus. In his model of the cosmos, Aristarchus imagined the Sun at the centre and the Earth and other planets revolving around it. At the time it was proposed, it was not received well. There were philosophical and scientific reasons for rejecting the model.

aristarchus-01
Aristarchus’s model of the heliocentric Universe

First, let us look at the philosophical reasons. In ancient Greek cosmology, there was a clear and insurmountable distinction between the celestial and the terrestrial. The celestial order and bodies were believed to be perfect, as opposed to the imperfect terrestrial. After watching and recording the uninterrupted waltz of the sky over many millennia, it was believed that the heavens were unchangeable and perfect. The observations revealed that there are two types of “stars”. First the so-called “fixed stars” do not change their positions relative to each other. That is to say, their angular separation remains the same. They move together as a group across the sky. Imagination coupled with a group of stars led to the conceiving of constellations. Different civilizations imagined different heroes, animals, objects in the sky. They formed stories about the constellations. These became entwined with cultures and their myths.

telescope_0013

The second type of stars did change their positions with respect to other “fixed stars”. That is to say, they changed their angular distances with “fixed stars”. These stars, the planets, came to be called as “wandering stars” as opposed to the “fixed stars”.

Ancient Greeks called these lights πλάνητες ἀστέρες (planētes asteres, “wandering stars”) or simply πλανῆται (planētai, “wanderers”),from which today’s word “planet” was derived.

Planet

So how does one make sense of these observations? For the fixed stars, the solution is simple and elegant. One observes the set of stars rising from the east and setting to the west. And this set of stars changes across the year (which can be evidenced by changing seasons around us). And this change was found to be cyclical. Year after year, with observations spanning centuries, we found that the stars seem to be embedded on inside of a sphere, and this sphere rotates at a constant speed. This “model” explains the observed phenomena of fixed stars very well.

The unchanging nature of this cyclical process observed, as opposed to the chaotic nature on Earth, perhaps led to the idea that celestial phenomena are perfect. Also, the religious notion of associating the heavens with gods, perhaps added to them being perfect. So, in the case of perfect unchanging heavens, the speeds of celestial bodies, as evidenced by observing the celestial sphere consisting of “fixed stars” was also to be constant. And since celestial objects were considered as perfect, the two geometrical objects that were regarded as perfect the sphere and the circle were included in the scheme of heavens. To explain the observation of motion of stars through the sky, their rising from the east and setting to the west, it was hypothesized that the stars are embedded on the inside of a sphere, and this sphere rotates at a constant speed. We being fixed on the Earth, observe this rotating sphere as the rising and setting of stars. This model of the world works perfectly and formed the template for explaining the “wandering stars” also.

These two ideas, namely celestial objects placed on a circle/sphere rotating with constant speed, formed the philosophical basis of Greek cosmology which would dominate the Western world for nearly two thousand years. And why would one consider the Earth to be stationary? This is perhaps because the idea is highly counter-intuitive. All our experience tells us that the Earth is stationary. The metaphors that we use like rock-solid refer to an idea of immovable and rigid Earth. Even speculating about movement of Earth, there is no need for something that is so obviously not there. But as the history of science shows us, most of the scientific ideas, with a few exceptions, are highly counter-intuitive. And that the Earth seems to move and rotate is one of the most counter-intuitive thing that we experience in nature.

The celestial observations were correlated with happenings on the Earth. One could, for example, predict seasons as per the rising of certain stars, as was done by ancient Egyptians. Tables containing continuous observations of stars and planets covering several centuries were created and maintained by the Babylonian astronomers. It was this wealth of astronomical data, continuously covering several centuries, that became available to the ancient Greek astronomers as a result of Alexander’s conquest of Persia. Having such a wealth of data led to the formation of better theories, but with the two constraints of circles/spheres and constant speeds mentioned above.

With this background, next, we will consider the progress in these ideas.

A stabilised image of the Milky Way as seen from a moving Earth.

 

The psychology of perception of time in elevators

As a technology, elevators were mandatory for having high rise apartments. You really don’t want to climb up 35 flights of stairs to just get home. My experience with elevators (or lifts as they are more commonly called in India) has been rather strange at times and continues to be so. And I am pretty sure, this is something most people also experience. If you look at it with scrutiny, it is not a strange experience per se, but I found it fascinating nonetheless. As the title of the post suggests, it is about how we perceive the passage of time when we are in an elevator. Now, typically, they would take less than a minute, sometimes perhaps 10-20 seconds to traverse the required distance. Now, here I am considering typical apartment buildings which I have lived in. Not the skyscrapers with 100s of floors. The lift takes about 12 seconds, as timed using a stopwatch to reach my floor if there are no other stops. Of course, if there are stops on intervening floors when people get in or get out, this is longer. So this is the minimum possible time for the lift to take this floor, both ways. That is from my floor to the ground floor and from the ground floor to my floor.

The distance between the ground floor and my floor is constant. The lift and its motor produce the same acceleration and hence same terminal velocity, and the time taken is the same (as measured with a chronometer). I used a quantum-temporal-displacement-chronometer to be sure about time measurement. So our experience of this short travel should also be the same. But this is far from the case. Traveling in the lift gives a variety of experiences. But most strongly it affects how we perceive the passage of time during this short journey. Sometimes it is as if the ground floor is touched as soon as you press the 0 button on the control panel, while at other times it seems time itself has slowed down and it is taking centuries to cover that trivial distance. You may look at the panel displaying the current floor several times during these few seconds and yet it somehow feels lift is moving too slowly. And at times when you are not looking at the panel, and are lost in your thoughts, it chimes to indicate the ground floor has arrived. And you are surprised that it took such a short time. So what kind of blackmagicfuckery is this you wonder? That we subjectively experience something entirely different in terms of time perception is nothing new, but in the case of an elevator, it is so much striking and a part of everyday experience.

I have concocted explanations for the two cases one in which we deem the lift going too slowly and one in which we perceive it be too fast. In the first case, when we perceive the lift to be too slow, we are perhaps not thinking about anything else. Our entire cognitive apparatus and sense organs (eyes and ears) are solely focussed on getting to the destination. Hence, we tend to only look at the floors numbers on the display panel again and again. Expecting it to change often, and our expectation time, the way our neurons are firing is much faster than the real-time. The anticipation is that it should go faster whereas it is going at its own pre-determined pace. Hence, there is a cognitive dissonance that we experience as lift going too slowly. This is even more pronounced if we are in a hurry to get somewhere or are already late. I have seen people press the buttons on the control panel again and again in the hope that it will get them there faster, but it doesn’t work that way. Objectively measured the lift will take the pre-determined time to reach its destination. You are only subjectively experiencing that it is taking longer. Perhaps two persons in the same lift will have a  completely different perception of time depending upon their mental states.

Now coming to the other case, in which we experience the time to be too short, perhaps our cognitive system is already too loaded. This is when before entering the lift we are deep in a thought chain that we are processing. In such a scenario, we expect the lift to just take us to the destination once we press the button. Our schema for the elevator is activated, we don’t have to do any cognitive processing once we press the button. The schema, as an automated response shaped by our experiences with elevators and induction, works seamlessly when not interfered with, assuming that the elevator is behaving in its normal manner. I have had experience of an elevator which could close the door as you were trying to enter. It was almost as if the elevator waited like a predator to catch its pray. Some logic circuits in this elevator were fried, and it won’t let you off you when it caught your leg. Or the elevator might itself have a severe case of fear of heights (vertigo?), as told in HHGTG and would not want to travel to heights. But these being extreme cases, most elevators are domesticated and docile, doing the deed they are designed to do depositing and delivering cargo to destinations, despite the draconian ways in which some travellers might treat them.

Coming back to the explanation for the former case, perhaps due to no cognitive load we are trying to screw with the automated schema. We are just running the simulation of the schema for elevators in our minds, and confusing it with the real world out there. Hence there is a cognitive dissonance. We are expecting something in the mind, while we are seeing something in reality. I have also tried this experiment sometimes when this happens. I close my eyes and mentally calculate the amount of time that might have passed and try to predict the floor that I might have reached. I open my eyes to check if I have guessed correctly but most of the times I am incorrect in the guess.

When we have company in the lift, the temporal experience can be altered and can be subjective as well. If you are with a person whom you find attractive or admire, you might feel that the time taken was perhaps too short. On the other hand, if it is somebody whom you find disgusting or un-attractive, the same journey might seem like a lifetime or a life sentence. In this case, perhaps the cognitive system has become completely Epicurean (when it is not?) in its approach and wants to maximise the good times and minimise the not-so-good ones.

But this does not end the discussion of the elevators. Experiments in elevators provide some useful insights in fundamental physics. This is related to the concepts of frames of reference and the so-called equivalence principle. Elevators are used in Gedanken experiments for thinking about the equivalence principle, which later gave rise to the general theory of relativity.

Apple falling inside a box that rests on the Earth. Indistinguishable motion when the appl is inside an accelerated box in outer space.

 

The equivalence principle states that to an observer in a freely falling elevator the laws of physics are the same as in the inertial frames of special relativity (at least in the  immediate neighbourhood of the centre of the elevator). The effects due to the accelerated motion and to the gravitational forces exactly cancel. An observer sitting in an enclosed elevator cannot, if he observes apparent gravitational forces, tell what portion of these correspond to acceleration and what portion to actual gravitational forces. He will detect no forces at all unless other forces (i.e., other than gravitational forces) act on the elevator. In particular, the postulated principle of equivalence requires that the ratio of the inertial and gravitational masses be M_i/M_g = 1. The “weightlessness” of a man in orbit in a satellite is a consequence of the equivalence principle. Pursuit of the mathematical consequences of the  principle of equivalence leads to the general theory of relativity.. –

From Kittel Mechanics – Berkeley Physics Course Volume 1

 

Another fundamental aspect of physics which uses elevators is the notion of inertial and non-inertial frames of reference. An inertial frame of reference is one in which the particle experiences no acceleration (either transitional or rotational).

Our ability to say whether or not a particular reference frame is an inertial frame will depend in a strict sense upon the precision with which we can detect the effects of a small acceleration of the frame. In a practical sense, a reference frame in which no acceleration is observed for a particle believed to be free of any force and constraint is taken to be an inertial frame.

Now an elevator moving with a constant downward acceleration will be no different than the gravity that we experience on the surface of the Earth. No dynamical experiments conducted inside the elevator will ever tell us whether the elevator is moving with constant acceleration or it is stationary at the surface of the Earth. To know what is the actual case we have to go and perform experiments / take observations outside the lift.

Screenshot 2019-07-31 at 8.21.51 PM

Thus the humble lift or elevator has more to offer to you than just taking you from point A to point B in your daily routine.

Just for fun or how to invite readers to immerse in your book

These problems are for fun. I never meant them to be taken too seriously. Some you will find easy enough to answer. Others are enormously difficult, and grown men and women make their livings trying to answer them. But even these tough ones are for fun. I am not so interested in how many you can answer as I am in getting you to worry over them.

What I mainly want to show here is that physics is not something that has to be done in a physics building. Physics and physics problems are in the real, everyday world that we live, work, love, and die in. And I hope that this book will capture you enough that you begin to find your own flying circus of physics in your own world. If you start thinking about physics when you are cooking, flying, or just lazing next to a stream, then I will feel the book was worthwhile. Please let me know what physics you do find, along with any corrections or comments on the book. However, please take all this as being just for fun.

From Preface of Jearl Walkers The Flying Circus of Physics

Science, a humanistic approach

Science is an adventure of the whole human race to learn to live in and perhaps to love the universe in which they are. To be a part of it is to understand, to understand oneself, to begin to feel that there is a capacity within man far beyond what he felt he had, of an infinite extension of human possibilities . . .
I propose that science be taught at whatever level, from the lowest to the highest, in the humanistic way. It should be taught with a certain historical understanding , with a certain philosophical understanding , with a social understanding and a human understanding in the sense of the biography, the nature of the people who made this construction, the triumphs, the trials, the tribulations.

I. I. RABI
Nobel Laureate in Physics

via Project Physics Course, Unit 4 Light and Electromagnetism Preface

Do see the Project Physics Course which has come in Public Domain hosted at the Internet Archive, thanks to F.  James Rutherford.

Flying Circus of Physics…

The Flying Circus of Physics began one dark and dreary night in 1968 while I was a graduate student at the University of Maryland. Well, actually, to most graduate students nearly all nights are dark and dreary, but I mean that that particular night was really dark and dreary. I was a full-time teaching assistant, and earlier in the day I had given a quiz to Sharon, one of my students. She did badly and at the end turned to me with the challenge, “What has anything of this to do with my life?”

I jumped to respond, “Sharon, this is physics! This has everything to do with your life!”

As she turned more to face me, with eyes and voice both tightened, she said in measured pace, “Give me some examples.”

I thought and thought but could not come up with a single one. I had spent at least six years studying physics and I could not come up with even a single example.That night I realized that the trouble with Sharon was actually the trouble with me: This thing called physics was something people did in a physics building, not something that was connected with the real world of Sharon or me. So, I decided to collect some real-world examples and, to catch her attention, I called the collection The Flying Circus of Physics.

via Jearl Walker | Flying Circus of Physics

The 5 Φ’s of Life

Life as I see it, has five essential `F’s’. Many people may not agree to them, but then this is my blog, so I will tell, whether you like it or not. I will give my reasons for each one, why it is esential according to me. You may agree, or disagree, or give no opinion, it does not matter. Since this blog is more like a personal diary, which I will not link to anybody, I think it is safe to write things here, which I would not like to be in public.

[But then am I not contradicting myself, when I am putting my personal thoughts in a public place?]

So the five F’s

  • Phood: Food is essential for our survival, this represents a living organisms most basic needs. This is what distinguishes us from non-living matter. But the food just should not be for sustenance. It should also be enjoyed. What is the point in eating something that you don’t like? No I don’t mean that we get to eat everything that we like, [I am definitely not suggesting that if you don’t have breads then you should eat cakes], but with whatever we have to eat, we should be enjoying it. If you make the food [not like the plants] but in the more human sense of the world. When you “make” food you get joy of creating something wonderful, if you do not then I am sorry for you. Also the cook should have the complete freedom to do with the food .
  • Philosophy: This is what distinguishes us from the other living beings, we have to have a philosophy of our own, or at least one that is taken from others. But what is essentially needed is to critically look at the aspects of life.
  • Phuck: Well what to say about this? I guess you understand my feelings!
  • Physics: Physics according to some people is the pinnacle of our achievement. Since I am a physicist by training, I have included physics here. Physics has given me a skeptical attitude towards things in life. Though this is not the only path which will lead you here nor that everyone who is a physicist by training will go along this path, but this was my path, hence I list is here.
  • Photography: I have included photography for two reasons.[I am still an amateur [literally and figuratively], as I have not been paid for anything that I have done so far.] One is that photography enables you to store moments, that you have for an extended period of time, and that too in a form that you can share with other people. The other reason is about the art of photography itself. When you are behind a camera, you start to see things differently, from differently perspectives and angles. Is this what not a skeptic needs? Photography in a way provides me with practical tools of implementing many philosophical ideas which would otherwise remain abstract.

Archimedes and the Law of Lever

Archimedes & The Law Of The Lever

The lever presents us with one of the most simple of machines that humans have invented. In fact the lever is one of the six simple machines, which are the building blocks of any complicated, mechanical equipment that we produce. The total six simple machines are:
▪ Lever
▪ Wheel and axle
▪ Pulley
▪ Inclined plane
▪ Wedge
▪ Screw

Simple machines are devices which use mechanical advantage to multiply force. Simple machines can be used to increase the output force, this is at the cost of a proportional decrease in the distance moved by the load. With the mechanical advantage that you get in a lever, you can lift large loads, with application of much less force. We use levers in a variety of ways in our daily life. Just to name a few, the weighing balance, see-saw, even when you lift a weight with your own hand! How many can you identify?

But being a simple machine does not mean that the secret of it can be derived very easily. Just given with a lever, some weights and no other knowledge it is hard for us to derive the mathematical expression for the law of lever. Whereas the Newtonian mechanics gives us a proof, we will see how this proof was presented first. The proof is by Archimedes. Archimedes was the first person to reason and build a theory about the lever, amongst many others things he did besides crying Eureka!! We will sketch the outline of his proof about the law of lever.

If two weights w, W are placed on a horizontal weightless stick, which rests on a support called the fulcrum. One of the following three scenarios will occur. Either the stick will tilt towards the right or the left side, otherwise it will remain horizontal. When the stick remains horizontal it is said to be in equilibrium position. Archimedes considered the question:

“If W is at a distance D from the fulcrum and w is at a distance d from the fulcrum, what condition on W, D, w, d corresponds to equilibrium?”


The answer known to almost all students of physics is that the products wd and WD should be equal. Archimedes did not give a proof to this law in this form. For it is said he would have been offended by multiplying two entirely different quantities such as weight and length. The balance is expressed by him in terms of equality of two proportions W : d = D : w. The statement is that the weights balance at distances inversely proportional to their magnitudes.

Archimedes made some assumptions, which were supposed to be self evident. The assumptions are:

  1. Equal weights at equal distances from the fulcrum balance. Equal weights at unequal distances do not balance, but the weight at the greater distance tilts the lever towards itself.
  2. When two weights are balancing and we add some weight to one of the weights, the weights no longer balance. The weight to which we add goes down.
  3. When two weights balance, we take some weight away from one, the weights no longer balance. The side holding the weight we did not change goes down.

Archimedes uses this assumptions to prove propositions which lead us to the law of the lever. The point in the proofs is that the assumptions should not contradict each other.

Proposition 1: Weights that balance at equal distances from the fulcrum are equal.

Proof: If they are not equal, remove the greater weight difference of the two weights. We have now two equal weights at equal distances from the fulcrum. But according to assumption 3 they do not balance. This contradicts assumption 1. Hence the proof.

Proposition 2: Unequal weights at equal distances do not balance, but the side holding the higher weight goes down.

Proof: Take out the difference between weights, by assumption 1, the remaining two weights balance. If we put back the weight that we have taken, then by assumption 2 weights now do not balance.

Proposition 3: Unequal weights balance at unequal distances from the fulcrum, the heavier weights being at the shorter distance.

Proof: Suppose that the heavier weight is W which is placed at A, and lighter weight w placed at B and the fulcrum is at C, consider the case that they balance each other. Remove W – w from the heavier weight W, thus we have two equal weights. Now by assumption 3, the remaining weights do not balance, but w goes down.

But this is not possible for the following reasons:
Either AC = CB, AC greater than CB, or AC less than CB. Archimedes rules out the first two possibilities. If AC = CB, by assumption 1 the remaining weights balance. If AC is greater than CB then again by assumption 1 weight at A will go down. Thus AC must be less than CB.

Proposition 4: If two equal weights have different centers of gravity, then the center of gravity of the two together is the midpoint of the line segment joining their centers of gravity.


[Archimedes does not define the term “center of gravity”, but approaches the term axiomatically. For a mathematical definition knowledge of integral calculus is required. But we can still understand the term in physical terms. We can assume that all the weight is concentrated at the center of gravity. Another ways is to visualize that the entire weight acts as if it is located at one point, which we know as centre of gravity. ]

Proof: Let the equal weights be w and W, with centers of gravity located at A and B. Let M be the midpoint of the segment AB. Assume that the weights balance at C, a point different than M.

The distance AC is not equal to the distance CB. Hence by Assumption 1 the weights do not balance each other, no matter how C is chosen, as long as it is different from M. This implies that M must be the balancing point.

Here a tacit assumption is made that any two weights have a center of gravity, that is, a balancing point. It is a restatement of Assumption 1 in terms of center of gravity.

Corollary: If an even number of equal weights have their centers of gravity situated along a straight line such that the distances between the consecutive weights are all equal. Then the centre of gravity of the entire system is located on the midpoint of the line segment joining the centers of gravity of the two weights at the middle.

The figure below illustrates the corollary. The center of gravity of the six weights is at C.

Proposition 5: Commensurable weights balance at distances from the fulcrum that are inversely proportional to the magnitudes of the weights. More precisely, if commensurable weights W and w are at distances D and d form the fulcrum, then:

D/d = (1/W)/(1/w) = w/W

[By commensurable it is meant that the ratio of w/W is rational, i.e. there is a third number m such that W = pm, and w = qm and w/W = q/p, and p and q are whole numbers.]


Proof: For convenience let us take a specific case let the ratio of the weights be known, lets say 2:5. So that w/W = 2/5. Let w and W be located at A and B. Let M be their balancing point. What is needed for the proof is that AM : BM = 5:2.

AM/BM = [1/2] / [1/5] = 5/2

We cut the segment AB in to 5 + 2 = 7 equal parts. Divide the weight W into 2*5 = 10 equal parts, and place 5 of them at the midpoints of the five sections just to the right of B, one in each section and five of them in congruent sections to the left of B. Similarly divide w, into 2*2 = 4 equal parts and place 2 of them on the left of A at the and 2 of them to the right of A.

Thus we have a collection of 10 weights each W/10, whose centre of gravity is as same as that of W i.e. B. Similarly the centre of gravity of the w/6 weights is same as that of w, i.e. A. We now have a system of 14 equal weights, which are equally spaced. This collection of 14 weights by the corollary to Proposition 4 balances around the midpoint of the segment holding the 14 weights. This implies the ratio of the lever arms is 5:2, since AM has 2 little weights, and BM has 5 of them. This completes the proof.

Proposition 6: Incommensurable weights balance at distances inversely proportional to their magnitudes.

Proof: Let the weights be W and w at respective distances D and d from the fulcrum. Assume that WD = wd and that the two weights do not balance each other. Assume that weight W goes down.

Remove a small amount of weight from W to obtain W’ such that W’ still goes down but W’ and w are commensurable. Since W’D < wd, W’ rises. This contradiction – that both W’ rises and goes down – proves the proposition.

Is that what prompted him to say


“Give me a place to stand and with a lever I will move the whole world.”

Given the things that he asks for i.e. a place to stand and a lever, will a single person be able to lift the earth?

Let us do some order of magnitude physics for this. We assume that we have a strong rigid rod which will act as lever, which is sufficiently long. Lets say that we have a strong person of who can apply a force of $10^3 N$. We know that the weight of earth is $10^{25} N$. We then calculate in order to displace the earth by 1 cm, how much distance Archimedes will have to move. So that $ \frac{10^{25} \, N}{ 10^3 N} \times 1\, cm \, = \, 10^{22} \,cm}$. Now $10^{22} \,cm \, = \, 10^{20} \, m$. This is too large a distance. So even if Archimedes moved at an fantastic rate of 100 m/s it would still take $10^{18} \, s $ to complete the arc of $10^{20} \, m$. This is equal to $ 3 \times 10^{10}$ years or 3 million years. So to do this fantastic job, Archimedes would have to be incredibly powerful, supplying $10^3 \, N$  force for such long, and also should have a fantastic age of 3 million years!!

But anyways Archimedes was a great man, no doubt about that. Archimedes is referred to as greatest mathematician, physicist and engineer of the antiquity. We will get to other wonderful discoveries of the Archimedes soon…

Mach gives albeit a somewhat different account of the derivation of this law.

References:

Archimedes: What Did He Do Besides Crying Eureka?
Sherman Stein

The Science of Mechanics
Ernst Mach

PS: Typing math without LaTeX is a pain….
PPS: Finally \ \LaTeX \ on the blog \ \ldots \