physics

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How big is the shadow of the Earth?


The Sun is our ultimate light source on Earth. The side of the Earth facing the Sun is bathed in sunlight, due to our rotation this side changes continuously. The side which faces the Sun has the day, and the other side is the night, in the shadow of the entire Earth. The sun being an extended source (and not a point source), the Earth’s shadow had both umbra and penumbra. Umbra is the region where no light falls, while penumbra is a region where some light falls. In case of an extended source like the Sun, this would mean that light from some part of the Sun does fall in the penumbra.  Occasionally, when the Moon falls in this shadow we get the lunar eclipse. Sometimes it is total lunar eclipse, while many other times it is partial lunar eclipse. Total lunar eclipse occurs when the Moon falls in the umbra, while partial one occurs when it is in penumbra. On the other hand, when the Moon is between the Earth and the Sun, we get a solar eclipse. The places where the umbra of the Moon’s shadow falls, we get total solar eclipse, which is a narrow path on the surface of the Earth, and places where the penumbra falls a partial solar eclipse is visible. But how big is this shadow? How long is it? How big is the umbra and how big is the penumbra? We will do some rough calculations, to estimate these answers and some more to understand the phenomena of eclipses.

We will start with a reasonable assumption that both the Sun and the Earth as spheres. The radii of the Sun, the Earth and the Moon, and the respective distances between them are known. The Sun-Earth-Moon system being a dynamic one, the distances change depending on the configurations, but we can assume average distances for our purpose.

[The image above is interactive, move the points to see the changes. This construction is not to scale!. The simulation was created with Cinderella ]

 

The diameter of the Earth is approximately 12,742 kilometers, and the diameter of the Sun is about 1,391,000 kilometers, hence the ratio is about 109, while the distance between the Sun and the Earth is about 149 million kilometers. A couple of illustrations depicting it on the correct scale.

 

 

The Sun’s (with center A) diameter is represented by DF, while EG represents Earth’s (with center C) diameter. We connect the centers of Earth and Sun. The umbra will be limited in extent in the cone with base EG and height HC, while the penumbra is infinite in extent expanding from EG to infinity. The region from umbra to penumbra changes in intensity gradually. If we take a projection of the system on a plane bisecting the spheres, we get two similar triangles HDF and HEG. We have made an assumption that our properties of similar triangles from Euclidean geometry are valid here.

In the schematic diagram above (not to scale) the umbra of the Earth terminates at point H. Point H is the point which when extended gives tangents to both the circles. (How do we find a point which gives tangents to both the circles? Is this point unique?). Now by simple ratio of similar triangles, we get

$$
\frac{DF}{EG} = \frac{HA}{HC}  = \frac{HC+AC}{HC}
$$

Therefore,

$$
HC = \frac{AC}{DF/EG -1}
$$

Now, $DF/EG = 109$, and $AC$ = 149 million km,  substituting the values we get the length of the umbra $HC \approx$  1.37 million km. The Moon, which is at an average distance of 384,400 kilometers,  sometimes falls in this umbra, we get a total lunar eclipse. The composite image of different phases of a total lunar eclipse below depicts this beautifully. One can “see” the round shape of Earth’s umbra in the central three images of the Moon (red coloured) when it is completely in the umbra of the Earth (Why is it red?).

When only a part of umbra falls on the moon we get a partial lunar eclipse as shown below. Only a part of Earth’s umbra is on the Moon.

So if the moon was a bit further away, lets say at 500,000 km, we would not get a total solar eclipse. Due to a tilt in Moon’s orbit not every new moon is an eclipse meaning that the Moon is outside both the umbra and the penumbra.

The observations of the lunar eclipse can also help us estimate the diameter of the Moon.

Similar principle applies (though the numbers change) for solar eclipses, when the Moon is between the Earth and the Sun. In case of the Moon, ratio of diameter of the Sun and the Moon is about 400. With the distance between them approximately equal to the distance between Earth and the Sun. Hence the length of the umbra using the above formula is 0.37 million km or about 370,000 km. This makes the total eclipse visible on a small region of Earth and is not extended, even the penumbra is not large (How wide is the umbra and the penumbra of the moon on the surface of the Earth?).

When only penumbra is falling on a given region, we get the partial solar eclipse.

You can explore when solar eclipse will occur in your area (or has occurred) using the Solar Eclipse Explorer.

This is how the umbra of the Moon looks like from space.

And same thing would happen to a globe held in sunlight, its shadow would be given by the same ratio.

Thus we see that the numbers are almost matched to give us total solar eclipse, sometimes when the moon is a bit further away we may also get what is called the annular solar eclipse, in which the Sun is not covered completely by the Moon. Though the total lunar eclipses are relatively common (average twice a year) as compared to total solar eclipses (once 18 months to 2 years). Another coincidence is that the angular diameters of the Moon and the Sun are almost matched in the sky, both are about half a degree (distance/diameter ratio is about 1/110). Combined with the ratio of distances we are fortunate to get total solar eclipses.

Seeing and experiencing a total solar eclipse is an overwhelming experience even when we have an understanding about why and how it happens. More so in the past, when the Sun considered a god, went out in broad daylight. This was considered (and is still considered by many) as a bad omen. But how did the ancient people understand eclipses?  There is a certain periodicity in the eclipses, which can be found out by collecting large number of observations and finding patterns in them. This was done by ancient Babylonians, who had continuous data about eclipses from several centuries. Of course sometimes the eclipse will happen in some other part of the Earth and not be visible in the given region, still it could be predicted.   To be able to predict eclipses was a great power, and people who could do that became the priestly class. But the Babylonians did not have a model to explain such observations. Next stage that came up was in ancient Greece where models were developed to explain (and predict) the observations. This continues to our present age.

The discussion we have had applies in the case when the light source (in this case the Sun) is larger than the opaque object (in this case the Earth). If the the light source is smaller than the object what will happen to the umbra? It turns out that the umbra is infinite in extent. You see this effect when you get your hand close to a flame of candle and the shadow of your hand becomes ridiculously large! See what happens in the interactive simulation above.

References

James Southhall Mirrors, Prisms and Lenses (1918) Macmillan Company

Eric Rogers Physics for the Inquiring Mind (1969) Princeton

 

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Galileo’s Experiments on Accelerated Motion

A short account of Galileo’s description of his own experiment on accelerated motion — a short account of it, the apparatus he used and the results he got.
The first argument that Salviati proves is that in accelerated motion the change in velocity is in proportion to the time (𝑣 ∝ 𝑡) since the motion began, and not in proportion to the distance covered (𝑣 ∝ 𝑠) as is believed by Sargedo.

“But for one and the same body to fall eight feet and four feet in the same time is possible only in the case of instantaneous (discontinuous) motion; but observation shows us that the motion of a falling body occupies time, and less of it in covering a distance of four feet than of eight feet; therefore it is not true that its velocity increases in proportion to the space. (Salviati)

Also, he proves that the increase in proportion is not of simple doubling but larger. They agree upon a definition of uniformly accelerated motion,

“A motion is said to be equally or uniformly accelerated when, starting from rest, its momentum receives equal increments in equal times. (Sargedo)

To this definition Salviati adds an assumption about inclined planes, this assumption is that for a given body, the increase in speed while moving down the planes of difference inclinations is equal to the height of the plane. This also includes the case if the body is dropped vertically down, it will still gain the same speed at end of the fall as it would gain from rolling on the incline This assumption makes the final speed independent on the profile of the incline. For example, in the figure below, the body falling along𝐶 → 𝐵, 𝐶 → 𝐷 and 𝐶 → 𝐴 will attain the same final speed.

This result is also proved via a thought experiment (though it might be feasible to do this experiment) for a pendulum. The pendulum rises to the height it was released from and not more.
After stating this theorem, Galileo then suggests the experimental verification of the theorem. of The actual apparatus that Galileo uses is an wooden inclined slope of following dimensions: length 12 cubits (≈ 5.5 m, 1 cubit ≈ 45.7 cm), width half-cubit and three-finger breadths thick . In this plank of wood, he creates a very smooth groove which is about a finger thick. (What was the thickness of Galileo’s fingers?) The incline of this plank are changed by lifting one end. A bronze ball is rolled in this groove and time taken for descent is noted.

“We repeated this experiment more than once in order to measure the time with an accuracy such that the deviation between two observations never exceeded one- tenth of a pulse-beat.

Then Galileo performed variations in the experiment by letting the ball go different lengths (not full) of the incline and “found that the spaces traversed were to each other as the squares of the times, and this was true for all inclinations of the plane”. Each variation was repeated hundreds of times so as to rule out any errors. Also, the fact that for different inclines the times of descent were in noted and were in agreement with the predictions.
Since there were no second resolution clocks to measure time, Galileo devised a method to measure time using water. This was not new, water clocks were used earlier also.

The basic idea was to the measure the amount of water that was collected from the start of the motion to its end. The water thus collected was weighed on a good balance.This weight of water was used as a measure of the time. A sort of calibration without actually measuring the quantity itself: “the differences and ratios of these weights gave us the differences and ratios of the times”

Galileo used a long incline, so that he could measure the time of descent with device he had. If a shorted incline was used, it would have been difficult to measure the shorter interval of time with the resolution he had. Measuring the free fall directly was next to impossible with the technology he had. Thus the extrapolation to the free fall was made continuing the pattern that was observed for the “diluted” gravity.

“You present these recondite matters with too much evidence and ease; this great facility makes them less appreciated than they would be had they been presented in a more abstruse manner. For, in my opinion, people esteem more lightly that knowledge which they acquire with so little labor than that acquired through long and obscure discussion. (Sargedo)

 

Reference

Dialogues Concerning Two New Sciences

William Francis Magie A Source Book in Physics, New York, Harvard University Press 1935
The Mitr

Book Review: Pendulum: Léon Foucault and the Triumph of Science by Amir D. Aczel

The book traces Leon Foucault’s ingenious approach to solving the problem of providing a terrestrial proof of rotation of the Earth. The pendulum he devised oscillates in a constant plane, and if properly engineered (as he did) can actually show the rotation of the Earth. The demonstration is one the most visually impressive scientific experiments. Also, Foucault gave prediction, an equation which would tell us how the pendulum will behave at different parts of the Earth. The pure mathematicians and physicists alike were taken aback at this simple yet powerful demonstration of the proof which eluded some of the most brilliant minds, which includes likes of Galileo and Newton. Rushed mathematical proofs were generated, some of the mathematicians earlier had claimed that no such movement was possible. That being said, Foucault was seen as an outsider by the elite French Academy due to his lack of training and degree. Yet he was good in designign things and making connections to science. This was presented to the public in 1851, and the very next year in 1852 he created another proof for rotation of the Earth. This was done by him inventing the gyroscope.. Gyroscope now plays immense role in navigation and other technologies. Yet he was denied membership to the Academy, only due to interest of the Emperor Napolean III in his work in 1864. The pendulum is his most famous work, but other works are also of fundamental significance.

  • He was first person to do photomicrography using Daguerreotype
  • Accurate measurment of speed of light using rotating mirrors –
  • Devised carbon arc electric lamp for lighting of micrcoscope
  • One of the first to Daguerreotype the Sun
  • Designed the tracking systems used in telescopes
  • also designed many motors, regulators to control electrical devices

There are a couple of places in the book where Aczel seems to be confused, at one point he states parallax as a proof for rotation of Earth around its axis, whearas it is more of a proof of Earths motion around the Sun. At another place he states that steel was invented in 1800s which perhaps he means to say that it was introduced in the west at the time. Apart from this the parallels between the rise of Napoleon III, a Nephew of Napolean, to form the second Empire in France and Foucault’s own struggle for recognition of his work and worth is brought out nicely.

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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.

The Mitr

Review of His Master’s Voice by Stanislaw Lem

 

The book is an autobiographical tale by one of the mathematician-scientist who looks at a mysterious signal from the cosmos. The title is from the title given to the top secret project which tries to decipher this signal. The signal in the form of a neutrino stream is discovered accidentally and is hidden well in the general noise of neutrino signal. Only if you know where to tune in to is the signal readable/recordable/visible. The signal is attacked upon by a team of experts from different domains like physics, chemistry, biology, language, mathematics. They are able to know that the signal has an “alphabet” but are not able to crack the code as a whole. Though they discover some properties of the signal to interact with matter. For example, they discover that this letter from cosmos has a positive effect on the formation and consolidation of large protein molecules. They also discover a “recipe” for building a substance which is dubbed as “Frog Eggs” and “Lord of the flies”. This substance with a consistency of frog eggs can absorb energy from radioactive fission within itself and has some peculiar properties.

even though, receiving the message from the stars, we did with it no more than a savage who, warming himself by a fire of burning books, the writings of the wisest men, believes that he has drawn tremendous benefit from his find!

That not withstanding, the entire operation is under government supervision and there are plots and counter-plots of bureaucracy enmeshed within the narrative. This also includes an effect termed as “TX” in which a small nuclear detonation can have its energy transmitted to another place. But large scale implementation fails as the energy is dissipated over a very large area rendering any weapons created from them unusable. After these initial success, there is not further “code-breaking” possible and things come to a standstill

We are proceeding like a man who looks for a lost thing not everywhere, but only beneath a lighted street lamp, because there it is bright.

They also discover there is another parallel team working on the same problem but under the command of the military. Finally, the two units are merged.

At this point, various theories are put forth which try to explain the origin of the “letter”. Doubts are even raised to know if the signal is “natural” or “artificial”.  One of the military members uses the oscillating universe model to suggest that the neutrino signal is information from the past universe, from a ‘fissure” in between the universes, to the current one. One more hypothesis is given in the form that the frog eggs naturally evolved and the neutrino signal is just a by-product and the “organisms” do not know if this signal is being sent. A closer example of this is plants doing photosynthesis, they are not aware that their photosynthetic activity is helping other organisms grow, they do it nonetheless.

And surely it was unintentional on the part of the grass to give us the opportunity to exist!

While the author genuinely believes that the signal is from a very old and highly evolved “civilisation”, and we are at a stage such that we cannot understand the letter fully. We are not meant to, not at this stage of our technological evolution. The signal has been there for billions of years, and it takes an enormous amount of power (at least by our standards) to send it, so whoever (or whatever)  is sending it must have a purpose, just that we don’t know ( and perhaps will never know) what the purpose is.

 We will make it undecipherable for all who are not yet ready; but we must go further in our caution — so that even a false reading will not be able to supply them with any of the things that they seek but that should be denied them.

The book is an interesting take on the status of technological progress and its ramification for civilisation as a whole. Some of the themes that one can identify is the survival of the species and not of a particular nation. The concerns expressed over the “TX” discovery make the smaller group privy to this very anxious as we would then have a weapon which at the speed of light can deliver an atomic explosion anywhere. Some of the musings about the senders of the signal and the kind of evolution the civilisation that must have are interesting to read.

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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

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Can general laws of physics explain everything?

Many scientists look on chemistry and physics as ideal models of what psychology should be like. After all, the atoms in the brain are subject to the same all – inclusive physical laws that govern every other form of matter. Then can we also explain what our brains actually do entirely in terms of those same basic principles? The answer is no, simply because even if we  understood how each of our billions of brain cells work separately, this would not tell us how the brain works as an agency. The “laws of thought” depend not only upon the properties of those brain cells,but also on how they are connected. And these connections are established not by the basic, “general” laws of physics, but by the particular arrangements of the millions of bits of information in our inherited genes. To be sure, “general” laws apply to everything. But, for that very reason, they can rarely explain anything in particular.
– Marvin Minsky in The Society of Mind pp. 26