Wednesday, September 12, 2007

Gravity

Man has always been aware of gravity, even if that conception was as simple as "what goes up, must come down". Before the advent of the scientific revolution, the study of science was merely qualitative: it was a matter of speculation and philosophical debate rather than evidence and testing hypotheses. Aristotle for example, believed that all things moved downwards because it was their natural place.

Although ideas of gravity were pretty well developed outside of Europe (although we always pay our attentions in the history of science to European scientists, we must not forget that scientists in the Middle East, India and China were often much more advanced than Europe at various times, especially during the Dark Ages), it was in Europe that major theoretical advances were made.

In the early 16th century, a Catholic cleric named Nicolaus Copernicus, who was also a highly skilled astronomer wrote a book named De revolutionibus orbium coelestium. Legend has it that on the day Copernicus died, the book was delivered to his hands, and died peacefully after he had seen it. This book was the first of its kind offering an alternative to the geocentric theories (the idea that the Sun revolved around the Earth). Copernicus, being a Catholic cleric was wary enough to state very clearly that his work was merely a fantasy, and not something that he believed was true, because to believe in heliocentricity (the Earth revolving around the Sun) was considered contrary to the Bible.

In his book, he proposed that the planets, including the Earth, revolved around the Sun in perfect circles, and in so doing was able to explain some phenomena that were never very well explained by the geocentric theory, such as the fact that Mars does not move in a continuous motion throughout the sky at times, sometimes going forwards, coming to a halt, and then moving backwards (known as Retrograde motion)

However his theory wasn't able to match up fully with celestial observations at that time. It would take subsequent modification by Kepler (who did away with the perfectly circular motion and replaced it with simply elliptical motion) to become a coherent set of rules that could easily predict the motion of the planets.

Kepler developed his own three laws of planetary motion to explain how the planets revolved around the sun (click here to find out more).



For example, one of Kepler's laws state that if a planet, at two different segments, sweeps out an area (as shown in blue) with equal area, then they must have travelled that segment in the same amount of time.

Notice that his laws are useful in the prediction of the motion of planets, but yet do not tell us fundamentally the reasons for their motion: This is why his laws are considered laws of kinematics: we can describe the motion, but we don't really know why, much like the laws of kinematics that you have studied.

Notice also that his laws do not involve gravity, whereas today we know the reason for their motion is because the Sun's gravity acts on them. It may seem obvious to us today, but before Isaac Newton was knocked on the head by an apple, it was difficult to realise that the same gravitational force that brings apples to the ground also kept the planets in motion!

Our first truly useful theory of gravity was provided by who else but Isaac Newton. He hypothesised that gravitational force could be computed as follows:



The force of gravity between two objects is given by the mass of one object multiplied by the mass of the other, divided by the distance separating them squared, times the universal gravitational constant G.

What this equation says is profound: firstly, it states that any, and truly any two masses exert a gravitational force on one another. Right now, whether you know it or not, even though I am halfway across the world, I am exerting a gravitational pull on you now, although of course it is so small as to not affect you at all. (There is something profound in this as well. Clearly it is very absurd that my movement right now can somehow affect you halfway across the world: how is the force transmitted?)

Also, notice that the denominator of the equation is squared: what this means that for an increase in distance between two objects, the force drops off significantly, which suggests that the force of gravity is very weak.

And finally, one of the most excellent strokes of brilliance by Newton, his universal gravitational constant was a constant that he believed applied everywhere in the universe: an extremely bold conclusion! But yet, a few hundred years later, he was proven right: the value of G was found to be 6.67 x 10-11, as first determined by the Cavendish Experiment.

This equation was put to great effect by Newton in explaining the motion of the celestial bodies, successfully incorporating all three of Kepler's Laws under this simple equation. Today, this equation is no longer regarded as the best way to describe gravity: it has been supplanted by General Relativity. But still, it is extremely useful in the prediction of celestial objects in the Solar System, since the difference between Newton's law of gravitation is not significantly different from General Relativity in our region. When I start my lessons in relativity I may come back here and talk more!

Saturday, August 25, 2007

Energy and Newton's Laws

I'm sure you are finished with Newton's Laws by now, and will be moving on to energy. The question I would like to pose here is: what is the point of learning about energy?

Think about it: Newton's three laws cover every aspect of motion, so why can't we understand all motion by the three laws alone? Why do we have to come up with an extra concept called energy? What is energy anyway?

I'm not sure if you have thought about before, but can you define what a force is? Or for that matter, can you define mass? Some of you may have heard that by definition mass is the amount of substance in an object, but that is distinctly untrue! As chemistry students, I would expect you to know that the measure of the amount of substance is the MOLE, not the kilogram.

So what is mass? Would it bug you if I told you that there is no clear way of defining it? And for that matter, there's no real way of defining force either? And energy?

So why do we bother with all these things that can't be defined? The point is, as I've pointed out many times before, physics is a way of quantifying physical behaviours and analysing them using mathematics, and it just happens the quantity of "force", "energy" and "mass" seem to obey some kind of law. A good follow up question would then be why do they bother to follow some kind of law in first place?

I guess this is one of the most fundamentally interesting things about physics and nature: that for some reason these seemingly meaningless (undefinable!) quantities appear to have some kind of logical relationship with each other. In 1918, the female mathematician Emmy Noether published a mathematical paper, showing that if we assume that the laws of physics are invariant over time (meaning that the laws of physics don't change with passing time, a reasonable assumption), then the quantity known as "energy" is conserved, meaning that the total amount of energy always remains constant. This is of course something that you've already learnt, but here is a real explanation as to why the quantity of energy is so important.

But then we return to the question, why is Newton's Laws not sufficient for us to understand everything about motion? Why must we introduce energy?

In fact, Newton's Laws are sufficient in studying motion. But the problem is, in many cases they are incredibly cumbersome. When you begin your study of energy, note that in many cases, the questions that you are solving tend to involve information only about the initial and the final state of the object. In another words, you may be solving questions in which you know what's happening at the start, and what's happening at the end, but you may not exactly know all the details of what's happening in between. In such cases, because you don't really know what goes on in the middle, Newton's Laws become very difficult to use, and sometimes even impossible.

Energy is equivalent to Newton's Laws, but they allow the physicist to make conclusions about something just by looking at the initial and final stages, without bothering about what is going on in between. This is of course a very powerful tool.

So here's a tip: if you are unsure whether a question deals with energy or Newton's Laws, the best way to tell is what kind of information do you have? If you have no information about the process of something, or if you have no information about the time taken for a process, it most probably involves energy. As simple as that.

Remember, energy is not separate from Newton's Laws: they are like two sides of the same coin, all part of a big picture.

Saturday, August 18, 2007

Flying and the Third Law

Greetings from Cornell University! Haven't really been updating recently, been too busy packing and doing my orientation stuff, but now I finally feel quite inclined to do this.

I don't know if you feel the same thing I do, but every time I take a plane, its take off is always a test of my faith in science and the triumph of human thought. Just think about it: Something like 400 people, sitting in what is effectively a fortuitously well-designed metal tube lifting off from the ground. It defies all human experience and all human logic: it is not surprising that Wilbur Wright told Orville Wright, the two great pioneers of human flight, that man would not fly in a thousand years.

So what exactly makes a plane take off? Well, there have been plenty of misconceptions and poor explanations of the real reasons for lift, and I've also realised that my understanding of it was rather poor before doing a little research for this post. Strangely, the "Bernoulli Principle" that people like to throw around when they explain how an aerofoil generates lift is not really very appropriate.

There are two reasons to explain lift, and the first is pretty straightforward. Imagine hitting a ping-pong ball in a game of table-tennis, or a tennis ball in a game of tennis: if you incline your racket downwards so that it is tilted towards the ground when you hit the ball, you are going to generate a downward force on the ball, and by Newton's 3rd Law, the ball generates an upward force on the bat. This works as well for the airplane wing. As long as it is inclined in an angle to deflect air downwards, lift will be generated!

The other reason is pretty complicated, and you'll just have to take my word for it, because I'm taking the website's word for it! As air passes across the aerofoil, it tends to "stick" to the metal of the aerofoil, and because the aerofoil is curved, the air is forced to curve around the aerofoil. Apparently, when air is forced to make a curve, it will be flung outwards, just like passengers in a car are flung to the side of the car when the car makes a turn. Thus, the air is flung outwards, and because it is stuck to the wing, it pulls the wing along with it.

Sorry for the quick post, but time's really quite tight at the moment. I'll come back with more the next time!

Monday, July 30, 2007

The Horse and The Cart (IN SYLLABUS!!)

Let's discuss something that is commonly brought up in a class on Newton's 3rd Law to test the concepts of students. If you find that what follows doesn't really make sense to you, then please try to find out what's happening.

Newton's 3rd Law states that for every action, there is an equal and opposite reaction. This basically means that forces always occur in pairs. In any situation where one object is exerting a force on the other, then the other object is also exerting a force that is equal in magnitude but opposite in direction on the initial object. So for example, The Sun exerts a gravitational pull on Earth, and likewise, the Earth exerts the same gravitational pull on the Sun. But since the mass of the Earth is so much smaller than that of the Sun, the effects of the force on the Earth is much greater, and so we orbit around the Sun, while the Sun merely does a small wobble in reaction to the forces exerted by the Earth.

We come now to a very famous puzzle which features a horse and a cart.



The horse(I know that's a donkey, but the idea's the same.) is pulling on the cart to try to pull the cart forward. But by Newton's third law, doesn't the cart pull on the horse as well? So if the horse pulls on the cart, and the cart pulls back on the horse, how does the whole thing move at all? Shouldn't they be locked in a never-ending battle that is guaranteed, by Newton's 3rd Law, to always end in a tie?

Now before you read on, please pause for awhile to figure out this problem on your own. If you've already had your lesson on Newton's 3rd Law, and you thought you had no problems with the 3rd Law, then you should be able to figure this out on your own. If you can't, then perhaps your understanding is not as clear as you thought!

Figured it out yet?

Answer:

If you couldn't figure out what was wrong with the paradox presented above, then you have not understood the true spirit of Newton's 3rd Law. Let us state Newton's 3rd Law again, this time using the modern phrasing: If Body A exerts a force on Body B, then Body B exerts a force that is of equal magnitude but in the opposite direction on Body A. Now let's look at the horse and cart system again:



By Newton's 3rd Law, the horse exerts a force on the cart, and the cart exerts an equal but opposite reaction on the horse. That is true. But this does not mean that no motion is possible. Why? Because the pair of forces are acting on different bodies.

Let us look at the horse alone. What are the forces exerted on it? There are only two forces: the force exerted on it by the cart, and the force exerted by the ground. Remember also that the force exerted on the horse by the ground is equal and opposite to the force exerted on the ground by the horse. So long as the horse is able to exert a force greater than the force exerted by the cart on him, then there will be a nett force, and the horse will begin moving.

Looking at the cart alone, there are two forces acting on it: the force exerted by the horse, and the force exerted by the ground. So long as the force exerted by the horse is greater than the force exerted by the ground, then the cart will start moving as well.

So there is no paradox! If you thought there was a paradox, that was because you thought that the force acting on the horse by the cart and the force acting on the cart by the horse cancel each other out. But they do not because they are acting on different things. So once again, Newton saves the day!

Remember, you may think you understand certain concepts in physics, but a more rigorous examination will prove otherwise. Even after 6 years of studying physics, there are aspects Newton's laws that I still have not fully understood. Don't be satisfied with what you've learnt in class and believed was correct. Test them out yourselves!

Wednesday, July 18, 2007

Determinism

Science (especially physics) has plenty in common with philosophy. This may seem strange, because in school they seem radically different from each other: science involves the demonstration of certain rigid facts and truths that are immutable, whereas philosophy involves thinking about life, and is fraught with grey areas.

But the perception that science is about facts and truths is a dangerous one. In fact, as scientists, we know of nothing that is a fact or a truth. You probably already know this: no inductive statement can be made in the certainty that it is absolutely and totally correct. What are inductive statements? They are general conclusions that are believed to be true, true made after some observations. For example, you've seen the Sun rise in the east your whole life, and so you claim that "the Sun always rises in the east". That is a conclusion you have arrived at, because you've seen it happen again and again, but of course, that is no guarantee that the same will happen tomorrow. Of course, you could say, "yesterday, the Sun rose in the east," but science has no use for these statements: they are mere observations, which are distinct from conclusions.

So as you can see, whatever you've been taught, e.g. the angle of incidence and the angle of reflection are always equal in the reflection of light, are not facts. They are merely statements that have been tested repeatedly, possibly millions of times, and have not once been found to be wrong.

What has been said so far is just a small glimpse into the philosophy of science, and unfortunately pretty much all I'm confident enough to speak on about this huge subject. If you look at the two subjects of physics and philosophy, you can see why they are so intertwined: one is the scientific study of the universe, while the other is thinking about the true nature of the universe. Actually, the subject matter at hand is pretty much the same in both subjects! Just that the approach is different.

So I leave you with a philosophical tidbit to ponder over, and, in accordance with your syllabus, it's related to Newton's Laws. I hope that as you are studying Newton's Laws, you will realise that these laws, together with Newton's Law of Gravitation, are meant to describe any kind of motion, whether it's normal motion down here on Earth, or the motion of stars and galaxies. They encompass everything in the universe. Why things stay still, why things move at constant velocity, why things have a change in their velocity, and how do we know how much it changes by, how objects interact with each other, these are just some of the questions answered by Newton's Laws.

Imagine that we knew what every single particle in the universe was doing at this exact moment, where every particle was and how it was moving. Since we know how each and every particle is going to move (since they move according to Newton's Laws), if we fed all of this information into a gigantic supercomputer, wouldn't the computer be able to work out the exact future of each and every particle? In other words, predicting the future is possible if we knew precisely what every single particle is doing in the whole universe.

Now, think about what implications this has on human beings. Aren't we composed of particles ourselves, that undoubtedly obey Newton's Laws? If there was someone somewhere out there running a supercomputer that really did know the exact position and motion of every single particle in the universe, wouldn't he be able to look into our futures? Doesn't that mean that our futures are already cast in stone?

With the advent of Newton's Laws, many people began to believe that the universe is deterministic: the idea that there is only one possible future to this universe. This idea posed a great challenge to European thought, which was still closely associated with Christian beliefs in the 17th and 18th centuries.

Nowadays, with the advent of quantum theory and the theory of relativity, which in effect are more accurate ways of looking at the universe as compared to Newton's Laws, the universe is commonly regarded as being indeterministic, although much debate still rages on over what the horribly complicated math of quantum mechanics actually suggests. I leave you to read about the Heisenberg Uncertainty Principle and the many interpretations of quantum mechanics.

Monday, July 9, 2007

A Newton's 1st Law Thought Experiment

One of the most powerful weapon that a physicist can have is the thought experiment, or gedankenexperiment, which was the original German term coined by the physicist Hans Christian Oersted (which you should subsequently meet: he discovered that a compass, when placed near a wire with electric current, is deflected, the first known link between electricity and magnetism!).


The powers of the thought experiment are immense: some of these experiments can never be performed (e.g. a bucket of water suspended in an entirely empty universe!), but nonetheless thinking about them reveal some very important loopholes in thought. The bucket of water thought experiment (see: Mach's Principle) and the arguments of Newton and Mach helped formulate General Relativity for Einstein. Einstein, Podolsky and Rosen's famous thought experiment, known as the EPR Paradox is another example of a thought experiment that helped bring to attention the failings of theories, and to re-order our thinking.


I'm going to present a very simple thought experiment that leads us naturally to Newton's 1st Law. Newton's 1st Law states that any object in a uniform state of motion will continue in that state of motion unless acted upon by a force. This means that any object will go on doing whatever it was doing (either travelling in a straight line, or remaining stationary) unless someone or something decides to do something about it.


This fact is by no means straightforward to deduce: when a horse pulls a cart, the cart starts moving, but when the horse stops pulling, the cart stops moving. In fact, this fact was so difficult to deduce, that for many years the thinking was that a force was required to produce and to sustain motion (of course, you would know that the reason the cart stops moving is due to friction).


I hope you see the difference here: before Newton came along, people were questioning the reasons for motion, and many believed that motion was a result of a force. But Newton realised that motion itself has no reason. An object that is moving uniformly continues to move uniformly because that is the way of the universe. Only changes in motion could be explained by the presence of a force.


One way to arrive at this conclusion is a very neat thought experiment that really impressed me when I first encountered it. It is so simple that there is no denying the accuracies of its conclusions and the correctness of Newton's 1st Law.



So, imagine a ball rolling down a ramp. If we make the ramp really really smooth, like bowling alley smooth, and drop the ball off from a certain height, you can ascertain that the ball ascends to somewhere around its original height. One thing you can be sure: the ball never stops somewhere along the bottom of the ramp.


I hope you can see where this is going. We just have to make the ramp longer:

And by the same argument, the ball should rise up to its original height again. And now, the prestige of the trick!



If we were to have an infinitely long ramp, what would the ball do?


Naturally the ball wants to return to its original height, as we have argued in the previous two diagrams, but in an infinitely long ramp, the ball has no choice but to roll on forever! And thus, what we have shown from a simple thought experiment is that uniform motion is a natural state, and if undisturbed, goes on forever.


Of course, thought experiments in no way prove anything. The EPR Paradox mentioned earlier was cited as a way of debunking Quantum Mechanics, because it gave incredulous results in theory that goes against "common sense". But, when physicists actually got around to performing the experiment cited in the EPR paradox, the results that came back totally went against what scientists had long considered to be irrefutable.


But nonetheless, they are extremely useful ways of thinking about physics and reality.

Saturday, June 30, 2007

Vectors

I've heard that you guys have already started vectors (so fast!), so here's something to help you understand the concept better. Disclaimer: Some parts of the following were self-formulated, so they may not be conceptually very sound, but it helped me to look at things this way.

In physics, we are extremely interested in physical quantities. These are measurements taken from an object, such as mass, force, speed, displacement, etc. that have relationships amongst each other that leads to an understanding of reality. To a certain extent, these quantities are extremely difficult to define (try asking yourself what is mass, force, and displacement. You will find that many times you only have an intuitive grasp of the concept that cannot be put into proper words!) or are defined based on one another.

To understand the fundamental difference between scalars and vectors (which may seem rather easy, but you will encounter some really confusing quantities as you trot along in your physics career), we must see how we add these quantities together.

Let's look at a scalar quantity, like mass. If we have a piece of plasticine of mass 3 g, and we mash it together with a piece of plasticine of mass 4 g, what is the final mass of the plasticine? Of course, the answer is 7 g. Even without knowing how the two pieces were mashed together and other sordid details, all we just had to do to obtain the final "sum" of the two masses was to add two numbers together.

What this means is that scalar quantities are added together or subtracted from each other in the usual sense: we just add or subtract the required numbers and voila, you get your answer. No new mathematics required here.

How about for a vector quantity? Let's have a look at displacement, since you should already be very familiar with the concept. If I travel a displacement of 3 m and then subsequently travelled a subsequent displacement of 4 m, what is my total displacement?

Clearly, what you require now are some details, and the details that are required are the directions in which the displacement occurred. It is clearly because the direction of the quantity matters that they become vectors. Depending on the directions of travel, the answer could be anywhere between - 1 m and 7 m. Literally anywhere.

What you will immediately realise is that the adding of vectors require some higher level mathematics, whereas for scalars, we just simply use arithmetic to perform our computation.
So what kind of higher level mathematics is required? Well, not much more than what you have learnt in co-ordinate geometry!

Let's look at the above example: say you travelled 3 m east, and then 4 m south. Let's super-impose an x-axis and a y-axis over your displacement, with one unit on the "graph" representing one metre:
Now, to find the total displacement at the end of the journey, we have to find the magnitude of the blue vector (which can be found by Pythagoras' Theorem) and its direction, angle t (which can be found by application of your knowledge of trigonometry). Now, you can see clearly that the blue vector can be easily represented by 3 units east, and 4 units south, or, since we have drawn the axes on, 3 units right on the x-axis, and 4 units down on the y-axis. We can represent such a vector as follows:

The entry at the top represents the x-axis (positive means right, negative means left) and the entry at the bottom represents the y-axis (positive means up, negative means down). The advantage of this notation is that for any vector, if you wish to calculate its magnitude or direction, you can easily obtain it from the values.

What is even more remarkable is what comes next:

When we add the two parts of the displacement together by adding each individual entry to each other, we can get the sum total of the two vectors.

So here's the overall conclusion:

1. Any vector can be expressed in the form of a matrix (that's what we call that fancy notation), with the top entry being the x-axis, and the bottom entry being the y-axis.

2. In order to add vectors together, all we have to do is just add up each individual entry to get the resultant vector, for example,
What you will realise is that for scalars, we just have to add the numbers together. Vectors however, aren't that much harder: we just have to add the various entries up together. So what you will realise is that vectors are nothing but 2 or more scalars being added together at the same time.
Now, since the world is pretty much in 3 dimensions, how do we extend this to reality? Well, if you can imagine your x and y-axis being drawn on a flat piece of paper sitting on your table, just picture a new z-axis sticking directly out of the paper, pointing at you. Vectors will now have three entries, not two, but the principles underlying vector addition and subtraction remain the same.
Vectors are basically a mathematical tool for exploring quantities that are directional. You can not only add or subtract vectors from each other when required, you may also multiply them! Please go ahead and check out the dot product and the cross product. For people like CoffeeCoke who has been bullied into studying higher level physics, this is a must!! The vector is one of the most important mathematical tools used in physics!

Monday, June 25, 2007

This Blog is Back Online!

School re-opens today. Haha. Hope you guys had a good day in school, but, in other news, this blog is back! Let's begin with something light.

I'm not sure if you've ever wondered how physics is organised: how is it broken down into different categories, and how is physics taught in general? Actually, from the lessons you've had so far, you should probably have a rather good idea of how some of this organisation is done. You have encountered one important branch of physics already: optics, or the study of light.

Actually, the physics that you will be learning for the most part is known as classical physics, or physics that was thought to be correct up till the start of the 20th century. Yes, I know it can be quite upsetting to hear that the physics that you are learning now is "incorrect" (actually, a sizeable portion of it still stands, but a significant number of important facts that you are taught are in fact either outrightly incorrect, or are approximations of something more correct), but what is taught gives correct results in every day experiments.

Classical physics can be divided into three really gigantic groups: classical mechanics, electromagnetism and thermodynamics.

As an indication of how gigantic and complex classical mechanics is, the things that you are learning now constitute what is known as Newtonian mechanics: so all the kinematics stuff, Newton's laws, energy work and power etc. fall under Newtonian mechanics. Mechanics is a study of motion, whether it is the motion of a point particle (particle mechanics), a ball (mechanics of many-particle systems), a rotating disk (rotational mechanics), two galaxies colliding (celestial mechanics), a sound wave or air over an aerofoil (fluid mechanics).

In itself Newtonian mechanics is riddled with complexities, but, can you believe it, mechanics has been reformulated twice into Lagrangian mechanics and Hamiltonian mechanics. These are basically different versions of mechanics that can be used in more rigorous and complex ways, that are fundamentally the same as what Newtonian mechanics says.

Electromagnetism is the study of electricity, magnetism and their combined phenomena, e.g. electromagnetic waves. Optics actually falls under this category, but at our level, we treat it like a typical wave. Electromagnetism is essentially a study of charges and the kind of effects associated with stationary (electrostatics) and moving (electrodynamics) charges. Charged particles also produce electric and magnetic fields, and the study of the interaction of these two fields with each other is an integral part of electromagnetism.

Thermodynamics is a study of how large systems of particles (with too many particles to account for one by one: think of how many particles there are in one mole!) respond to changes in their surroundings. This branch of physics deals with heat, temperature, pressure and volume, and is an attempt to describe a huge and complicated system with each an every particle in the system behaving differently with simple laws. An important mathematical tool here that is not commonly found in the other two classical disciplines is statistics. In statistical mechanics (a very closely related field to thermodynamics) is the physics of linking microscopic properties of the atoms and molecules, like energy and velocity of each particle, to macroscopic properties of the whole system, like temperature, pressure and volume.

These are the three main branches of classical physics: you will find that Sec 3 will deal mostly with mechanics and optics, while Sec 4 will deal with thermodynamics and electromagnetism, plus a little bit of modern physics.

So if classical physics is "classical" and rather out of date, what is modern physics? And how is it organised? Curiously, you'll find that topics in classical physics tend to overlap each other, even at the fundamental level, but modern physics can be clearly divided into two: Relativity, the theory that deals with everything gigantic, and quantum mechanics, which deals with the minute. Of course, they do overlap, resulting in (dear God) relativistic quantum mechanics, which I don't pretend to understand in the least bit, but there is at this moment some conflict of interest going on between quantum mechanics and relativity. You will, of course, deal with a little bit of quantum mechanics in Sec 4, but don't be afraid, it's really simple stuff: just an introduction to radioactivity.

If you do a little bit of reading about classical physics, you will realise something interesting about each of the three categories: they have a set of laws governing each, and interestingly, there are four most most fundamental laws for each of the three main categories (they are: Newton's three laws and his law of gravitation for mechanics, Maxwell's Equations of Electromagnetism (there are four) for electromagnetism, and the Laws of Thermodynamics (from zeroth law to third law)). Don't think that they are distinct and separate: they overlap on many occasions, and indeed, their unification is one of the driving forces behind the advancement of physics as a science.

Indeed, if there's one thing you should realise, now that you know the categorisation and all, it is that each category is governed essentially by a few very simple laws that can give rise to many results. Please, do check out the fundamental laws that I've listed out for you earlier, and then appreciate this fact that I've quoted from a book as you study this science. This is why I constantly tell you guys there's no need to study for physics!

"Applying the laws of physics can give rise to challenging problems whose solutions call for clever insight and mathematical agility. The challenge of problem solving is what gives physics some of its intellectual interest and also its reputation as a difficult subject. But if you approach this course thinking that physics presents you with numerous difficult things to learn, you're missing the point. Because it is so fundamental, physics is inherently simple. There are only a few basic laws to learn; if you really understand those laws, you can apply them in a wide variety of situations. We wrote this book in a spirit that emphasises the underlying simplicity of physics by reminding you how diverse examples are really manifestations of the same underlying physical laws. You should come to understand the basic laws thoroughly so you can apply them confidently in new situations. As you read the text and work the problems, remember the simplicity of the underlying physical principles. Ask yourself how each problem you approach is really similar to other problems and to the text examples. And you will find that similarity, because the many problems and examples really do involve only a few underlying laws. So physics is simple - challenging, too - but with an underlying simplicity that reflects the scope and power of this fundamental science." - Physics For Scientists and Engineers, Wolfson & Pasachoff.

Friday, May 25, 2007

The Last Days

There are times when you wake up and feel that you can do nothing right, and you just want to return to your sleep and dream away the rest of the day, without having to think and make the right decisions all the time, just the way we like it. There are days when you wake up and you ask yourself what's the point of getting up and subjecting yourself to the same old things again and again.

But for the past 6 months, despite the incredibly early get up timing of 5.45 am (Okay. I know what you are going to say. You do that everyday, and you've been doing that for the past (insert double digit) years. But no other job I could have picked requires this!) after shaking off the initial lethargy, there's a sense of purpose to the day, and more than that, a sense of meaning in life that is very hard to come by.

I try my best every lesson to interest, inspire and at the very least help you learn the bare essentials in a subject that you may not enjoy thoroughly, but still have to take anyway. I tried, and many times I think I failed. I looked at other physics teachers and felt pressure looming over my head, as I watched them plan flawless lessons and whip out incredible applets and other resources for use in the classroom, and sometimes you feel so emotionally drained: because you try your best, and you don't know whether that is enough. You feel for the subject, you know exactly what the concepts you are trying to get across are, but you can't convey it in that particular way that everyone instantly gets.

This has got to be the most frustrating and life-consuming job ever: there was never a lesson that I walked out of without feeling drained and tired. There were a few days when I would be in the physics lab teaching pin-sighting method, and then I'll trot down to the chemistry lab to teach how to test gases, and then back to the physics lab again.

But at the end of everything, what keeps a teacher going is the students. And strangely, despite all the rather frustrating times, and scolding (though I'm sure most of it bounced of you, though I mean it all the time) and marking scripts with god-awful mistakes that make you feel like quitting the job because obviously you have got absolutely nothing across, I enjoyed myself in all the classes, each with their own weird concoction of terrors. Ha. Just kidding. It was fun, and, for lack of a better way of putting it, you guys were entertaining, and in many cases surprising both on an intellectual level and maturity.

In some ways, we are tied together. There is something inexplicably strong about the bonds formed in the process of learning. I hope you'll remember something in this 6 months, and hopefully it's something fundamental, although I'm being sickeningly self-righteous here, believing what I try to teach is the right thing to learn. But every teacher inherently thinks he's correct. So indulge me.

Good luck to all of you guys, avid (ha, avid) readers of this blog. I wish you the best of luck with your time here in RI, because I swear this place is the place that will define who you are in the future. Don't give your new physics teacher a torrid time! Though I trust that he will do a better job than me. I seriously think you all are getting short-changed in some aspects. Other teachers seriously have some cool stuff up their sleeves.

Don't worry, I'll keep updating this place to follow your syllabus and all: I can roughly remember what the next few chapters are. As my English teacher (Mrs. Selvan) emailed me when I sent her my farewell email to the RI staff:

"This is not a farewell. It's a till then.
So till then!
(I don't care, our paths must cross again!)"

Till then!

Saturday, May 19, 2007

Relativity! Galilean Relativity, that is.

I'm sure you are getting a hang of how important a physicist Galileo is. Recently we had a very splendid argument in 3F about whether a ball that is dropped by a person that is walking will fall in front of, behind or at exactly the same spot at which the ball would have dropped if the person were completely stationary with respect to the person.

Actually the answer should be immediately apparent to everyone if you just imagine yourself playing basketball: how would it be possible to dribble if everytime you let go of the ball the ball falls somewhat in front of you, or somewhat behind you? Another thing to try: place a wastepaper basket in front of you and try to drop a paper ball into it as you walk by the basket. You will find that you have to let go of the paper ball somewhat before you arrive next to the basket for the ball to go in. This is because you must time the paper ball to arrive at the paper basket at the same time as when your feet arrive at the basket.

Returning to the dropping ball experiment described earlier: note that whether the person is stationary or moving with some constant velocity (this is important: I will come to it later) the ball still drops next to the person's feet. Nothing impressive here, but as usual, there's a physics twist to it.

Imagine now that you are in a very very very dark room that is extremely quiet, but the ball is glow in the dark. You drop the ball. Where will it land? Now the answer is, obviously, that it will land at your feet. But notice something: I've given you absolutely no details as to what speed you are moving inside the room, because it doesn't matter at all!

So when you are within the room, say you were drugged and placed in the room by a sinister physicist, and then ordered to bounce the ball, you would have no way of telling whether you are stationary within the room, or riding on one gigantic travelator travelling at constant velocity.

Why am I so insistent on constant speed? Because even if you were severely drugged, you would immediately know if you were not travelling at constant velocity. You would feel your stomach lurch if you were suddenly heading downwards, or your feet suddenly feeling heavier if you were to rush upwards, just like in an elevator. You would be thrown to your left if you were suddenly turned to the right, just like in a car. So you can be sure that you are moving if you were not travelling at constant velocity. But if you just happened to be travelling at constant velocity, no matter how you bounced the ball, you would never be able to tell whether you were moving or were truly stationary.

If you can remember your last trip on a plane, imagine that few hours after take off and before the landing preparation while you were getting from one place to another: was there any real way to tell that you were moving? For all you know, the plane could have been stationary! You can walk with the usual ease of walking on solid ground, the stewardesses can serve their food and drinks without worrying about spillages and trolleys rolling away.

Another thing to think about: do you feel like you are moving now? Well you are! Because the Earth is moving at a constant speed of 30 km per second around the Sun!

So why don't we feel this motion of the Earth, on the plane, or when we're dropping glow-in-the-dark balls on a travelator that may or may not exist in a dark room because some weird-ass physicist told us to? The answer is in a property of all motion: Galilean Relativity.

Galilean Relativity is not hard to understand, unlike its more famous counterpart, Einstein's Theory of Relativity. Basically, it states there is something somewhere in this universe that is truly stationary: whatever it is, wherever it is, we aren't interested, but all of the laws of physics that we know, like Newton's Laws of Motion, would apply at that place. Galilean Relativity then goes on to state that if a physicist were to watch any object, say a car travelling at constant velocity from that truly stationary place, then the laws of physics would apply in the car as well. To phrase it in another way, we will never be able to tell the difference between constant velocity travel and stationary states.

Of course, the immediate question is, doesn't the laws of physics apply in ALL situations? Sadly, for the basic laws that we have encountered and will encounter in secondary school physics, they don't. Think about it: you are in a car that suddenly comes to a stop. What happens? well, of course, you are thrown forward, but a good follow-up question to ask is why? Normally physics teachers would explain it away as a phenomenon known as inertia: something that is moving will continue moving unless a force acts on it.

But imagine for a second you are back in that dark dark room standing on a travelator that you didn't know was there, and suddenly, the travelator stops, and you fall down. You would be totally caught by surprise, and the immediate thought that would come into the mind of any self-respecting physicist or science student doing the test would be that a force made him fall down. That is what Newton's first law dictates: any kind of change in motion must have been caused by a force.

However, only the sadistic physicist planning the experiment would know the truth: it was the travelator that came to a halt. So the truth was there was no force! You can see from here that an observer in the dark dark room who can't see what's going on would make a wrong prediction, because the laws of physics does not hold when the travelator doesn't move with constant velocity.

So this is why you can walk around with ease on an airplane, and you won't feel yourself moving with the Earth, because both the airplane and the Earth are travelling with constant velocity, which immediate ensures that the laws of physics will apply perfectly to us, and we will not be able to tell the difference between constant velocity travel and stationary states.

Of course, there is a crucial problem at work here: note that Galilean relativity requires that there be some truly special place in the universe which is truly and exactly stationary. Traditionally in solving physics problems we usually take the Earth as truly and exactly stationary, which of course is not true. Perhaps the Sun then? Unfortunately it is also performing an orbit around the Milky Way, which in itself is performing an orbit around our local galaxy cluster etc. etc. ...

Philosophically, before and during Galileo's time, it was commonly assumed that the Earth was that special place, being a place of God's creation, but since that time, physicists generally believe that there is no special place where the laws of physics truly hold, about which everything moves and revolves at their true speeds. But Galilean relativity requires just such a point.

Today we know that Galilean relativity is merely a rough approximation fo the theory of relativity proposed by Albert Einstein early last century, but the full argument is rather involved, but generally boils down to a belief by scientists that there is no special point at which the laws of physics will hold. Today, Einstein's relativity believes that the laws of physics holds no matter how we move: a conclusion that has far reaching consequences that have been briefly expounded. Please, do find out more for yourselves.

Friday, May 11, 2007

Doppler

The Doppler effect is an effect that we encounter in our lives rather often: When an ambulance or any vehicle with a siren approaches us, we can hear the pitch of the sound (i.e. the frequency of the sound) increase as the vehicle approaches, reaching a peak frequency when it is directly beside us, and then taking a swift plunge to a lower frequency when it recedes from us.

Let's have a look at what's going on. So far in class we have only been looking at stationary sources, like in a ripple tank. The source of the wave, i.e. the dropper or the damper in the ripple tank does not move with respect to the wave. It just taps the water surface, that's all.

The Doppler effect comes into play when the source begins to move. Here's a diagram to give a quick illustration:


You can see very clearly the bright spot is the where the source of the wave is, and the source is moving towards the left. As the source moves, you can see how each wave produced is centred slightly to the left, and so the wavelength of the waves on the left hand side is comparatively smaller to the wavelength of the waves on the right.

If you happen to be standing somewhere on the left, you would hear (if the waves were sound waves) a shorter wavelength sound, i.e. higher frequency sound, since the speed of sound is rather fixed in air. As the source passes you, and you suddenly end up to the right of the source, the frequency takes a plunge. This explains the ambulance approaching and receding you perfectly.

Now what is so splendid about this? A quick flick of the mathematics wand will give you some rather involved equations that in principle are quite simple, but in reality some pretty tedious stuff. Anyway, the cool thing about this effect is that it leads to some wonderful applications, such as in the radar.

As you probably can remember (or should remember), radar works by shining radio waves in all directions, and waiting for the reflected radio waves to come back to produce a blip on the screen. As the radio waves arrive at the airplane, they reflect off the surface of the moving airplane, which gives the effect of a moving source. The Doppler effect in this case will actually increase the frequency of the approaching aircraft, and a quick calculation (done preferably by the computer) gives us the speed of the airplane. This procedure is also used by the traffic police to determine the speeds of vehicles.

This is also how missiles decide when you to explode when they are in midair. Contrary to popular belief, missiles do not go off when they come into contact with the plane. Instead, they usually have a radar of their own built into the missile. As the missile approaches the plane, the frequency increases, and then suddenly drops when the plane starts to recede. This is one of the signals for the missile to detonate itself.

A common occurrence when jet planes fly past is the sonic boom. This is the Doppler effect at its extreme: the source of the wave is moving at pretty much the same speed or faster than the wave itself:

You can see that the waves are unable to move away from the source, because the source is travelling at the same speed as the waves being produced, and by the time the waves reach the observer, all the waves will simultaneously arrive and be heard, which would probably make for a rather loud sound.

Finally, I've been talking so much about sound, sound and sound, but the Doppler effect actually applies across the board to all forms of waves, including light waves.

One of the most important discoveries made in the 20th century was that galaxies and other distant objects in our universe were actually moving away from the Earth. You have to realise that before the 20th century at the advent of advanced astronomy techniques, many people held a firm belief in the static nature of the universe. Scientists thought that the universe was eternal and never changing, until a few brilliant ideas (see Olber's Paradox) and the Doppler effect observed in light helped us realise that the universe was expanding.

Using spectroscopic techniques, scientists back then were able to deduce what elements were present in gas clouds in space. When the light arriving from these gas clouds were dispersed by a prism (split up into the rainbow colours) astronomers were able to see certain missing colours, which corresponded to certain elements.

They soon realised that many of the lines did not appear where they were supposed to, but rather they all seemed to be squashed towards the red side of the spectrum:

This meant that when the light arrived on Earth, the wavelengths were longer than expected. If you observe the Doppler effect carefully, this would mean that the object that was being studied was moving away from the Earth. Astronomers called this phenomenon the redshift, because the spectral lines were all shifted towards the red side of the spectrum.

What was even more incredible was the fact that this redshift became more pronounced the further away astronomers looked, which suggested that the further away an object was, the faster it was moving away from us! An excellent astronomer named Edwin Hubble (of Hubble Telescope fame) was the discoverer of the law in 1912 (known as Hubble's Law) that the speed v at which something in our universe was receding from the Earth, is directly proportional to the distance d from the Earth:


Where H0 is the Hubble's constant. Let's consider a galaxy that is a distance D from us right now. Here's something amazing: since we believe that everything began with the Big Bang which basically says that everything in our universe started from a single point, the distance D is very roughly the distance travelled by the galaxy since the time when the galaxy and the Earth were still together at the start of the Big Bang. The velocity of the galaxy, v, let's assume to be constant. If we take D divided by v, we should get the time taken for the galaxy to go from the Big Bang to where it is now, i.e. we would roughly get the age of the universe!

So from the equation, t = D/v = 1/(H0). From astronomical observations, we do have a rather good idea of the value of H0, and it turns out that our estimate of the age of the universe from this method is about 15 billion years old.

Incredible? I hope you think so!

Thursday, May 3, 2007

Sonoluminescence and Other Creatures

As I've mentioned a few times in class, physics is an open science. The number of unsolved problems in physics are quite stupendous, and some of them seem to be questions that we really ought to know about.

One really fine example of how limited our knowledge is in physics is the fact that we are currently unable to provide a complete understanding of how fluids flow. Physicists are unable to supply a law that governs the behaviour of turbulence (which we sometimes encounter during airplane trips). The extremely famous physicist, Werner Heisenberg, who pioneered quantum physics and gave us the infamous Heisenberg Uncertainty Principle once quipped, "when I meet God, I am going to ask him two questions: Why relativity? and why turbulence? I really believe he will have an answer for the first."

Many of the open questions in physics are rather esoteric, but some of them can be rather spectacular. You can check out a list of unsolved problems in physics at Wikipedia, and you can scroll down to click a few of the links at the bottom which give more insight into the nature of these problems.

I'd just like to point out something weird and intriguing in that whole list: sonoluminescence. Sonoluminescence arises when a liquid is excited by ultrasound. Bubbles are formed within the liquid in the presence of the ultrasound (remember that sound waves are pressure waves? They create differences in pressure in the medium, which in this case is the liquid. The rarefactions are areas of low pressure, and in special cases a bubble can form at a rarefaction, which will pop when it becomes a compression), and when they burst, somehow or another they release light.

This process is really very strange if you think about it: why would a collapsing bubble give off light? It becomes even weirder when some scientists actually believed that the temperatures within the bubble actually reached as high as one megakelvin, or 1,000,000 K.

Sonoluminescence is still an unsolved problem today, but you can check out some of the mechanisms that have been proposed by physicists in the Wikipedia entry. In the mean time, don't forget that although what you are studying seems to be written in stone, physics as a science is rather lost at the moment. Everything seems to make sense at some level, but each individual piece is difficult to string together to give a definite picture of the universe, which is more or less the ultimate goal of physics. I hope you can click some of the links available in this post here. Most of the open questions are explained in nice and easy terms.

Happy reading!

Thursday, April 26, 2007

The Greenhouse Effect

In conjunction with the recently concluded Earth Week, let's take a physics look at the Greenhouse Effect.

The Greenhouse Effect doesn't only occur on Earth: it is a phenomenon that is present on both Venus and Mars. It is a purely natural effect first studied in detail by a scientist named Svante Arrhenius. If you remember your chemistry, this fellow also provided us with our theory of acids and bases: acids are substances that donate H+ ions, bases are substances that donate OH- ions. Sounds familiar?

Certain kinds of gas molecules, such as water, carbon dioxide and methane, are very adept at absorbing infrared radiation, more so than other kinds of gas molecules. Simply speaking, these gas molecules are able to absorb infrared radiation and start vibrating, whereas others like oxygen and nitrogen, dislike infrared radiation, because they don't really have any way of properly dealing with the energy if they absorb it.

In a planet completely devoid of an atmosphere, the ground absorbs solar radiation and heats up. Remember I mentioned about how every object gives off infrared radiation? At the same time that the ground is absorbing radiation from the sun, the ground is also giving off its own infrared radiation. As the ground gets hotter and hotter, the amount of infrared radiation it emits gets more and more too.

Eventually, it will reach a stage where everything is nice and balanced: the amount of solar radiation absorbed is equal to the amount of infrared radiation emitted, and the ground will remain at a constant temperature thereafter. It turns out that without the atmosphere, the surface temperature of the Earth would be 25 degrees celsius lesser than what it is now. Think of the temperature of Singapore being 0 degrees celsius!!

The thing that comes to our rescue (indeed, the thing that allows life to thrive on Earth) is our atmosphere. Our atmosphere actually acts like a blanket, with the greenhouse gases like water, carbon dioxide and methane absorbing the infrared radiation emitted by the ground that would have gone into outer space, and then re-emits it back to the ground.

This means that on top of the solar energy incident on the Earth, there is also a significant amount of energy being emitted by the atmosphere back to the ground, causing the temperature of the ground to rise to a much higher temperature than expected.

Of course you would already know that the level of carbon dioxide and water is controlled by the carbon cycle and the water cycle respectively. The gist of it is that carbon dioxide is mainly gotten rid of by photosynthesis, and water basically gets rid of itself by precipitation. These cycles maintain a fine balance, and are now unable to keep up with the growing levels of carbon dioxide being emitted by burning fossil fuels.

The increase in greenhouse gases necessarily means more radiation being returned to the ground, which means that the ground gets heated up more, leading to higher global temperatures, or global warming.

This I guess you should already know, but something of greater interest is what we know as the runaway greenhouse effect. The atmosphere of Venus comprises mainly carbon dioxide, which leads to a huge greenhouse effect on the planet. The surface temperatures are estimated to be about 500 degrees celsius (high enough to melt lead and zinc), which is higher than that of Mercury's surface temperature despite being so much further away from the Sun. The greenhouse effect also causes the day and night temperature of Venus to be of no significant difference, despite the fact that a Venusian night lasts about 116.75 Earth days, giving it a long time facing away from the Sun.

It has been theorised that Venus once had water on the surface, much like Earth does today. But water is a greenhouse gas, and somehow Venus was hot enough such that quite a bit of the water on the surface evaporated over time. Since water accelerates the greenhouse effect, the temperature of the surface of Venus must have risen, which caused more water to evaporate, which in turn causes the temperature to rise further, and... you get the point. This is known as the runaway greenhouse effect, and the end result is a planet that would immediately incinerate most living organisms.

Some trivia about the greenhouse effect: one of the most important contributions to the greenhouse effect on Earth is the methane gas, which is given out by cows when they fart. So, it's not all about carbon dioxide from fossil fuels!

Another thing: the name greenhouse effect is quite a misnomer. Greenhouses work by allowing light to pass through the glass, heating the ground. The ground then heats the air above it, which rises up, but is unable to escape from the greenhouse because of the roof, of course. It's pretty much like boiling water while keeping the lid on.

Another fine example of scientists naming things poorly.

Friday, April 20, 2007

The Elusive Photon

I'm sure most of you have heard some of your classmates mention "photons" in class, or at least people mentioning how light is also a particle, and also see me ignoring them.

So here I will address the question: what is a photon?

First of all, the reason why we know so clearly that light is a wave arises from this wonderful ingenious experiment (I might have mentioned this in class) known as the Young's Double Slit Experiment. I won't go into the details of the experiment, but you can check it out yourself here. But what the experiment demonstrates very clearly is that light undergoes a process known as (you can check it out too)interference. This process is one very special phenomenon associated with waves, much like reflection and refraction are special phenomenon associated with waves.

So a whole generation of physicists studied electromagnetic waves and made many conclusions about how electromagnetic waves should behave, and sure enough almost all phenomena associated with light could be easily explained by the fact that light is a wave.

One would think that the answer is immediately apparent from the experiments: light is a wave, and no more arguments. But there were a set of three physics problems at the beginning of the 20th century that were thought to be minor issues at that time. They defied the physicists' attempts at explanation.

One of them was known affectionately (or rather unaffectionately) as the ultraviolet catastrophe. A simplified version of the problem goes like this: when light was assumed to be an electromagnetic wave, it was predicted that if a person stands in front of a stove, he would be exposed to an infinite amount of radiation, and would therefore instantly burn to death, which is of course not true! Such a gross error definitely needed prompt correction.

The other two problems are the famous Photoelectric Effect and the less famous Compton Effect . In both cases, simply ignore the math.

All three of the problems had one thing in common: they could all be solved if we allow ourselves to envision light existing as billiard ball-like entities. Don't concern yourself with the technical details here, unless you are willing to crack your heads, but these three problems were all resolved by assuming that light existed in little chunks at a time called quanta (hence quantum physics). You can think of it as the light ray being composed of billions and billions of microscopic particles called photons.

What's the difficulty here? Well, fundamentally particles and waves are really different. For one thing, in a wave, the energy is continuous throughout the wave. When you shine light on a surface, the energy arriving from the light will gradually accumulate on the surface, and heat it up. But if you think of light as little billiard balls, then energy transfer comes in short "spurts": when a billiard ball hits the surface, the energy is immediate transferred to the surface, so there is a sudden "spike" in the total energy of the surface.

What is even more troubling is the fact that waves undergo refraction and interference, whereas particles absolutely do not. How can we explain the wave-like properties of light if we treat them as particles? On the other hand, how do we explain the three problems without resorting to particle-like properties?

The problem gets even worse!

After awhile, physicists were actually able to detect wave-like properties in electrons, which we definitely thought were particles. They were able to perform Young's double slit experiments on electrons, and showed that electrons can behave as waves too! How can that be?

A French prince named Louis de Broglie came up with a most proposterous suggestion, that turned out to be right. He suggested that everything in the world had two "faces": wave-like behaviour, and particle-like behaviour. And he really meant everything.

So even you can behave like a wave, undergo interference and all, but because of your incredibly great mass (in comparison to the electron, for example) the wave-like properties are not detectable, and the particle-like properties dominate.

It may not sound sensible at all, but if you read more about this incredible theory and its history and how the ideas behind quantum physics first developed, you would see that we were forced into a corner, and were basically forced to concede something that makes little common sense, but yet was right, because that was what the experiments were saying.

How can something be both particle and a wave? The modern understanding, is seriously mind-blowing. I don't truly understand it on some level, but what I understand is this:

Before you observe anything, you will know absolutely nothing about that thing, and so you will never know whether something is a particle or a wave until you decide to look at it. Now, in order to find out if something is a wave or a particle, we must make an observation, which involves some kind of measurement. It turns out that depending on what we choose to measure and what the experiment set up is, you will get different conclusions! If you design an experiment to measure the wave-like properties of something, you will detect its wave-like properties. If you design an experiment to measure the particle-like properties of something, you will detect its particle-like properties!

Sounds silly right. But it is a lot more ingenious and subtle than it sounds. On top of that, it works pretty damn well too.

An excellent example of this idea would be the extremely famous Schrodinger's Cat . You can read it yourself here, or click on the external links: there are some that give a very quick introduction.

For those who are interested in pursuing this further, you can check out the book "Who's Afraid of Schrodinger's Cat?", or the comic-like book "Introduction to Quantum Physics": the series with introductions to a lot of different subjects in comic form.

Like in all things complicated in physics, skip the math. It can wait. You can still understand most of this stuff at a basic level if you read the right stuff!

The School Next Door

Okay, today let's take a break from physics. As you can imagine, I'm sitting here in the staff room staring at your practical worksheets and have decided not to mark them for the time being: the task is too daunting. And I'm sick of preparing for your lessons as well. Teachers need breaks too.

My time in RI was definitely the best period of my life, better than the time spent in RJC, and definitely definitely much better than time spent in NS. You will find that the friends that you make here in RI will last for a very long time, typically longer than the friends you will make in RJC. These are the people you grow up with, and the RI crowd tends to be a very weird crowd. JC is all about being "normal", being part of the mainstream culture, whereas in RI differences are very much more acceptable.

In JC you get to drop the subjects you dislike, although you must be very wary because many subjects morph into monsters. I took physics, chemistry, mathematics and further mathematics, and even though I did well, I still hate chemistry to this day. Chemistry and biology are two subjects that will rear their ugly heads over in JC, although I can see that the current Raffles Programme chemistry and biology are already gearing you towards JC style chem and bio.

Subjects like economics were very popular in my time, and is now incredibly popular, with virtually everyone taking it as their humanities, but I can't say too much about it, other than the fact that it does seem rather interesting if you are interested in all that.

As to physics and mathematics, there shouldn't be much problem. Physics is really quite manageable, because they can't push the standards up too high without bringing in higher level mathematics. Mathematics will be built on what you already know, plus some more stuff that are rather interesting, but really quite manageable. Further mathematics is problematic, but since they took that out, you don't really have to worry.

There used to be this thing called Special Papers in my time. I took two, in physics and mathematics. Nowadays, they call them H3 subjects, and from what I'm hearing many people are taking them in either math or physics. I'm not too clear on what this is really about, but special paper is just harder questions set to test your math and physics abilities in a more stringent manner, with physics incorporating some calculus (although almost every student manages to skip through this part by not answering those questions).

You will encounter really irritating subjects, like General Paper, the most useless and pretentious thing in the world. I still don't understand how they expect any of us to write anything sensible for their essays, because very often the problems that they are discussing are so complex, and so multi-dimensional that it is impossible to say anything smart about it without some good hard evidence and at least 200 pages of explanation and report.

Project work is hell. Completely pointless. You'll see what I mean. Good luck for that.

School wise, you will have to start taking your own notes, which is what I've been telling you people in class during lectures. Some periods will be lectures, where you will go to the lecture theatre and listen to the tutor teach about the topic in the subject, while others will be designated tutorials, where you will go to classrooms and discuss the questions set in the tutorials with your tutor. Nothing very new here, except that the stuff that you learn isn't going to come from the classroom, but from the lecture theatre. What you learn during the lecture is entirely up to you, and sometimes even if you don't turn up (which is illegal, of course, but I don't pretend to being an angelic student back then) they don't really care.

To tell you honestly, I used to skip school, skip classes, especially S-paper classes. But one thing that I never skipped was chemistry, because that was my worst subject. I was, and still am terrible at chemistry practical, and so those practical lessons I would definitely go. But physics and mathematics, especially S-paper, were really useless lessons.

In all, academically you are pretty much on your own. You have to learn the skills of studying by yourself to some extent, without the help of teachers to spoon-feed you notes and all, and definitely you will need to discipline yourself. You can choose not to do any of your tutorials, which is precisely what I did for physics, but at the end of the day you must get your grade.

Socially, JC is more geared towards being "cool". Although I know it sounds very silly, but that in fact is what it is. There are set behaviours and things to say that are considered normal by each class, and if you don't happen to be that particular thing, then you won't fit in. I didn't like my class, to tell you the truth, and I don't think they really liked me as well, but we were really different people. Thank god for RI friends and CCA.

Of course there's also the matter of there being girls in RJC, and well, okay, that's nice and all. But a word of advice, stay away from all those till after your NS, because I think no one is able to emotionally handle any relationship properly at that age, and under those circumstances. You know too little. I know it sounds kind of weird coming from me, and I still can remember how I resented the way other people say things like that, but once you get through NS, you will see what I mean.

Teachers, well, you won't be as close to them as you are to some of your teachers now. My class was very close to my RI teachers, and even up till now they still invite us over for Chinese New Year, or we have dinner together or something. When I came back to RI to teach after 5 years away from this school, all my old teachers could still remember my name, and other weird details associated with me. They are amazing. In RJC, you can't expect this level of closeness, because very often they spend too little time with you, and in that short window of time that they are talking to you, they are talking business.

Things are different over there, and I think some of you will adapt really well to the environment, while others will probably not. When it comes, don't be too sad if you're not "in", because there are many people out there who are like you. I think what is most frustrating about being in RJC is that people tend to forget the true purpose of being in school, which is to learn. I hope throughout the course of the time you are spending in RI you will appreciate how amazing knowledge in any field is, not only physics, and not forget that when you go to RJC, and not finish the JC course feeling uninspired and unsure of what you want to do with your life.

Because JC is the period of time that shapes your ambition and the future "you". Participate actively in CCAs (there were months when I never reached home before 8, and would lose every single Saturday to CCAs) and pursue your academics and everything else with passion. Don't get too caught up with the glamour and girls and fun associated with JC, all the socialising and going out and parties, because those are empty things in the end.

In the mean time, enjoy your time in RI. It probably doesn't get any better in the next 5 years!

Tuesday, April 17, 2007

Resonance

Quick review:
Frequency: number of oscillations per second.
Period: time taken for one oscillation.
Amplitude: maximum displacement from equilibrium position (rest position) of the particle or pendulum.

We've all heard urban legends of people being able to break glass, or more usually crystal by singing at high pitches. Some of you would probably also have seen this picture:




This is the famous Tacoma Narrows Bridge just before its collapse, which started tilting left and right on a particularly windy day. If you can find the video clips detailing the collapse, it's quite amazing.


All of these phenomena, and much more, can be attributed to the natural phenomenon of resonance. So what exactly is resonance?


Every object, when excited in some way, vibrates at a very special frequency known as its natural frequency. For example, in the lab experiment you just did with the pendulum, the pendulum has a natural frequency that can be calculated. Those of you who still haven't handed up you lab report, you can verify that this formula is true:


Where T is the period of the pendulum, pi is just pi, l is the length of the pendulum, and g is the acceleration due to gravity, which is 10 m/s^2.

So a pendulum when set into motion will automatically start moving at this period or frequency called its natural frequency: the frequency at which it oscillates when it is not disturbed by anything. Anything that is able to oscillate has a natural frequency, a kind of preferred frequency at which they like to oscillate.

This also happens when you blow air across a half-filled bottle to produce a sound: this sound has a certain frequency that is the natural frequency of the air inside the half-filled bottle. When you let your arms swing naturally by your sides, they should be swinging near the natural frequency. When you hit a drum, or a string, or when you produce a note on a wind instrument, you are making the skin, string or air vibrate at its natural frequency.

But of course, in a pendulum, I can try to force it to oscillate at a faster or slower frequency by trying to control the motion with my hand: I can grab the string above the pendulum bob and try to make it go faster or slower by moving my hand faster or slower than the natural frequency.

You can try this at home with a broom stick. Grab the top of the broom stick with one hand, and allow it to swing naturally, without you exerting any force. It will start oscillating at its natural frequency. Now, you can try to make it swing faster by applying a force through your hand. Observe what happens to the amplitude of the swing as you try to make it go faster. You should see the amplitude getting smaller and smaller, and the broom swinging less and less.

If you try to make it oscillate really really fast, by moving your hand back and forth really really quickly, you would actually see the broom not moving at all! This is because it has no time to react to your motion before the direction of motion suddenly changes, so it'll just sit there and watch you.

Now let it swing freely again, and then try to oscillate it really really slowly with your hand, and compare the amplitudes when it's swinging naturally, and when you are trying to oscillate it really really slowly. You'll probably notice a slight difference: when you try to interfere with the oscillation, sometimes you are going against the natural movement of the broom, and hence you are slowing it down and decreasing the amplitude.

It all sounds pretty confusing, but when you try it, it's quite apparent.

Now try to move your hand back and forth at the same frequency at which it is naturally swinging. I'm sure you'll see the difference: the amplitude of the swings will increase greatly, because your force is always going with the direction of motion, so you are making the oscillations greater.

So this is the lesson we learn: if we try to oscillate anything at its natural frequency, it will produce an oscillation with very large amplitude, that will increase with time if you continue to make it oscillate. This phenomenon, where you try to make something oscillate at its natural frequency to produce really big oscillations is known as resonance.

So here's how we break glass with our voices: we tap the glass to hear the sound it produces: this sound is the natural frequency of the glass. We then match our voices to that natural frequency (we just have to match the pitch of the note: what we hear as pitch is actually the differences in frequency) and try to oscillate the glass molecules at its natural frequency. Due to the phenomenon of resonance, the glass molecules will develop very high amplitude oscillations. If these waves are large enough to break the bonds between molecules (just like they were strong enough to break the bridge apart in my opening picture) then the glass will shatter!

I don't think it's easy to do, but at least it is possible in theory.

Thursday, April 12, 2007

Radio

In class I mentioned a little bit about radio waves. Here I will mention a little bit more.

If you could see all the radio waves around you, you would literally be blinded: not only are there all the radio signals coming from all the different TV and radio signals, there are also radio waves transmitted by the police, the army, the civil defence, the air force, commercial flights, GPS signals and much much more.

Each of these signals are the form of a typical transverse wave. The information is encoded in two ways: AM or FM. AM stands for amplitude modulation, which basically means the different information is encoded in the varying amplitudes of the wave. FM, frequency modulation, means that the changing frequencies of the wave can be interpreted and translated into real pieces of information.

A radio set typically consists of two things: a transmitter, and a receiver. Their names should tell you what they do: a transmitter takes some information, let's say something that is being said in the form of a sound wave, and turns it into a radio wave (either FM or AM). A receiver does the exact opposite. Both of these things make use of antennae to transmit or receive the info.

A transmitter is pretty easy to imagine: basically you do what would be the equivalent of turning a switch on and off again and again to produce an electric current wave. This electrical wave is then sent to an antenna, which is essentially just a straight wire, where the current moving in this antenna will be converted into radio waves.

Of course the electric current wave will have either its amplitude or its frequency changed over the course of time, which encodes information in the wave. So, when I receive the wave, the radio can produce the sound of music or some kind of information by observing the changes in amplitude or frequencies over time.

When radio waves arrive at an antenna, it will cause an electric current to flow within the antenna. This current will then flow to the receiver. There are thousands of radio waves arriving at the antenna, but the receiver can be adjusted electronically to filter all but one frequency. How? You can read up about the concept: it's known as resonance, which is also the process by which a singer is able to break a crystal glass, although in a very different way.

The current that is filtered through is then passed through a demodulator, a series of complicated electrical components that will break up what is encoded in the wave and then sends the appropriate electrical signals to the speakers. Along the way, the signals are amplified, so that we can hear the signal clearly.

This is of course much too simplistic. For more information, you can check out howstuffworks: http://electronics.howstuffworks.com/radio.htm, which gives a good explanation of radio, and a good explanation of many other things that you can check out for yourself.

Process

Acquiring knowledge is a difficult process. I understand that feeling. Even up till now after studying 4 years of official physics, together with some outside reading, I know virtually nothing, and that is really frustrating sometimes.

While teaching you guys physics, I too am learning physics, but unlike you I have a teacher that I can only turn to through e-mails, and can get to see only when I'm free, and currently, I'm not very free. He told me rather bluntly last meeting that what I was struggling to understand was the very basics of quantum mechanics, and even though I know he didn't mean it as a "you suck" kind of statement, it still hurt rather deeply.

I'm incredibly grateful for someone like him to take the time off to teach a complete beginner like me, but sometimes it really is awful sitting down next to him, watching him write things that you only have the slightest clue of, and not knowing whether to pretend to know, or admit that you know absolutely nothing. I know that feeling all too well.

I know what it feels like to flip through a textbook, or to see things that hardly make any sense. It's a sinking feeling, a feeling of being like a drop in an ocean, of being absolutely stupid and ignorant. Of being defeated, and tired.

And it's absolutely true too. We are too stupid to comprehend physics. We are too ignorant of the things that go on. Most things are too complicated for our small minds.

And guess what, I'm really glad you've realised. If you don't realise that you know absolutely nothing, then that is truly foolish behaviour. No single person is smart enough to be familiar with the entirety of anything.

Many people turn away from anything remotely to do with knowledge once they get to my age. Every single thing is eclipsed by the practical nature of the world: a need to be financially secure or wealthy, a need for social status and prestige, a need for power even. You don't remember what you liked in school, you don't remember what inspired you, you only see what lies ahead. No one has time to ponder the stars. Or music.

But just for one moment I hope you can see how much bigger than the self physics is. Not only physics, but any discipline you can name: mathematics, chemistry, biology, history, geography, literature, music, art etc., each discipline gigantic in itself. They are an attempt to describe something much bigger than yourself. Something that will still be here forever when you die and cease to exist. Something that trillions of people after you will come across again, and hopefully with each person that encounters it, it will become more complete.

So don't turn away because you can't understand. Turn towards it because it is truly greater than you, and the things that you take away from it will help you see the world around you in an entirely new light. Look beyond yourself, and hate the ignorance that you have. Cure it. We all know absolutely nothing, so it is not surprising that everything looks so difficult. But fight the ignorance. Think till you feel like you can't think anymore, question till you get all the answers, look for the answers till you just can't bear to search any more, and when you've finally understood, it's an amazing feeling. You'll feel like you know a dark secret that has been told only to you, and you'll realise how incredible the world around you is.

Tuesday, April 10, 2007

Doe a deer, a female deer.

To all readers, even if you are totally estranged from music, which is difficult to believe, you would have heard of the infamous Sound of Music song from which the title comes. So, even armed with as basic an idea as that (there are 7 basic notes that go into making a simple song, and these 7 notes make up what we call a scale), you can probably understand a good part of what's coming.


You can probably tell the difference between a poor singer and a good one, as they are singing to music being played. You don't really know how you can tell, (musically, you can say that the singer is out of tune, or perhaps if you can tell you would say he is singing "flat"i.e. too low, or singing "sharp" i.e. too high) but you can just tell when someone isn't singing well. There's an almost physiological response to poor singing, or to music that is simply out of tune, and likewise there is a feeling of pleasure when you hear an orchestra that is perfectly in tune with each other, or two singers singing in harmony.


Have you ever wondered why we respond to harmony? How do we know when a singer is singing well? How are people able to tell whether an orchestra is playing well, or if a certain instrument is in tune or not? The answer, and I must admit I never fully realised this until quite recently, lies in physics.


If you ask the nearest choir members, or person with an instrument, to play or sing a low "do" and a high "do" together (remember "that will bring us back to doe!", so after "ti" in the music scale, the next note is called "do" again.) you would hear an amazing thing: the two notes sound like one!


As a physicist, if you were able to dissect the voice box of the said choir members, or just simply observe the instruments in action, you would observe something very interesting about the sound produced: The frequencies of the sound would be in the ratio of 1:2.


A quick example for more music familiar people: A concert A has a frequency of 440 Hz. The A above that, A', would have a frequency of 880 Hz. Likewise for any two notes an octave apart. The higher note will be twice the frequency of the lower one.


Now, it turns out that any two sound waves that are produced that have a very nice ratio, for example 2:3 (you can do this by playing "do" and "so", "re" and "la", or "mi" and "ti" together, known as perfect fifths), or even 4:5 ("do" and "mi", "fa" and "la", "so" and "ti" known as major thirds) sound nice. If you don't believe me, you can go try it on any instrument, or simply sing it. On virtually any music that you can hear on the radio, every time you hear two singers singing, most of the harmonies will be in either perfect fifths or major thirds. (Of course this is outrageous to the musically very literate: what of minor thirds, minor and major sixths, and perfect fourths? Undeniably present. But don't forget, in music we can't simply have harmonies, that would make for very boring music. Music is made interesting by disharmonies that are then subsequently resolved into harmonies. So even notes that don't sound good together are useful!)


Okay.


Now for the more musically literate: This is what I'm actually talking about so far: the harmonic series:



Okay. So what is this diagram? Starting from the really low C on the left, if I double its frequency, I will hear the C one octave above it. If I triple the frequency, it would turn out to be the G third from the left, etc. So each multiple of the original frequency will produce a note that gets increasingly less harmonious with the note before it. Of course, 1 and 2 are extremely harmonious: the octave. 2 and 3 form the perfect fifth, 3 and 4 the perfect fourth, 4 and 5 the major third, 5 and 6 the minor third, 6 and 7 the major second, 11 and 12 the semitone. You can see for yourself that the harmonies get less and less pleasant. Dissonances like the tritone occur at 8 and 11 (terrible ratio, 8:11), and augmented fifth (8:13) etc. etc.

We can construct the harmonic series from any starting note of course. Let's try it from A (concert pitch)

A: 440 Hz

A': 880 Hz

E'': 1320 Hz

A'': 1760 Hz

C#''': 2200 Hz

E''': 2640 Hz

G''': 3080 Hz

Okay. Now let us do something here. If G''' is 3080 Hz, then G'' must be 1540 Hz, G' 770 Hz, and G itself 385 Hz. Let us construct a harmonic series from this G:

G: 385 Hz

G': 770 Hz

D'': 1155 Hz

If D'' is 1155 Hz, then D' must be 577.5 Hz, and D must be 288.75 Hz. Now, let us construct a harmonic series from this D:

D: 288.75 Hz

D': 577.5 Hz

A': 866.25 Hz

This means that the A constructed from this harmonic series is 433.125 Hz, a whole 7 Hz different from the original A at 440 Hz!!

What has happened? Long ago, musicians realised this problem: that starting a harmonic series from one note will form notes that are unique to that harmonic series. If you wish to change keys, say from A Major to D Major, if we insist on using the notes formed in the harmonic series of A to play in D Major, the sound would be horribly off key. Modulation was rendered impossible (not to mention to problems this caused for the french horn and other brass instruments: a separate topic, but well worth exploring!), and without modulation, music got boring fast.

Until a certain Johanne Sebastian Bach came into the picture. He had a wonderful solution to this problem: why not make every single note equally out of tune in such a way that ALL keys will sound equally out of tune? Of course, this sounds disagreeable: if it's out of tune, wouldn't we know it's out of tune and hate the music being played?

Turns out that isn't true. He wrote the Well-Tempered Clavier, a set of 24 preludes and fugues in all 24 keys to show that it was possible to do such a thing. Initially, I would think people were rather uncomfortable with the poor tuning, but today we are so used to his system that what we think is in tune today, would probably have been considered badly out of tune in his time!