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!