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.