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.