Before we dive any further into Newton, forces and other things, lets go a bit into the fantastic world of vectors.
first, a definition. Vectors are defined as things that possess a magnitude as well as direction, in contrast to lowly scalars, which only get magnitude.
"So what is the big deal?", I am sure you ask. Well, it does not seem like much, but let me try to explain it like this.
If you wanted to lift a rock, pushing down on it would not lift it. Instead , you would have to probably pull the rock up, and there you go, one rock lifted. Here, the direction of the force is just as important as the magnitude. So, we can then say that force is a vector. Now how about travelling. Let us say that I have taken my car, driven down from Bangalore all the way to Chennai, and back again. So I have covered 600 (or so) kilometres doing this. My tyres and my petrol tank make that clear. But the fact of the matter is that I have not really moved 600 kilometres from my home...I am back there, lounging. So while the distance travelled might have been 600km, my displacement from my initial condition was zero.
So for now, just remember that a vector is rather more complicated than a scalar, and although the vector may be very strong (has high magnitude), it may not do much to you (beacuase the direction of th high magnitude is low.
We wont even go into tensors, and where they fit into the situation. We can worry about tensors we we come face to face with one!
Friday, July 6, 2007
Zeno's Paradox
I must say, I did intend to complete a discussion of Accelaration and Newtons laws today, but in trying to figure out the principle of differentiation, I came across this pretty interesting one which is a sort of introduction to limits and infinity (and beyond too!).
I promise that this will help while we discuss differentiation.
This one is about the greeks. There was this philosopher named Zeno, who was generally regarded as hot stuff in the philosophical circles of Greece. What did he come up with? Well, mainly he was of the school of thought that said that all motion is impossible, or some such thing. In order to prove it, he came up with these paradoxes, which at first sight seem terribly convincing. For example, there is his famous one about Achilles and the tortoise. (For those who came in late, Achilles was a mighty greek hero. He was a great runner. And showed a clean pair of heels to all his competitors. And was shot in the heel, causing his death. For the unabridged version, read the Illiad, by Homer). Below is a quote from the Wiki (always useful).
In the paradox of Achilles and the tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is so fast a runner, Achilles graciously allows the tortoise a head start of a hundred feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox.
Now, try to spot the hole in that logic. Its actually quite tough. But I shall spare you, the reader from thinking this through. The problem with Zeno's logic is this. He assumes that for there are an infinite number of steps for Achilles to take to overtake the tortoise (Infinitely small steps). And since it takes a finite time to complete each step, it will take infinite time to complete infinite steps. In functions, this is called a one-to-one correspondence. But this is not necessarily true. Let us quantify the tortoise and Achilles problem like this.
In the case of Achilles and the tortoise, suppose that the tortoise runs at a constant speed of v metres per second (ms-1) and gets a head start of distance d metres (m), and that Achilles runs at constant speed xv ms-1 with x > 1. It takes Achilles time d/xv seconds (s) to reach the point where the tortoise started, at which time the tortoise has travelled d/x m. After further time d/x2v s, Achilles has another d/x m, and so on. Thus, the time taken for Achilles to catch up is
The above value is a finite number, which means that Achilles will eventually catch up with the tortoise. Notice, that without a definition that implies that distance and time can be related, this problem cannot be solved at all.
p.s: Almost all the material is lifted from the Wiki. The link is given below for those who wish to have a full explanation (and to get thoroughly confused...I was).
here
I promise that this will help while we discuss differentiation.
This one is about the greeks. There was this philosopher named Zeno, who was generally regarded as hot stuff in the philosophical circles of Greece. What did he come up with? Well, mainly he was of the school of thought that said that all motion is impossible, or some such thing. In order to prove it, he came up with these paradoxes, which at first sight seem terribly convincing. For example, there is his famous one about Achilles and the tortoise. (For those who came in late, Achilles was a mighty greek hero. He was a great runner. And showed a clean pair of heels to all his competitors. And was shot in the heel, causing his death. For the unabridged version, read the Illiad, by Homer). Below is a quote from the Wiki (always useful).
In the paradox of Achilles and the tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is so fast a runner, Achilles graciously allows the tortoise a head start of a hundred feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox.
Now, try to spot the hole in that logic. Its actually quite tough. But I shall spare you, the reader from thinking this through. The problem with Zeno's logic is this. He assumes that for there are an infinite number of steps for Achilles to take to overtake the tortoise (Infinitely small steps). And since it takes a finite time to complete each step, it will take infinite time to complete infinite steps. In functions, this is called a one-to-one correspondence. But this is not necessarily true. Let us quantify the tortoise and Achilles problem like this.
In the case of Achilles and the tortoise, suppose that the tortoise runs at a constant speed of v metres per second (ms-1) and gets a head start of distance d metres (m), and that Achilles runs at constant speed xv ms-1 with x > 1. It takes Achilles time d/xv seconds (s) to reach the point where the tortoise started, at which time the tortoise has travelled d/x m. After further time d/x2v s, Achilles has another d/x m, and so on. Thus, the time taken for Achilles to catch up is
The above value is a finite number, which means that Achilles will eventually catch up with the tortoise. Notice, that without a definition that implies that distance and time can be related, this problem cannot be solved at all.
p.s: Almost all the material is lifted from the Wiki. The link is given below for those who wish to have a full explanation (and to get thoroughly confused...I was).
here
Thursday, June 14, 2007
An Introduction to Science
Today, I decided to begin a new (and hopefully regular) section. As someone who has always been interested in Physics, I thought that my inimitable qualifications (ie: none at all) enabled me to give some gyan on things that I thought interesting.
Well, that is enough of an introduction. Now on to what I think is the fundamental principle of mechanics. Forces.
The first question to ask is, "What on earth is a force?" This is what my 8th standard Physics Book had to say on the subject.
"A force is anything that changes, or tends to change the state of rest or of uniform motion of a body in a straight line." Hmm...fairly categorical that. But what does that statement actually say?
By using the Bala (tm) method of analysis (ie: reading the sentence), we can break it down to its bare meaning which every one of us has seen or felt. To move something you have to push!
Now consider the meaning of the statement in more detail. Newton (or whoever) says that for something to change its position, some effort (force) has to be applied. Why is this so? Now we come to the concept of Inertia.
"Inertia is a property of a body which resists its change from a state of rest or of uniform motion in a straight line" (Yes, you guessed it. Its from the same textbook!)
So, coming back to English, we can define Inertia as the tendency of a body to resist a change in its state, whatever it may be. Why should a body resist this? I don't know. And I doubt too many people do. Anyone who claimed to know why a body resists change has invariably given up in despair or has stormed away after enough "But Why?" thrown in his direction. But the fact of the matter is that for some reason, nobody likes being moved. So there!
Now for a quick recap of what gyan I have given. Force is that which tries (successfully or not) to change a body's current state (of rest or of motion). What this means in reality is that you need some force in order to start a body moving and you need some force to stop it from moving once you get it going.
This is seen in real life. To move a car, we need to use some force (whether from the petrol engine or from push-starting the dashed thing). And to stop it, we need to press the brakes, which in turn apply a force on the wheels. So that is the first fundamental concept.
Next time, I shall start giving gyan on momentum; after touching upon velocity, vectors, along with polysyllabic words like differentiation and conservation...And all with equations too.
Oh...and welcome to the blog! :)
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