The Principle of Homogeneity: Checking Equations

Transcript

0:00Introduction

Hello everyone! Okay, imagine this. You are in the JEE exam hall, and you need the period of a pendulum. Two versions come to your mind. First: T equals two pi, root of l by g. Second: T equals two pi, root of l into g. Which one is right? No book, no derivation, and the clock is running. Hey, don't panic! By the end of this class, one simple principle will settle this in a few seconds. And look, JEE loves this idea: checking an equation, throwing out wrong options, even correcting a formula. So, let's begin.

0:41The principle of homogeneity

So, let me start with a very simple question. Can you add a length to a time? Some metres, plus some seconds? No, you can't! What would the answer even be? Neither metres, nor seconds. It makes no sense at all. You can add a length only to another length, okay? And this is exactly the principle of homogeneity. Only quantities with the same dimensions can be added, subtracted, or equated. Homogeneous simply means of the same kind. Clear? Now think what this means for an equation. An equation is just terms that are added, subtracted, and set equal, right? So in a correct physical equation, every single term has the same dimensions. Not just the left side and the right side. Every term. In symbols: if A equals B plus C, then the dimensions of A, B and C must all be equal. The square brackets you see on screen mean, dimensions of, and the double arrow means, so it follows that. That's it, this one line is the whole principle.

1:45Checking an equation, term by term

Now, let's use it. Suppose a student writes this equation for the distance s covered in time t: s equals u t, plus a t squared. Here u is the initial velocity, and small a is the acceleration, okay? Let us check it, term by term. First term, s. It is a distance, so its dimension is just L, for length. Easy enough, isn't it? Next, u into t. Velocity into time. What are its dimensions? Yes, length again! Velocity is L T to the minus one, time is T to the power one, and the time parts cancel. So u t is also a length, okay? Last term, a t squared. Acceleration is L T to the minus two, and t squared gives T squared. Multiply, and again the time parts cancel. Length! So every term is a length, and the equation passes the test. Perfectly consistent. So we are done now, aren't we?

2:48What the test can and cannot tell you

So the equation passed the test. That means it is correct, right? Most of you are nodding yes. No, no, careful! This is the trap. Let us be very clear about what this test can tell you. If an equation fails the test, it is certainly wrong. No doubt at all. And here is a topper's trick. In JEE, when four formulas are given as options, first throw out every option with the wrong dimensions. It saves a lot of time. But if an equation passes, it may still be wrong. Passing only means that it can be right, okay? Look at our equation. It passed, but the correct one has a half in front of the a t squared. So our student's equation is wrong, and dimensions could not catch it. Why not? Because a half is a pure number. Pure numbers like a half, or two pi, have no dimensions, so this test simply cannot check them. Remember this. Got it? It will come back in a minute.

3:50Example 1: which pendulum period formula can be right

Now, back to our puzzle from the start. Which of these can be the period of a pendulum? T equals two pi, root of l by g, or T equals two pi, root of l into g? Here l is the length of the pendulum, and g is the acceleration due to gravity. Pause the video, and try it first. Done? Okay, let's check. Two pi is a pure number, so set it aside. Small l is a length, dimension L, and g is an acceleration, L T to the minus two. So l by g is L over L T to the minus two. The lengths cancel, and we get T squared. Its square root is T to the power one, a time. Perfect for a period! Now the second one, l into g. What does its square root give? Look, l into g is L squared, T to the minus two. Its square root is L T to the minus one. Wait, that is a speed, not a time! So only the first one can be the period. Puzzle solved, in a few seconds, without any derivation! With me? But hold on. Why did I say, can be, and not, is? Because T equals four pi, root of l by g, also passes this test. Dimensions cannot tell two pi from four pi, right? That pure number is exactly what the test cannot check.

5:15Example 2: check and correct a planet's orbit formula

Now a harder one, JEE level. Here, a student writes the period of a planet that goes around a star, in an orbit of radius r. T squared equals four pi squared r cubed, over G M squared. Capital M is the mass of the star, and G is the gravitational constant, okay? Check it, and if it is wrong, correct it. Pause and try. Done? Okay, let's check. Four pi squared is a pure number, so set it aside. The left side is T squared, so the right side must also be T squared. Now we need the dimensions of G, okay? They are M to the minus one, L cubed, T to the minus two. So, r cubed over G M squared. On top, L cubed. In the bottom, G times M squared, which is M, L cubed, T to the minus two. The L cubed cancels. The M in the bottom goes up as M to the minus one, and T to the minus two in the bottom becomes T squared on top. Result: M to the minus one, T squared. Now tell me. Is this a time squared? No! There is an extra M to the minus one sitting there. The two sides have different dimensions, so the equation is certainly wrong. So how do we fix it? How many powers of M should sit in the bottom? Just one, yes! G into M is L cubed, T to the minus two, and the mass dimension is gone. So r cubed over G M has dimensions T squared. Exactly right. So the corrected formula is: T squared equals four pi squared r cubed, over G M, with just one M, okay? And indeed, that is the true result. But notice: dimensions fixed the power of M, while the four pi squared is a pure number the test cannot check. Same warning as before, remember?

7:12Quick revision: what the dimension test can and cannot show

So, quick revision. One: every term of a correct equation has the same dimensions. Two: if an equation fails the test, it is certainly wrong; if it passes, it may still be wrong. Three: pure numbers like a half and two pi cannot be checked. Now go and try the practice problems on this page. Check every term, and you will find them easy. All the best!