Dimensions of Constants Inside Formulae

Transcript

0:00Introduction

Hello everyone! Okay, one small question before we start. Your calculator happily gives you sine of thirty degrees. But tell me, what is sine of five metres? Or e to the power two seconds? They make no sense at all, do they? And look, this simple idea is a superpower. In JEE, they give you a formula full of strange constants, alpha, beta, a, b, and ask for their dimensions. Today, with just two facts, you will crack these in a few lines. So, let's begin.

0:36Fact 1: what goes inside sin, cos, e and ln

So, fact number one. Look at sine, cosine, e to the power something, and the natural log. Whatever sits inside them, we call the argument, okay? And the argument is always dimensionless: no metres, no seconds, no kilograms, just a pure number. That is why sine of five metres made no sense. Clear? Good, let's use it. Here is a wave: y equals capital A, sine of, omega t minus k x, okay? Omega is this Greek letter, like a curly w. Small x is a position, and t is time. Now tell me. Omega t is sitting inside the sine. So what must its dimensions be? Nothing at all, yes! It is a pure number. We write it like this. These square brackets mean the dimensions of, and equals one means dimensionless, okay? So the dimensions of omega t equal one. The arrow means therefore: omega has dimensions T to the power minus one, one over time. Makes sense, right? Omega into time must cancel the time. Same story for k x. It also sits inside the sine, so k x is a pure number. So k has dimensions L to the power minus one, one over length. Got it? And here is a bonus. Divide omega by k. T to the power minus one, divided by L to the power minus one, gives L T to the power minus one. That is a speed! The wave speed. See, the formula itself is telling you something physical.

2:14The value of such a function is a pure number too

Alright, now the second half of fact one. Not only the argument: the value of the function is also dimensionless. Sine of anything, e to the power anything, always a pure number, okay? Let's see it work. Here a force is written as, F equals alpha, e to the power minus beta x squared. Alpha and beta are the two Greek letters on screen, and x is a distance. First, the inside. Beta x squared must be a pure number. So beta is L to the power minus one, right? No, no, careful! Look again. It is x squared, not x. Now x squared has dimensions L squared. So beta must cancel L squared: beta has dimensions L to the power minus two. Miss that square, and you lose the marks. Classic JEE trap! Now the outside. The whole exponential is a pure number, so it carries no dimensions. So alpha must carry all the dimensions of force: MLT to the power minus two. Really simple, right?

3:23Fact 2: only like quantities add or subtract

Now, fact number two. Tell me, can you add five metres and three seconds? Of course not! Eight what? Eight metres? Eight seconds? Neither makes any sense! So, only like quantities can be added or subtracted. Metres with metres, seconds with seconds. So in capital A plus capital B, both terms have the same dimensions. And in a bracket like x plus capital B, the constant B must have the same dimensions as x, whatever x is. Clear? Now, a formula with three or four constants looks scary, right? So tell me, where should you start? Think. The constants in front, or the stuff inside? Yes, from the inside, always! That is the topper's trick for these questions: work from the inside out. Step one: first fix the arguments of sine, cosine, e and log, and any constants added inside a bracket. These are the easiest, okay? Step two: only then, the constants standing in front. Once the inside is known, the constant in front just has to match the rest. Got it?

4:34Example 1: dimensions of A, B, C in v = At + B/(t + C)

Okay, now your turn. The velocity of a particle is, v equals capital A t, plus capital B divided by, t plus capital C, okay? Find the dimensions of A, B and C, all three. Remember the trick: inside out. Pause the video, and try it first. Done? Okay, let's check. Where is the inside? The bracket, t plus C, okay? C is added to time, so C must be a time. Dimensions T, just time. Next, capital A t. It is added to the other term, and the total is a velocity. So capital A t has the dimensions of velocity, L T to the power minus one. Divide by time once more, and capital A is L T to the power minus two, okay? Now capital B, okay? The second term is also a velocity. So capital B is a velocity, right? No, no, wait! Not B alone. It is B divided by, t plus C, that is a velocity. And t plus C is a time. So capital B equals velocity into time: L T to the power minus one, into T, okay? The T cancels, and B is just L, a length. So the answers: A is L T to the power minus two, B is L, and C is T, all done.

6:02Example 2: dimensions of a/b in P = (a - t^2)/(bx)

Now a slightly harder one, real JEE style. Read it with me. Pressure P equals, small a minus t squared, divided by, b x, okay? Here t is time and x is a distance. Find the dimensions of a divided by b. Pause and try. Hint: inside out, as always. Done? Okay, let's check. Inside first: small a minus t squared. Only like quantities can be subtracted, so small a is like t squared: dimensions T squared. Not T, careful, T squared! Now b. Turn the formula around: b equals, a minus t squared, divided by, P into x. So the dimensions of b are T squared, divided by the dimensions of pressure into distance. Pressure is M L to the power minus one, T to the power minus two, and x is just L, okay? Now look at the bottom: L to the power minus one, into L, okay? What happens? Yes, they cancel! The bottom is just M T to the power minus two. So b is T squared, divided by M T to the power minus two. The M goes up as M to the power minus one, and dividing by T to the power minus two is multiplying by T squared. So b is M to the power minus one, T to the power four. Last step: a divided by b. T squared, divided by M to the power minus one, T to the power four. So M to the power minus one, T to the power minus two, right? No, no! Look carefully. When M to the power minus one sits in the denominator, it flips and goes up as M to the power plus one. And T squared divided by T to the power four is T to the power minus two. So a by b has dimensions M T to the power minus two. Inside out, a few lines, done. See how fast that was?

8:07Quick revision: inside sin, e and ln, and added terms

So, quick revision. One: whatever sits inside sine, cosine, e or log has no dimensions, and the value is a pure number too. Two: terms that are added or subtracted always have the same dimensions. Three: work inside out. Arguments and added constants first, then the constants in front. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!