Limitations of Dimensional Analysis

Transcript

0:00Introduction

Hello everyone! Okay, one small question before we start. Using only dimensions, we found the period of a simple pendulum: T equals k, into root of l by g. Beautiful, right? But tell me, what is this k? Is it one? Is it two? Is it two pi? Oh, dimensions simply cannot tell us! It was the experiment that gave k equals two pi. So look, dimensional analysis is a superpower, but every superpower has its weak spots. Today, we will see the five things it cannot do. And JEE loves to test exactly these blind spots. So, let's begin.

0:41Limit 1: pure numbers are invisible

So, limit number one. Dimensional analysis cannot find pure numbers: a half, two pi, root two, numbers like the ones on screen, okay? Let's see why. Back to our pendulum. Period T equals k, into root of l by g, where l is the length and g is the acceleration due to gravity, okay? Here k is a pure number: no metres, no seconds, no kilograms, nothing. Now tell me, what are the dimensions of two pi? None at all, exactly! It is just a number. These square brackets mean the dimensions of, and equals one means dimensionless, okay? So put k equals one, or two pi, or root two: the dimensions of the right side do not change at all. So matching dimensions can never find k. Clear? That two pi came from experiment, not from dimensions. So remember: dimensions give you the form of the formula, never the number in front. Got it?

1:43Limit 2: no sums, no sin, log or exponential

Next, limit number two. Remember how the method works? We guess Q equals k, into x to the power a, y to the power b, z to the power c. One single product of powers, okay? Now think. Can one single product of powers ever give you two terms, with a plus sign between them? No, never! One product always stays one term. So a relation like s equals u t, plus half a t squared, is out of reach. Here s is displacement, u the initial velocity, a the acceleration and t the time. It is a sum of two terms. We will see exactly what goes wrong in example two, okay? Same problem with sine, log, or e to the power x of a variable. None of these is a product of powers, right? So the method can never produce them, simple as that.

2:41Limit 3: too many variables

Alright, limit number three. In mechanics, we have only three base dimensions: M, L and T, that is mass, length and time, okay? When we match powers, each base dimension gives us one equation. So, three equations, and the arrow means so: at most three unknown powers. Now suppose a quantity depends on five mechanical variables. Five unknown powers, but only three equations. Can you find all five? No way, impossible! The powers cannot all be found. So the rule is simple. Whenever the quantity depends on more variables than there are base dimensions available, the method fails.

3:26Limit 4: same dimensions, different quantities

Clear? Now limit number four. Look at torque and energy. Both have dimensions M L squared, T to the power minus two. Exactly the same, okay? So torque and energy are the same thing, right? No, no, careful! This is the trap. Torque is a vector: it has a direction. Energy is a scalar: no direction at all. Completely different quantities. But dimensions see only M, L and T, nothing more. So they cannot tell a vector from a scalar, and they cannot tell apart two quantities with the same dimensions. Got it?

4:02Limit 5: dimensionless variables

And the last one, limit number five. Some variables are themselves dimensionless. For example, the amplitude of a pendulum, the angle you pull it through before letting go. An angle has no dimensions, okay? If a quantity depends on such a variable, dimensions say nothing about that dependence. Why? That is your first example. Let's see.

4:25Example 1: pendulum period and its angular amplitude

Alright, your turn now. Why cannot dimensional analysis find how the period of a pendulum depends on its angular amplitude, theta nought? Theta nought is the Greek letter on screen: the angle of the swing, okay? Pause the video, and try it first. Done? Okay, let's check. Theta nought is an angle, so it is dimensionless: its dimensions are one. Now suppose the period carries some extra factor, f of theta nought, a function of the angle. What does it do to the dimensions? Nothing at all, correct! It is a pure number. So f of theta nought, into root of l by g, has the same dimensions as root of l by g alone. Any function you like, it always passes the check. So dimensions cannot detect f of theta nought at all. Just like our k from the start: invisible. That is why the angle dependence stays hidden. That's it, done.

5:25Example 2: can dimensions derive s = ut + 1/2 at squared?

Now a slightly harder one, real JEE style. Read it with me. Can dimensional analysis derive s equals u t, plus half a t squared, if we assume that s depends on u, a and t? Pause and try. Hint: count your equations. Done? Right, let's check. The method always guesses one product: s equals k, into u to the power x, a to the power y, t to the power z, okay? Now the dimensions. Displacement is just L, a length. Velocity is L T to the power minus one, acceleration is L T to the power minus two, and time is just T, okay? Three unknowns: x, y and z. And we have M, L and T, three equations. So we can solve it, right? No, no, careful! Look again. Where is M? Nowhere! The M equation just says zero equals zero. Only two real equations. Powers of L: x plus y equals one. Powers of T: minus x, minus two y, plus z equals zero, so z equals x plus two y. Two equations, three unknowns. So try y equals zero. Then x is one and z is one: s equals k u t. Perfectly allowed! Now try x equals zero. Then y is one and z is two: s equals k a t squared. Also allowed! So one free parameter remains, and the method cannot decide between u t and a t squared. One single product can never give their sum anyway. So the answer is no. And that half in front? Just a pure number, limit one again! See how it all connects?

7:21Quick revision: five things dimensions cannot tell you

So, quick revision. One: dimensional analysis cannot find pure numbers, sums of terms, or sine, log and e to the power x relations. Two: it fails when there are more variables than base dimensions. Three: it cannot tell a vector from a scalar, and it says nothing about dimensionless variables like an angle. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!