Same Dimensions and Dimensionless Quantities

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Torque and work have exactly the same dimensional formula: M L squared, T to the power minus two. So tell me, can I measure torque in joule? Sounds logical, right? Same dimensions, same unit? Hold that thought; by the end of this class, you will know why the answer is no. And look, in JEE, these same-dimension questions are quick marks if you know the families. So, let's begin.

0:32Same dimensions: the big families

So, first family. Work, energy, torque and heat: M L squared, T to the power minus two. The square brackets on screen mark a dimensional formula, okay? Heat and work are both energy. But torque is the interesting member; we will come back to it. Second family: pressure, stress, Young's modulus and bulk modulus. M, L to the power minus one, T to the power minus two. Stress is force per unit area, and so is pressure. Remember all four as one group. Clear? Now a surprise member of this family. Half epsilon nought E squared, and B squared over two mu nought: the Greek letters with a small zero are constants, E is the electric field and B the magnetic field. Both are energy densities, energy per unit volume. So think: energy divided by volume, what dimensions? M L squared T to the minus two, over L cubed. M, L to the minus one, T to the minus two. Exactly the pressure family! Third family: angular momentum, Planck's constant and angular impulse: M L squared, T to the power minus one. Remember this one specially: Planck's constant has the dimensions of angular momentum. Impulse and momentum: MLT to the power minus one. Of course, right? Impulse is just the change in momentum. And surface tension, spring constant and surface energy: M, T to the power minus two. Look, no L at all! Easy to remember, and easy to get wrong in a hurry.

2:07More families to remember

Alright, next. Frequency, angular velocity, velocity gradient, and decay constant: all T to the power minus one, something per second, okay? Now a quick one. Light year. Is it a unit of time? Many students say yes. Hmm, careful. No, no! Light year is a distance, dimension L, like wavelength and the radius of gyration. The word year fools many students, okay? And latent heat, with gravitational potential: L squared, T to the power minus two. No mass here. Both are energy per unit mass, so the M cancels. It is also the dimension of the square of a speed. Got it?

2:48Topper's trick: electrical combinations

Now a topper's trick for electricity questions: don't grind out the dimensions, just remember these combinations. R C, L by R, and root of L C: all three are time. R is resistance, C is capacitance, and careful: this L is inductance, not length, okay? Next: c, the speed of light, then one over root of mu nought epsilon nought, and E by B, all three a velocity. Clear? And one more. Root of mu nought by epsilon nought. What could it be? Resistance! Now, write these three lines in your formula notebook. They save minutes in the exam.

3:27Dimensionless quantities: pure numbers

Alright, now the other side of today's topic: quantities with no dimensions at all, pure numbers. Angle, solid angle, strain, refractive index, and relative density. Why is strain a pure number? Change in length divided by the original length: length over length, so nothing is left. Simple as that! Then the dielectric constant, epsilon r, and relative permeability, mu r: the Greek letters with a small r. See how often the word relative appears? Coefficient of friction, Poisson's ratio, and the Reynolds number from fluids. All pure numbers. Okay? Power factor, quality factor, and this big expression: e squared, over four pi epsilon nought, h bar, c. It is the fine-structure constant. Here e is the electronic charge, c the speed of light, and h bar is the h with a little bar across it. So many constants, still a pure number. Interesting, right? Now a very important rule. Sine of x, e to the power x, natural log of x. Can you take the sine of a length, sine of some metres? Not possible! The argument of every such function must be dimensionless, and the value it gives is dimensionless too. Remember this whenever a constant sits inside a sine or an exponential.

4:52The trap: dimensions do not identify a quantity

Now, back to our puzzle. Torque and energy: both M L squared, T to the power minus two. So torque can be written in joule, right? Same dimensions, so same unit, yes? No, no, careful! This is the trap. Torque is a vector, and it is measured in newton metre, never in joule. So dimensions alone do not tell you which quantity it is. Got it? Second surprise. Angle has no dimensions, but we still write its unit, the radian. So, no dimensions does not mean no unit, okay? And fuel economy, in kilometre per litre: length over volume, so it works out like metre to the power minus two, one over an area. But it has nothing to do with area! So, the unit alone does not tell you the quantity either.

5:42Example 1: which pair has the same dimensions

Now, our first example. Which pair has the same dimensions? Option a, impulse and torque. Option b, stress and energy density. Option c, angular momentum and work. Option d, moment of inertia and moment of force. Pause the video, and try it first. Done? Okay, let's check. Option a. Impulse is MLT to the power minus one; torque is M L squared, T to the minus two. Different. Cut it. Option b. Stress is force by area, energy density is energy by volume, and we just saw: both are M, L to the minus one, T to the minus two. Same! But let's quickly check the other two. Option c. Angular momentum is M L squared, T to the minus one; work is M L squared, T to the minus two. Close, but the power of T differs. Hurried students fall right here! Option d. Moment of inertia is M L squared, no T at all, while moment of force is just torque, M L squared, T to the minus two. Different. So the answer is option b. See? Know the families, and it takes seconds.

6:52Example 2: electric over gravitational force, a pure number

Now a harder one, JEE level. Show that e squared, divided by four pi epsilon nought, G, m e, m p, is dimensionless. Here e is the electronic charge, G the gravitational constant, and m e, m p the masses of the electron and the proton. Pause and try. Hint: think of forces. Here is the thing: most students start writing the full dimensions of epsilon nought. Long, and easy to slip. Is there a smarter way? Yes, there is! Think of the hydrogen atom. The electric force between the electron and the proton is e squared over four pi epsilon nought r squared, where r is the distance between them. The gravitational force between them is G m e m p over r squared. Both are forces, right? So multiply each by r squared. e squared over four pi epsilon nought, and G m e m p: both are force into r squared, so the same dimensions. Force is MLT to the minus two. Times L squared gives M, L cubed, T to the minus two, for both. With me? Same dimensions on top and bottom, so the ratio is a pure number. Dimensionless! How big? About two point three, times ten to the power thirty nine. Wow, the electric attraction in a hydrogen atom beats gravity by this huge factor! That is the trick: use a formula you know, instead of expanding every constant.

8:22Quick revision: dimension families and pure numbers

So, quick revision. Learn the families, and same-dimension questions take seconds. Same dimensions do not mean the same quantity: torque is in newton metre, never in joule. And pure numbers: angle, strain, and whatever goes inside a sine, an exponential or a log. Now go and try the four practice problems on this page. You know the families now, so they will feel easy. All the best!