Converting Units with Dimensions

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Imagine a new system of units. Mass is measured in blocks of ten kilograms, length in kilometres, and time in minutes. Now take one joule of energy. In this new system, will its number be bigger than one, or smaller? Hmm, not so easy, right? Three units changed together! By the end of this class, you will answer it in three lines, with one formula. And look, in JEE, changing a quantity like G or Young's modulus from S I to CGS is a favourite question. So, let's begin.

0:37Same quantity, different number

So, first, one idea from our very first chapter. When you change the unit, the quantity itself does not change; only the number changes, okay? Number into unit stays the same: n one u one equals n two u two. So the number n is proportional to one over u. The double arrow on screen means, implies, and this open loop sign means, proportional to. Now tell me. If the unit becomes bigger, what happens to the number? Smaller, yes! Exactly. One kilometre equals one thousand metres. Big unit, the kilometre, small number, one. Small unit, the metre, big number, one thousand. Clear? Alright, now the new part. Suppose a quantity has the dimensional formula M to the power small a, L to the power small b, T to the power small c. Then in any system, its unit is that system's mass unit, length unit and time unit, raised to these same powers. The small u written below each letter means, unit of. For example, force is MLT to the power minus two. So in S I, its unit is kilogram, metre, second to the power minus two, and we call it the newton. Change the base units, and the unit of force changes with them. Got it?

1:58The conversion formula

Now put this unit into n one u one equals n two u two. Here M one, L one and T one are the mass, length and time units of the first system, and M two, L two, T two those of the second. On the left, the first system: n one, into M one to the power small a, L one to the power small b, T one to the power small c. On the right, the same thing with twos, okay? We want n two, the new number. So what do we do? Simple, divide both sides by the units of the second system! And we get the conversion formula. n two equals n one, into M one by M two to the power small a, into L one by L two to the power small b, into T one by T two to the power small c. Write this down. This one line is the heart of today's lesson. Here, one is the system you are given, and two is the system you want. So in every ratio, the old unit sits on top and the new unit at the bottom. Old over new. Clear?

3:08Topper's method: three steps, two traps

Now, here is the method toppers use, in three steps. Step one: write the dimensional formula of the quantity, and read off the three powers, small a, small b and small c. Step two: write the three ratios, old unit over new unit. And put the top and the bottom in the same unit before dividing, okay? Step three: raise each ratio to its own power, and multiply them all with n one. That's it, done! Now, one question. What if you write the ratios the other way, new over old? Oh no, then every factor comes out upside down! So the whole factor flips, and your answer is wrong. This is the first JEE trap. Old over new, always. Second trap: negative powers. Time very often comes with the power minus two, right? T one by T two, to the power minus two, equals T two by T one, squared. The minus sign flips the fraction. Forget the minus sign, and the time factor comes out upside down. With me?

4:13Example 1: newton to dyne and joule to erg

Right, Example one, an easy one. Use the conversion formula to express one newton in dyne, and one joule in erg. Dyne and erg are the CGS units of force and energy. Pause the video, and try it first. Done? Okay, let's check. We go from S I to CGS, so S I is system one. Now tell me: M one by M two, is it one thousand, or one by one thousand? One thousand, correct! One kilogram on top, one gram at the bottom, and one kilogram is one thousand grams. So, the arrow on screen: from S I to CGS, okay? M one by M two, one kilogram over one gram, is ten to the power three. L one by L two is one metre over one centimetre: ten to the power two. And T one by T two? Second over second, just one. Really easy. Force is MLT to the power minus two, so small a and small b are both one. n two equals one, into ten to the power three, into ten to the power two: ten to the power five. So one newton is ten to the power five dyne. Remember chapter one? We did this by hand there. Now the formula does it in one line. Now energy. Same answer, ten to the power five erg, right? No, no, careful! Energy is M L squared T to the power minus two. Small b is two! So the length factor is ten to the power two, squared: ten to the power four. Ten to the power three, into ten to the power four, is ten to the power seven. One joule is ten to the power seven erg. Clear?

6:03Example 2: 1 joule in a system of 10 kg, 1 km, 1 minute

Now, remember our puzzle from the start? Let's finish it. In a new system, the unit of mass is ten kilograms, the unit of length is one kilometre, and the unit of time is one minute. What is one joule in this system? Pause and try. Hint: write every unit in kilograms, metres and seconds first. Done? Okay, let's check. S I is system one, and the new system is system two. M one by M two: one kilogram over ten kilograms, one by ten. L one by L two: one metre over one thousand metres, one by one thousand. T one by T two: one second over sixty seconds, one by sixty. Joule is energy again: M L squared T to the power minus two. So, n two equals one, into one by ten, into one by one thousand, squared, into one by sixty, to the power minus two. Now look at that last bracket. What is it? Yes, the minus sign flips it! It is sixty squared, three thousand six hundred, and it multiplies. If you wrote one by three thousand six hundred, that is trap number two, right? So: one by ten, into ten to the power minus six, which is one by one thousand, squared, into three thousand six hundred. That gives three point six into ten to the power minus four new units. So the answer to our puzzle: smaller than one, much smaller! But notice, the bigger time unit actually pushed the number up, three thousand six hundred times. The bigger mass and length units pulled it down much more. Only the formula tells you who wins.

7:46Quick revision: n1u1 = n2u2, always old over new

So, quick revision. Same quantity, so n one u one equals n two u two, and the number goes as one over the unit. To convert, multiply n one by each ratio, old unit over new unit, raised to its power from the dimensional formula. And watch the signs: a negative power flips the fraction. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!