Changing the Base Quantities

Transcript

0:00Introduction

Hello everyone! Let me start with one small question. Till now, we have built every quantity in mechanics from three bricks: mass, length and time. But tell me honestly, who decided that? Suppose I take mass away from you. Your bricks are now force, velocity and time. Can you still write the dimensions of mass? Mass, made out of force? Sounds strange, right? By the end of this video, you will do it in two lines. And look, this is not a side topic. JEE Main has asked it again and again, and JEE Advanced too. So, let's begin.

0:38Base quantities are a choice

So, here is the big idea of today. Mass, length and time, capital M, L and T in square brackets, are only our usual choice of base quantities. Nothing forces us to use them, okay? In fact, any three independent quantities can be chosen as the base. Force, velocity and time. Velocity, acceleration and force. Even the speed of light, Planck's constant and the gravitational constant. All allowed. Clear? But wait. What does independent mean? Let me test you. Can I choose force, mass and acceleration as my three base quantities? Many students say yes: three quantities, done. No, no, careful! Independent means none of them is built from the others. And force is mass times acceleration, right? It is built from the other two. So that choice is not allowed. Always check this first. Once you have three independent ones, every mechanical quantity can be written in terms of them. Mass, energy, surface tension, anything. Now the real question: how? There are two methods.

1:50Method 1: write the old bricks in the new ones

Method one. Very natural. Step one: write the old base quantities, mass, length and time, in terms of the new ones, okay? Say force is now one of your bricks. Then how will you write mass? Yes, from F equals m a, Newton's second law! Mass equals force divided by acceleration. We just turned the familiar formula around. Step two: substitute these into the quantity you want. And that's it. The old letters disappear, and only the new bricks remain. Simple, right?

2:26Method 2: match the powers

Method two. This one works even when the new bricks are strange constants. Call the new base quantities capital X, Y and Z, okay? Write the quantity you want, Q, as X to the a, times Y to the b, times Z to the c, okay? Now write X, Y and Z in our old mass, length and time. You already know their dimensional formulae, right? Alright, tell me. We have three unknown powers: a, b and c, okay? How many equations do we need? Three! And we get exactly three: one from the powers of mass, one from length, and one from time. Just like we did when we derived relations by dimensional analysis. Then solve the three equations for a, b and c, and the answer is ready. Got it?

3:22Example 1: mass and energy with F, V, T as base

Now, remember our puzzle from the start? Let's solve it. If force F, velocity V and time T are the base quantities, find the dimensions of mass, and of energy. Pause the video, and try it first. Hint: use method one. Done? Okay, let's check. Mass is force over acceleration. And acceleration? Velocity per unit time, so V over T, okay? So m equals F, divided by V over T, okay? Dividing by V over T is the same as multiplying by T over V, right? So the dimensions of mass are F, V to the minus one, T, okay? Mass, made out of force. Puzzle solved, in two lines! Now energy. Energy is force times distance, just like work. And distance is velocity times time, that is V T, okay? So energy is FVT, all three to the power one. Nice and neat. But how can you be sure your answer for mass is right? Here is a topper's trick: put the old dimensions back in, and see what comes out. Force is MLT to the minus two. V to the minus one gives L to the minus one, times T, and then we multiply by one more T, okay? Multiply: the lengths cancel, the times cancel, and only mass is left. So the answer is correct. Always do this ten second check, got it?

4:59Example 2: Planck length from c, h and G

Now a harder one. This type has come in JEE Advanced too. If the speed of light c, Planck's constant h, and the gravitational constant G are the base quantities, find the dimensions of length. Pause and try. Hint: method one gets messy here, so use method two. Done? Okay, let's check. Let length be proportional to c to the x, times h to the y, times G to the z, okay? Three unknown powers: x, y and z, so we need three equations. Now each brick in mass, length and time. The speed of light is a speed: L, T to the minus one. And what about h and G? Yes, we found them in the last chapter. Planck's constant is energy over frequency: M L squared, T to the minus one. And G, from Newton's law of gravitation, is M to the minus one, L cubed, T to the minus two. Now match the powers. Mass appears only in h and G, so y minus z equals zero. Time: minus x, minus y, minus two z, equals zero. And length: x plus two y plus three z equals one, because the left side is just L to the power one. Solve. The mass equation gives y equals z, okay? Put that in the time equation: minus x, minus three z, equals zero, so x equals minus three z, right? Now the length equation: minus three z, plus two z, plus three z, equals one. So z is one half. Then y is also one half, and x is minus three times one half: minus three by two. Fractions in the powers? Don't worry, that is completely normal here. So, now write the answer. Many students quickly write: length equals h G over c cubed. Right? No, no, careful! This is the JEE trap. Look at the powers on screen again. They are not one, they are halves. The powers are a half, a half, and minus three halves, so the whole thing sits under a square root. Length is proportional to the square root of h G over c cubed. Put in the values, and it comes to about four times ten to the power minus thirty five metre. This is called the Planck length. Just think about it: from three constants of nature, a length came out. That is the power of changing the base quantities!

7:37Quick revision: two ways to change the base quantities

So, quick revision. Any three independent quantities, none built from the others, can be the base. Method one: write mass, length and time in the new quantities, then substitute. Method two: write Q as X to the a, Y to the b, Z to the c, and match the powers of mass, length and time. And always check your answer by putting the old dimensions back. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!