Dimensions and Dimensional Formulae in Mechanics

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Think of the air pressure inside a cycle tyre. Now think of the Young's modulus of a steel wire, the number that tells you how hard it is to stretch that wire. Completely different things, right? But write down what each one is made of, in mass, length and time, and you get exactly the same answer. The same? How is that possible? By the end of this video, you will prove it yourself. And look, dimensions are one of the most useful tools in JEE physics. So, let's begin.

0:38Three bricks: mass, length and time

So, here is the big idea. Every derived quantity, every single one, can be written as a product of powers of a few base quantities. And in mechanics, we need only three: mass, length and time. Capital M, capital L and capital T, each inside square brackets, okay? Think of them as three bricks. Now you tell me. Velocity is metres per second. Which bricks are inside it? Yes, exactly! Metres is a length, and per second means divided by time. So velocity is L, T to the power minus one. Per means a negative power, okay? And the rest of physics? Just three more bricks: current, written as capital A for ampere, temperature, written as capital K for kelvin, and amount of substance, mol. But for mechanics, mass, length and time are enough.

1:30Dimensional formula and dimensions

Now the official language. Take any quantity, call it capital Q, okay? Q inside square brackets, read as the dimensions of Q, equals M to the a, L to the b, T to the c, and then A to the d and K to the e for current and temperature. Writing a quantity like this is called its dimensional formula. Okay, here is a quick question, and many students get it wrong. What exactly are the dimensions of this quantity? The letters M, L and T, right? No, no, careful! The dimensions are the powers: a, b, c and so on. The powers to which the base quantities must be raised to represent the quantity. Remember this definition, all right? So, for velocity: mass to the power zero, length to the power one, and time to the power minus one. Its dimensions are zero, one and minus one, in mass, length and time. And when we write it as a full equation, v in square brackets equals L, T to the minus one, that is called its dimensional equation. Simple, right?

2:38How to find any dimensional formula

Alright, now the method. It works for every quantity. Step one: take any equation that defines the quantity, or contains it. Step two: replace every other quantity by its dimensions, and simplify. That's it! Let's try force. Force equals mass times acceleration. Mass is just capital M, okay? And acceleration? Velocity per unit time: L, T to the minus one, divided by time, gives L, T to the minus two. Multiply them: force is MLT to the power minus two. Write this one down, we will use it again and again. Now, if a formula has a two, or a pi, or a half in it, do these numbers change the dimensions? No! Pure numbers have no dimensions. And angles also have no dimensions. And the next trap hides right here.

3:33JEE trap: an angle brings no dimension

Now, the JEE trap. Angular velocity, omega, the Greek letter that looks like a curly w, equals theta over t: the angle turned, divided by the time. Theta is measured in radian. So, does the radian enter the dimensional formula? Many students say yes, the radian must add something. No, no, careful! Angles have no dimensions. So the dimensional formula of omega is just T to the power minus one, okay? And notice, frequency is also T to the minus one. Angular velocity and frequency: same dimensions. Remember this pair. And angular acceleration, alpha, is omega divided by time: T to the power minus two. Got it?

4:20Topper's trick: start from force [MLT⁻²]

Now let me share a trick that toppers use. Don't mug up the whole table. Learn just force, MLT to the minus two, and then multiply or divide. Momentum is mass times velocity, and impulse is force times time. So multiply force by time: MLT to the minus one. Both the same! Multiply force by a length, and you get torque, r times F, okay? M L squared, T to the minus two: exactly the dimensions of work and energy. Torque and energy, same dimensions. Another favourite pair, right? Now divide force by a length. Surface tension is force per unit length, and the spring constant k, from F equals k x, is force per extension. So, a soap film and a spring, same dimensions? Exactly! Both are force divided by a length: M, T to the power minus two. Clear? And angular momentum, m v r, is momentum times a length: M L squared, T to the minus one. One formula to remember, so many answers. Really powerful.

5:25Example 1: dimensions of force, work, power and pressure

Now, example one. Find the dimensional formulae of force, work, power and pressure. Pause the video, and try it first. Hint: each one uses one you have already found. Done? Okay, let's check. Force we already know: m a gives MLT to the power minus two. Work is force times displacement, F times s. Displacement is a length, so multiply by L: M L squared, T to the minus two. Power is work divided by time, W over t. So divide by time: M L squared, T to the minus three, okay? And pressure is force over area, F over capital A, okay? Wait. Capital A, our brick for current? No! Here it means area, and area is length times length, L squared. So, MLT to the minus two, divided by L squared: M, L to the minus one, T to the minus two. And see, capital P is power, small p is pressure. Don't mix them up! Four answers, one chain.

6:36Example 2: stress, strain, Young's modulus, compressibility

Now a harder one, and it solves our puzzle. Find the dimensions of stress, strain, Young's modulus, and compressibility. Pause and try. Hint: one of these is a ratio of two lengths. Done? Okay, let's check. Stress is force over area, F over capital A, just like pressure. So, M, L to the minus one, T to the minus two. Strain is delta l over l: the change in length, divided by the original length. The little triangle, delta, means change, okay? So what are its dimensions? Length over length: none at all! Strain is dimensionless, just like an angle. Now, Young's modulus, capital Y, is stress over strain. Dividing by a pure number changes nothing, right? So Young's modulus is M, L to the minus one, T to the minus two, exactly the same as pressure! And there it is, puzzle solved: the air in a tyre and the stiffness of a steel wire, same dimensions. Last one, compressibility: one over the bulk modulus. Bulk modulus is a modulus, so it has the dimensions of pressure. So compressibility is also M, L to the minus one, T to the minus two, right? No, no, careful! This is the trap. One over something flips the sign of every power. So, M to the minus one, L to the plus one, T to the plus two. Compressibility is a modulus turned upside down. Got it?

8:11Quick revision: M, L, T powers and building from force

So, quick revision. In mechanics, every quantity is M to the a, L to the b, T to the c, and those powers are its dimensions. To find them, use a defining equation, and remember: pure numbers and angles have no dimensions. And start from force, MLT to the minus two, then multiply or divide by length and time. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!