Measuring Very Large and Very Small Distances

Transcript

0:00Introduction

Hello everyone! Okay, two small puzzles before we start. First one. There is a tree on the far bank of a wide river. No bridge, no boat, no tape. Can you find how far away it is, without crossing the river? Sounds impossible, right? Second one. How thick is a single molecule? No ruler can measure that! By the end of this class, you will solve both: the tree with two points and one angle, and the molecule with one drop of oil. And remember, in JEE these methods come straight into numericals. Let's begin.

0:36Large distances: the parallax method

So, look at the figure. Far away on the right is the object S, maybe our tree, or even the Moon. We cannot reach it. But we can look at it from two points on our own side, A and B, okay? The distance between A and B is called the basis, and we write it as small b on screen. This one we can measure easily, right here on our side. Clear? The two lines of sight meet at S and make a small angle there: theta, the little Greek letter on screen. This is the parallax angle. Now tell me. If S moves farther away, does theta grow or shrink? Shrink, exactly! The two lines become almost parallel. So a smaller angle means a larger distance. But how exactly are they connected? Remember the radian: angle in radian equals arc divided by radius. Imagine a circle centred at S, with radius D, okay? For a small angle, arc and chord are almost equal, right? So the basis b is almost a piece of that arc. So theta is approximately b over D, okay? Turn it around: D is approximately b divided by theta. This wavy equals sign means approximately equal. But listen carefully: theta must be in radian. Put degrees here and your answer is gone. This is the number one JEE trap in this topic. Got it?

2:01Angular size and echo methods

Alright, now the same idea, the other way round. Suppose we know the distance D of an object, like the Sun, and want its size, which we call small d, okay? It makes a small angle at our eye, alpha, this Greek letter. Then alpha is approximately d by D, so d is approximately D into alpha. Nothing new! Now a completely different trick: the echo. We send out a pulse, it hits the object and comes back. We measure the time t, okay? So the distance is just speed into time, v into t, right? No, no, wait! In that time, the pulse went to the object and came back. It covered the distance twice, didn't it? So we divide by two. D equals v t by two. Forget this two, and your answer becomes double. Classic JEE trap! Which pulse do we send? Radar uses radio waves. Sonar uses sound in water, say from a ship. And laser ranging uses laser pulses. Different waves, exactly the same formula. Clear?

3:06Small distances: how fine can we measure?

Now let's go to the other extreme: very small distances. Vernier callipers measure down to zero point one millimetre, okay? The screw gauge goes ten times finer: zero point zero one millimetre. But a molecule is far smaller. Use a microscope? But tell me, how small can an ordinary optical microscope go? Not that small! Only to about the wavelength of light, about zero point five micrometre. The wavy line means about, and mu, this Greek letter, means micro. The electron microscope goes further, to about one angstrom, the letter with a small circle on top. But look, for molecules there are two neat tricks that need no microscope at all.

3:51The size of a molecule

First, the oil film method. Take a tiny drop of oleic acid whose volume, capital V, we know. On water, it spreads into a film just one molecule thick! Now think. If a sheet has area capital A and thickness small t, what is its volume? Area into thickness, yes! So, V equals A into t, okay? So the thickness t is approximately V divided by A, and that thickness is the size of one molecule. One drop, one area, done. As simple as that! Second method, from density. Take a substance of molar mass capital M and density rho, the Greek letter that looks like a p. One mole has volume M by rho, and holds N A molecules, where N A is Avogadro's number. So each molecule occupies M divided by rho N A, okay? If the molecules are packed closely, each one sits in its own tiny cube of this volume. The side of that cube, the cube root, is roughly the size of the molecule. With me?

4:55Example 1: distance of a tree across a river by parallax

Now, remember our tree? Let's find it. The tree is sighted from two points one hundred metre apart on this bank, and the two lines of sight differ in direction by two point zero degrees. Estimate its distance. Pause the video, and try it first. Done? Okay, let's check. Basis is one hundred metre, angle is two degrees. So the distance is one hundred by two, fifty metre, right? No, no, careful! This is the trap. Two is in degrees, and our formula needs radian. Remember, one degree is about zero point zero one seven four five radian. So two point zero degrees is about zero point zero three four nine radian. That small circle is the degree sign. Now D is approximately b by theta: one hundred divided by zero point zero three four nine, in metre. That is about two point nine into ten to the power three metre, which is two point nine kilometre! See? Fifty metre was completely wrong. And we never crossed the river. Puzzle one, solved!

6:02Example 2: size of an oleic acid molecule from a film

Now a harder one, JEE level, and it solves our second puzzle. One cubic centimetre of oleic acid is dissolved in alcohol to make twenty five cubic centimetre of solution. One cubic centimetre of this is diluted to forty cubic centimetre. Ten drops of the final solution, each of volume zero point zero two cubic centimetre, spread on water into a film of area zero point two zero square metre. Estimate the size of an oleic acid molecule. Pause and try. Hint: how much of it is really acid? Done? Okay, let's check. One part acid in twenty five, then one part of that in forty. So the fraction of acid is one by twenty five into one by forty, that is one by one thousand, okay? Now, ten drops of zero point zero two cubic centimetre make zero point two cubic centimetre. So the acid volume is zero point two, right? Wait, no! That is the solution, almost all alcohol. Only one thousandth of it is acid. So V equals ten into zero point zero two into ten to the power minus three, which is two into ten to the power minus four cubic centimetre. Next, the area. Topper's trick: same units before you divide. Volume is in cubic centimetre, so take the area in square centimetre. Zero point two zero square metre is two thousand square centimetre. Got it? Now just divide. Thickness t equals V by A: two into ten to the power minus four, divided by two thousand. That is ten to the power minus seven centimetre, or ten to the power minus nine metre. See, one molecule, measured with a few drops of oil. Puzzle two, solved!

7:52Quick revision: parallax, echo D = vt/2, molecule size

So, quick revision. Far away: D is approximately b by theta, theta in radian, and size d is approximately D into alpha. Echo: D equals v t by two, because the pulse goes out and comes back. Molecules: thickness is approximately V by A, or the cube root of M by rho N A, okay? Now try the practice problems on this page. Convert to radian first, and you will find them easy. All the best!