Least Count and the Vernier Principle

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Take your metre scale. The smallest gap on it is one millimetre, right? Now pick up the vernier callipers in your physics lab. Its main scale also has marks just one millimetre apart. Same marks. And yet, the callipers can read one tenth of a millimetre! How? Where does that extra digit come from? Well, there is no magic here, just one beautiful little trick, from the year sixteen thirty one. And look, JEE loves this: find the least count of a vernier, then take a reading with it. So, let's begin.

0:38Least count: the smallest direct reading

So, first, one important word: least count. The least count, L C for short, is the smallest measurement an instrument can read directly. Directly means just by looking at its marks, okay? Now tell me. What is the least count of your ordinary metre scale? One millimetre, yes! The smallest gap between two of its marks. So for a metre scale, L C equals one millimetre. And notice, this number has a meaning. Any reading from that scale is uncertain by about one least count, so about one millimetre. Want a better reading? Then you need a smaller least count. Clear?

1:20Vernier's idea: a sliding scale

So, how do we beat one millimetre? In sixteen thirty one, Pierre Vernier had a clever idea. Keep the main scale as it is, and add a small sliding scale along it. On screen, the main scale is on top, and the sliding scale below it is the vernier. Now the clever part. The vernier divisions are slightly smaller than the main-scale divisions. Not equal, slightly smaller. Why would anyone do that? Well, because then the two sets of lines do not stay in step. As you move along the vernier, the gap between each vernier line and the nearest main-scale line shrinks a little, line by line, until one vernier line sits exactly on a main-scale line. In this figure, it is line number four, okay? So we only have to see which vernier line coincides. That one line tells us the tiny extra length.

2:12The vernier constant

Right then, let's put numbers on this. Two short forms first. MSD is one main-scale division, and VSD is one vernier-scale division, okay? Suppose n vernier divisions are equal in length to n minus one main-scale divisions. One fewer. That is the whole design. So one VSD equals n minus one by n, times one MSD, okay? Slightly less than one main-scale division, exactly as Vernier wanted, right? Now, the least count is just the difference: one MSD minus one VSD, nothing else. Work it out. One, minus n minus one by n. Yes, one by n! So L C equals one MSD, divided by n. Remember it the topper's way: least count equals one main-scale division, divided by the number of vernier divisions. Nice and simple, got it? Now, ordinary callipers. One MSD is one millimetre, and ten VSD equal nine MSD, okay? So here, n is ten. So L C is one millimetre divided by nine? Because there are nine main-scale divisions, right? No, no, careful! This is the trap. We divide by n, the number of vernier divisions, and that is ten. One millimetre by ten: zero point one millimetre, which is zero point zero one centimetre. And that is how the callipers read one tenth of a millimetre. Puzzle solved!

3:41Taking a reading

Now, how do we actually take a reading? Look at the figure again. First, find the vernier zero. Here it sits between two point one and two point two centimetres, okay? The main-scale reading, MSR, is the mark just before the vernier zero. So MSR is two point one centimetres. Next, scan along the vernier. Which of its lines sits exactly on a main-scale line? The fourth one, exactly! That number is the vernier scale reading, VSR, so VSR is four. Now put it together. Reading equals MSR, plus VSR times the least count, minus the zero error, if there is one. Zero error has its own lesson in this chapter. For now, take it as zero, okay? So here: two point one, plus four times zero point zero one. Two point one four centimetres. And notice one thing. We add four times the least count, not four millimetres. Clear?

4:41Example 1: least count when 20 VSD equal 19 MSD

Let's do Example one. In a vernier callipers, one main-scale division is one millimetre, and twenty vernier divisions coincide with nineteen main-scale divisions. Find the least count. Pause the video, and try it first. Done? Okay, let's check. Twenty VSD equal nineteen MSD, okay? So, what is n? Twenty, the number of vernier divisions. Not nineteen. Remember the trap! So L C equals one millimetre divided by twenty. Zero point zero five millimetre. See, twenty divisions instead of ten, and the least count becomes half as big. More vernier divisions, finer reading.

5:26Example 2: vernier reading with 0.5 mm main divisions

Now a harder one, JEE level. Read it with me. In a vernier, one main-scale division is zero point five millimetre, and twenty vernier divisions equal nineteen main-scale divisions. The vernier zero lies between three point four five and three point five zero centimetres, and the twelfth vernier line coincides. Find the reading. Pause and try. Hint: find the least count first, and watch the units. Alright, least count first. Twenty vernier divisions, just like Example one. So the least count is zero point zero five millimetre again, right? No, no, careful! Read the question once more. Here one main-scale division is only zero point five millimetre, half a millimetre. So L C equals zero point five millimetre, divided by twenty. Zero point zero two five millimetre. But the main-scale reading will be in centimetres, so change the units first. Zero point zero two five millimetre is zero point zero zero two five centimetre, okay? That is the second trap: mixing millimetres and centimetres in one sum. Now the reading. The vernier zero is between three point four five and three point five zero, so MSR is three point four five centimetres. And the twelfth line coincides, so VSR is twelve. Reading equals three point four five, plus twelve times zero point zero zero two five. Twelve times zero point zero zero two five is zero point zero three. So the reading is three point four eight zero centimetres. See? Find L C, fix the units, then read. That's it, three steps.

7:11Quick revision: least count and vernier reading

So, quick revision. Least count: the smallest measurement an instrument can read directly. For a vernier, the least count is one main-scale division, divided by n, the number of vernier divisions. And the reading is MSR, plus VSR times the least count, minus the zero error. Now go and try the practice problems on this page. Watch your units, and you will find them easy. All the best!