The Vernier Callipers

Transcript

0:00Introduction

Hello everyone! Here is a small puzzle before we start. On the main scale of a common laboratory vernier callipers, the marks are one millimetre apart. And yet, we read a length with it right down to a tenth of a millimetre. That is ten times finer than its own marks, and it sounds impossible, right? The secret is a small sliding scale, the vernier. This instrument comes up often in the experimental skills questions of JEE, so let's begin.

0:33The vernier principle

So, the instrument has two scales: the fixed main scale, and a small vernier scale that slides along it. Here is the whole idea. Let the letter n be the number of vernier divisions. These n divisions are equal in length to n minus one divisions of the main scale. On screen, VSD means vernier scale division, MSD means main scale division, and the brackets hold n minus one, okay? Now let me ask you this. Which one is longer, one vernier division or one main division? Yes, exactly, the main division is longer, because the same length is shared among more vernier divisions. So each vernier division is slightly shorter. In fact, one VSD equals n minus one over n, times one MSD, right? That small difference between the two divisions is called the least count, or the vernier constant, written L C on screen. So the least count equals one MSD minus one VSD, got it? Now put the second line into the third line, and subtract. What does the least count become in terms of n? The bracket, one minus the fraction n minus one over n, is just one over n. So the least count equals one MSD over n, the formula to remember. Now notice the condition: n vernier divisions must equal n minus one main divisions, as in our first line, okay?

1:59Least count with real numbers

Alright, now some real numbers. In the common laboratory callipers, one main scale division is one millimetre, the m m on screen, and ten vernier divisions cover nine millimetres. Clear? Now you tell me, what is the least count of this callipers? Let us work it out. Nine millimetres shared among ten divisions makes one vernier division zero point nine millimetres, okay? So the least count is one minus zero point nine, which is zero point one millimetres. In centimetres, the c m on screen, that is zero point zero one centimetres. With me? Now here is a trap. Suppose the main scale is in half millimetres, and fifty vernier divisions coincide with forty nine main-scale divisions. Many students write one millimetre over fifty. Is that right? No, it is wrong, and this is the trap. The top of the fraction is one main scale division, and here that is zero point five millimetres, while n is fifty. So the least count is zero point five millimetres over fifty, which equals zero point zero one millimetres. Got it?

3:10Taking a reading

Now let us read the instrument. The figure shows a callipers of least count zero point zero one centimetres, with the main scale on top, marked in centimetres, and the vernier scale below it, numbered from zero to ten. Leave the reading written beside it for now, we will reach it in a moment. Okay? First, the main scale reading, written MSR on screen. It is the main-scale mark just before the zero of the vernier. Here, the arrow at the top shows the vernier zero just past two point three, so the main scale reading is two point three centimetres. Second, the vernier coincidence, written V C on screen. Find the vernier division that lines up with some main-scale mark. Here, the arrow at the bottom points to the fourth division, so the vernier coincidence is four, okay? Now put them together. The reading equals the main scale reading, plus the vernier coincidence times the least count. This one line works whenever n vernier divisions equal n minus one main divisions, so write it down. Clear? So for this figure, the reading is two point three, plus four times zero point zero one, which equals two point three four centimetres, the same as the line in the figure, right?

4:24Three parts, three jobs

Now, the instrument has three measuring parts. The outer jaws, the lower ones, measure a length or an external diameter, like the diameter of a ball, okay? So tell me, which part would you use for the internal diameter of a tube? Yes, exactly, we use the inner jaws, the upper ones, because they measure an internal diameter, like that of a tube. And the strip at the end measures depth, like the depth of a beaker. Got it?

4:53Example 1: length with MSR 3.2 cm, 7th division coinciding

Now, example one. With a callipers of least count zero point zero one centimetres, the main scale reading is three point two centimetres, and the seventh vernier division coincides. We have to find the length. Pause the video and try it first. Done? Okay, let's check. Here is the reading formula again. So what is the vernier coincidence in this question? It is seven, that is correct, because the seventh division coincides. So the reading is three point two, plus seven times zero point zero one, right? Seven times zero point zero one gives zero point zero seven, and adding that to three point two gives three point two seven centimetres. That is the length, and it really is that simple.

5:44Example 2: reading, and the main mark the 6th division meets

Now a harder one, example two, with a twist. The zero of the vernier, with least count zero point zero one centimetres and a main scale in millimetres, lies between four point three and four point four centimetres. Its sixth division coincides with a main-scale mark. We have to find the reading, and say which main-scale mark the sixth division coincides with. Pause the video and try it first. Done? Okay, let's check. Which mark is the main scale reading, four point three or four point four? It is four point three centimetres, that is correct, because we take the mark just before the vernier zero. And the vernier coincidence is six, okay? So the reading is four point three, plus six times zero point zero one, which equals four point three six centimetres. That is our first answer, right? Now here comes the twist. Many students say the sixth division sits at four point three six. The main scale is in millimetres, so can that be true? No, it cannot, and this is the trap. The main scale has marks only at every millimetre, so there is no mark at four point three six. That number is where the vernier zero sits, okay? So we need the length of one vernier division. One VSD is one main division, zero point one centimetres, minus the least count, zero point zero one, which gives zero point zero nine centimetres. Right? The sixth vernier mark is six of these divisions beyond the vernier zero. So its position is four point three six, plus six times zero point zero nine. With me? Six times zero point zero nine gives zero point five four, and four point three six plus zero point five four equals four point nine. So the sixth division coincides with the four point nine centimetre mark. See what has happened here. The line-up happens at a plain millimetre mark, and yet it told us a reading of four point three six, finer than any mark on the scale. That solves our opening puzzle. Clear?

7:48Quick revision: least count, reading formula, three parts

So, quick revision. When n vernier divisions equal n minus one main divisions, the least count is one MSD over n, okay? The reading is the main scale reading, the mark just before the vernier zero, plus the vernier coincidence times the least count. The outer jaws measure external, the inner jaws measure internal, and the strip measures depth. Now go and try the practice problems on this page. You can do this, all the best!