Best way to find the Least Count of a Measuring tool

Hey People!

It's good to know, Understanding is also important!

Image by By antpkr, freedigitalphotos.net

A prominent concept to understand and measure readings properly (of a measuring instrument) is Least Count. It'll help you practically also. Today, I'd like to show to you, the best way to find it, which is of-course according to me. I'm not sure whether you will like it or not, although I hope you do!

I will try to explain the concept first & then give a few Examples about finding Least Count for Ruler, Vernier Calipers and then Travelling Microscope. Don't worry if you're not able to understand the concept completely, go through the examples (Ruler, Vernier Calipers & Travelling Microscope) to nicely understand.

THE CONCEPT

Least Count is a very important concept, introduced to properly measure the reading from a measuring tool, like Vernier Calipers, Travelling Microscope, Ruler OR a meter, like car's speedometer, ammeter, voltmeter, etc.. So it's a concept that'll help you not only in your theoretical life, but also Practically, in your everyday life. Learning and understanding it is utter.

Now, what is Least Count?
Least Count tells you the minimum reading or value that can be measured with a measuring tool or device.

Generally, simply multiplying Least Count with the number of divisions (like in ruler) or fraction of divisions (like in Vernier Calipers), we get our answer in the units specified. For Example, 21 divisions in a Ruler would mean 2.1 cm or 21 mm. Least count of a ruler is 0.1 cm or 1 mm (we'll understand how to find it, later in this article).

HOW TO FIND LEAST COUNT

I've seen many people confused about how to find Least Count. The method used by them might be slightly different, tough to remember, so even if they understand it once, the next time they try to do the same, they forget about it, which is not good.

The way I like to make it understandable is related to the definition of Least Count itself. Remember, Least Count gives you the minimum value that can be measured by by the instrument/device/tool. So considering that, Least Count will be:

You can take any number of divisions for finding Least Count, but those have to be the smallest ones. Let us take "n" small divisions.

        Value measured in "n" divisions
LC=                                                  
                             n

So overall, Least Count is based on the concept of Unitary Method.

Don't worry if you don't understand the above mentioned theory completely.

We'll use the same concept here to find the least count of Ruler, Vernier Calipers & Travelling Microscope, although things might change a little in case of Vernier Calipers & Travelling Microscope. I recommend to go through the examples below to understand properly.

Also we'll talk a little about the importance or Physical significance of Least Count, in case of a Ruler.

THE RULER

So for a ruler, that is, the scale, we use in daily life, we can find the Least Count, by the definition formula only, which we did.

Let us take 20 divisions for it (the value of "n". We know that a ruler measures 20 mm or 2 cm, 20 divisions.

        Value measured in 20 divisions
LC=                                                  
                             20
     = 2 cm/20 = 0.1 cm

Therefore Least Count of a Ruler comes out to be 0.1 cm or 1 mm. Remember, you could even take other values like { 1 cm / 10 }.

Now what is the importance of this value (0.1 cm)?

While calculating, we used 20 divisions. Now 10, 20, 30, 40, etc. are numbers that can be easily dealt with practically, rather than numbers like 13, 17, 27, etc.

What if you wanna calculate the value measured by 23 divisions of a ruler? Here Least Count becomes handy. You can simply multiply 23 with 0.1 cm (the Least Count) & get the answer 2.3 cm.

Therefore, 2.3 cm is the value measured by 23 divisions of a Ruler.

THE VERNIER CALIPERS

Now, Vernier Calipers, is similar to ruler, but a little more complex and can give more specific results. The accuracy of Vernier Calipers is much more than ruler, as the former can give results upto two decimal places (in case of centimeters), whereas the Ruler can only give upto 1.

Now let's talk about what's the Least Count of Vernier Calipers and how to find it.

In case of ruler, there was only 1 scale which gave us the readings, so we straight away could use Unitary method on that. But in Vernier Calipers, there are 2 scales, called "The Main Scale" & "The Vernier Scale". The Main Scale is similar to the Ruler we use, therefore "Main Scale" does the work of providing the results upto the first Decimal Place. (in centimetres)

Vernier Scale does the magic (near magical!) of providing us with the next Decimal Place, which imparts more Accuracy to the Vernier Calipers. Consider the Ruler, take the smallest division in it 0.1 cm. Now if we draw 10 more smaller divisions in that, theoretically those will give the second decimal Place.

But is that feasible practically? Of-course not! 0.1 cm (1 mm) is already very small and putting 10 more lines in that would be outrageous. So to do that, Vernier Scale is used. "Vernier Scale" can be considered as the magnified form of those small 10 lines that we were to draw in 0.1 cm of ruler.

So, the above should explain the concept of Vernier Calipers to you. Let's come back to find the Least Count of Vernier Calipers.

In addition to simply finding the minimum value (Least Count) given by Main Scale (ruler), we have to consider the Vernier Scale too (the magnified version of the smaller lines that we were to draw in the 0.1 cm of Ruler). What we'll do is first find the Least Count of the Main Scale and then move onto the Vernier Scale to solve our issue step by step.

The Main Scale (same as Ruler) will give the smallest Reading or Least Count of 0.1 cm, as in case of the Ruler. Next, the formula would be the same:

        Value measured in "n" divisions
LC=                                                  
                             n

where "n" are the SMALLEST DIVISIONS. Now I want you to imagine Vernier Scale as the smaller lines in 0.1 cm of the Ruler (Main Scale). What would you consider "n" to be? The smallest lines we've imagined, OR the smaller lines of an actual Ruler (without Vernier Scale). Of-course the first choice is correct. Since "n" are the SMALLEST DIVISIONS, we have to take up the imagined lines, which are drawn in the 0.1 cm gap.

10 are drawn in 0.1 cm gap, 20 in 0.2 cm gap & 30 in 0.3 cm gap. These give the corresponding values of "n" and values measured by n.

So you can take any of them. If you're writing in Exam I'll always recommend you to take up the first one in the series, so the Examiner doesn't get confused. So considering this, we'll take the first one:

n = 10 &
Value measured in 10 (smallest divisions, n) = 0.1 cm.

Correspondingly, The Least Count:

        Value measured in 10 divisions (smallest)
LC=                                                  
                             10

         0.1 cm

     =            

           10

which turns out to be 0.01 cm. Voila! it's the Least Count of Vernier Calipers. As a special case, the Least Count of a general Vernier Calipers (done above) is also called Vernier Constant.

I hope you got this! This Imagining concept will help you understand the Least Count a lot. It'll help you better know other devices also.

EXERCISE ON VERNIER CALIPERS

So, we've so far found the Least Count of a general Vernier Calipers. Let us exercise the same Imagining Concept on a different Vernier Calipers, having Least Count 0.002 cm, a lot more accurate. This accurate device has been shown in the image below:

Image by ArtMechanic, commons.wikimedia.org

The Main Scale, here is a same as the Ruler. But the Vernier Scale has been designed to be another ruler! (or Half-Ruler to be precise) Instead of simply 10 smallest divisions in the Vernier Scale, there are a total of 50 smallest ones. Therefore, summon the Least Count formula:

        Value measured in "n" divisions
LC=                                                  
                             n

Remember, Vernier Scale gives the smallest divisions "n", not the Main Scale. So Imagine the smaller 0.1 cm of the Main Scale being divided into 50 more divisions, such that 0.1 cm gap makes 50 smallest divisions, 0.2 cm makes 100, 0.3 cm gap makes 150, ....  So:

        Value measured in 50 divisions (smallest)
LC=                                                  
                             50

         0.1 cm

     =            

           50

which is 0.002 cm or 0.02 mm, the Least Count of the Vernier Calipers shown.

THE TRAVELLING MICROSCOPE

The Travelling Microscope offers even greater precision than the Vernier Calipers, that is, if you take up centimeters as units, The Travelling Microscope can measure upto 3 decimal places. This is the height isn't it? That is, 0.001 cm is the minimum value that can be measured by it! Oh! I've revealed the Least Count already! But what can I say, it's so nice.

Just think a little, 0.001 cm would mean 10 micrometres. That's such a high value of precision to be achieved practically.

Anyways, The Travelling Microscope works the same way as The Vernier Calipers. There is a Main Scale & then a Vernier Scale. The difference is just in the "n" (the number of smallest divisions considered) and the Value given by those n divisions.

The Imagining concept has to be utilized here too. The Main Scale Divisions are little different than Ruler. The 1 cm is divided into 20 divisions, instead of 10 (in case of a Ruler).

Therefore in the Main Scale only 1 cm gap means 20 divisions, 2 cm means 40 divisions, 3 cm means 60 divisions. Therefore, considering the Main Scale only, by unitary method:

1 Main Scale Division = 1 cm OR 2 cm OR 3 cm OR .....
                                                                                     
                                       20           40           60

All of them give the Answer = 0.05 cm.

Now, this is of-course not the right "n" or the Values measured by n, because "n" has to be the smallest divisions which are the Vernier Divisions, not the Main Scale ones.

Again the Imagining says that the Vernier Divisions have to be considered as the smallest lines drawn between the smallest lines, actually in the Main Scale (See Vernier Calipers Example for Details). Here, The Vernier Scale has 50 divisions, so imagine 50 lines drawn in that 0.05 cm gap of the Main Scale (Tough to even Imagine! :P).

Remember the definition formula for least Count:

        Value measured in "n" divisions
LC=                                                  
                             n

where "n" are the SMALLEST DIVISIONS.

The question is the same. What value of "n" to choose. The smallest divisions are the ones given by Vernier Scale.

Imagine them now. There are 50 smallest divisions in 0.05 cm gap, 100 (50 more) in 0.10 cm gap, 150 in 0.15 cm gap, ..... so on.

These are the things we needed. "n" can be takes as 50, 100 or 150 with the corresponding Value it tells. So:

         Value measured in 50 divisions (smallest)

LC=                                                  
                             50

         0.05 cm

     =            

           50

which gives the answer 0.001 cm, the Least Count of Travelling Microscope.

That's it!
That's how you find Least Count of a Measuring Tool/device/Instrument, which'll help you take up any Reading from it.

Note: As a disclaimer, I have to tell you that this is a thesis. Please don't be over-reliant on this article, about the exact information.
But this has been operated many-a-times and it works, so we should be good. 

I hope you liked the above post. Comment with your reviews, even a "Thank you" will be enough for me to know that I was able to help you.
You can ask your questions or to contact me, I can also be contacted by the "Contact Me" option on the website.

Also feel free to share the post using its link (URL).

Photo By Master isolated images, freedigitalphotos.net

ENJOY!!!