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When you want to do length measurements with more accuracy and precision, you would need to use a piece of equipment called a sliding Vernier caliper. A Vernier caliper is used widely in laboratories and its main use is to measure diameters (both internal and external) of objects.

On this page, we will look into the basics of a Vernier caliper, how measurements are marked on a Vernier caliper, some formulas around it, and finally, we will see how to measure internal diameter and depth of a given beaker using Vernier caliper readings.

The Vernier caliper was invented by the French scientist Pierre Vernier and is also named after him. A Vernier caliper consists of two parts: the main scale and the Vernier scale. The two parts are joined together where one part is stationary and the other part slides over it. The fixed part has the graduated main scale while the sliding part contains the Vernier scale. Both the scales are graduated along their longer lengths though they are divided into different magnitudes.

The main scale is graduated in mm and cm and has two fixed jaws (A and C in the image below) which are projected at right angles to the scale.

The sliding Vernier scale has two jaws, B and D, which are projecting at right angles to each other and also to the main scale.

There is a metallic strip E which, along with the jaw, is meant to measure the distance or diameter of objects.

The knob (P) is used to slide the Vernier scale along the main scale and the screw (S) is used when you want to fix the position of the Vernier scale at the required position.

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For a common scale, the least count is 1 mm and it is not easy to further divide the least count of the scale to better its accuracy. With a Vernier scale, this can be achieved. When the jaws of the Vernier scale touch each other, the zero of both the main scale and the Vernier scale coincide. The high accuracy of a Vernier scale comes from the fact that there is a very small relative pitch difference between the two scales (main and Vernier).

The design of a Vernier caliper is such that the difference between each division on the Vernier scale is a fraction of the distance between divisions of the main scale. Usually, 10 small divisions on the Vernier scale are equal to 9 small divisions on the main scale. So, when the zero points of both the scales coincide, the Vernier scale’s first mark is 1/10th behind the main scale’s first mark, the second mark is behind by 2/10th, and so on. So this way it's only the 10th mark which coincides exactly on both the scales.

Hence, for any length, the division on the Vernier scale that coincides with the main scale gives us the difference of that division with the main scale. This is the principle behind the Vernier caliper measurement which lets us measure lengths in decimals.

The difference between the magnitudes of 1 MSD (main scale division) and 1 VSD (Vernier scale division) is referred to as the least count of the Vernier caliper. The least count is the smallest distance that can be measured by any measuring instrument.

n * VSD = (n-1) * MSD

Below are some formulas that are used while taking measurements using a Vernier caliper:

Least count of Vernier caliper = magnitude of the smallest division on the main scale/the total count of divisions on the Vernier scale.

Final reading = Main scale reading + Vernier scale reading

Correction to compensate for zero error = Final reading - (+ Zero error)

1 MSD = 1 mm = 0.1 cm

Total count of VSD (n) = 10

10 VSD = 9 MSD

Vernier constant (VC) = 1 MSD - 1 VSD = (1 - 0.9) MSD = 0.1 MSD = 0.01 cm OR Vernier constant = 1 MSD/n= 1 mm/10 = 0.01 cm

The volume of the beaker (same as the volume of a cylindrical hollow object), V = π * r2 * h = (π * D2 * h)/4; where D is the internal diameter of the beaker, r is its internal radius, and h is the internal depth.

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Adjust the upper (A and B) of the Vernier caliper so that they touch the walls of the beaker from inside but do not exert pressure on it.

Now tighten the screws gently so that Vernier calipers are fixed in this position.

Next, carefully note down the position of the zero mark on the Vernier scale against the main scale. It will not coincide exactly with the main scale division.

Now take the main scale reading which is just to the left of the zero mark of the Vernier scale.

Start looking through the Vernier window to find the reading on the Vernier scale which coincides exactly with a reading on the main scale (from the left end to the right end). Note down that reading, let us say it is n.

Multiple “n” with the least count of the instrument and add this to the main scale reading which we found in step 3 above. Convert the result into proper units (usually it would be cm) so that the addition is valid.

Repeat steps 3 to 6 for two different positions (angular) of the beaker.

Keep the edge of the main scale of the Vernier caliper on its peripheral edge to determine the depth of the beaker. To do this make sure the tip of the strip is able to go down freely along the depth of the beaker.

Keep sliding the jaw of the Vernier caliper till it touches the base of the beaker. While doing this ensure the jaws are exactly perpendicular to the bottom surface of the beaker. Once it reaches the bottom, tighten the screws of the Vernier caliper.

Now repeat steps 4 to 6 to obtain the depth of the beaker. You must take the reading for depths at different positions of the beaker.

Record your observations in a tabular format as shown below, making sure all units are proper and apply zero corrections if necessary.

Find the mean of the corrected readings of the depth and the internal diameter of the beaker then express the results in suitable units and figures.

Mean diameter = ___ cm

Mean depth = ___ cm

Corrected diameter = ___ cm

Corrected depth = ___ cm

FAQ (Frequently Asked Questions)

Q1. What is a Micrometer Screw Gauge?

Ans 1 - Micrometer screw gauge is an instrument that can measure even smaller dimensions than a Vernier caliper. It uses an auxiliary scale (which can measure hundredths of an mm) marked on a rotary thimble. This instrument is basically a screw with an accurate and constant pitch (it is the magnitude by which the thimble moves backwards or forward in one complete rotation).

The thimble goes through a frame that has a scale in mm graduated to 0.5 mm. The small ratchet knob is used to rotate the thimble for adjusting the jaws. There is a friction clutch that prevents excessive tension from being applied. To open the jaws by 1 mm, the thimble needs to be rotated twice. To measure an object, it is placed between jaws, and using the ratchet the thimble is rotated till the object is secured in between.

Q2. Explain Negative Zero Error with an Example.

Ans 2 - When the upper jaws of a Vernier caliper touch each other, the zero on the Vernier scale must coincide with the zero on the main scale. If this is not the case then the instrument is said to have a zero error. In negative zero error, the zero of the Vernier scale shifts to the left of the main scale when upper jaws touch each other. In a negative zero error, the readings taken are less than the actual reading which needs a correction to be applied in proportion to the left shift of zero on the Vernier scale. In the image above we can see that the 5th Vernier scale reading is coinciding with the main scale reading. So, the zero error here is:

Zero error = 5 * least count = 5 * 0.01 cm = 0.05 cm.