Vernier calipers and the screw gauge

A ruler is not enough

Try to measure the thickness of a wire with a school ruler. The smallest mark is 1 mm. The wire is less than 1 mm thick. The ruler tells you only "less than one millimetre". A laboratory needs instruments that read a tenth or a hundredth of a millimetre. Two of them are in every school laboratory: the vernier calipers and the screw gauge.

Both use one idea. A second scale slides along or turns beside the main scale. It splits the smallest division of the main scale into equal parts, so that you can read a fraction of a division.

The least count

The least count (LC) of an instrument is the smallest length it can measure. Whatever you measure, it can be wrong by about one least count. So the least count also tells you how many digits are worth writing.

The vernier calipers

The calipers have a main scale, here in millimetres, and a vernier scale that slides along it. The vernier has $N$ divisions. They are made to fit into $N - 1$ divisions of the main scale. The usual choice is 10 vernier divisions (VD) in 9 main scale divisions (MSD).

So one VD is $\tfrac{9}{10}$ of one MSD. The difference between one MSD and one VD is the least count:

With 1 MSD = 1 mm and $N = 10$, the least count is $0.1$ mm $= 0.01$ cm. If $N = 20$, it is $0.05$ mm.

To read the calipers:

1. Read the main scale reading (MSR): the last main scale mark just before the zero of the vernier.
2. Find the vernier division that lines up with a mark of the main scale. Call it $n$. The extra part is $n \times \text{LC}$.
3. Add them: reading $= \text{MSR} + n \times \text{LC}$.

Why does it work? The $n$th vernier mark is $n \times 0.9$ mm from the vernier zero. When it lines up with a main scale mark, the vernier zero stands exactly $n \times 0.1$ mm past the mark before it. That is the extra part.

Slide the vernier to $4.7$ mm. Which division lines up? Now slide it to $4.0$ mm. Which one lines up this time?

The zero error

Close the jaws with nothing between them. The vernier zero should sit exactly on the zero of the main scale. Often it does not. The difference is the zero error.

• Positive zero error. The vernier zero is to the right of the main zero. If the $k$th vernier division lines up, the error is $+k \times \text{LC}$.
• Negative zero error. The vernier zero is to the left of the main zero. If the $k$th vernier division lines up, the error is $-(N - k) \times \text{LC}$. The vernier zero is $N - k$ divisions short of the main zero.

The rule for the result is: corrected reading = observed reading $-$ zero error. A positive error is taken away. A negative error is added.

A worked example. With the jaws closed, the 8th vernier division lines up. With a cylinder between the jaws, the main scale reading is $24$ mm and the 6th division lines up. Find the diameter.

1. The least count. $\text{LC} = \dfrac{1\ \text{mm}}{10} = 0.1$ mm.
2. The zero error. The vernier zero is to the left of the main zero, so the error is negative: $-(10 - 8) \times 0.1\ \text{mm} = -0.2$ mm.
3. The observed reading. $24\ \text{mm} + 6 \times 0.1\ \text{mm} = 24.6$ mm.
4. The corrected reading. $24.6\ \text{mm} - (-0.2\ \text{mm})$ $= 24.8$ mm $= 2.48$ cm.

The calipers also measure an inside diameter with the upper jaws, and the depth of a hole with the thin rod at the end. The reading is made in the same way.

The screw gauge

A screw gauge has a screw with a fine thread. When the thimble (the turning part) makes one full turn, the screw moves forward by a fixed distance, the pitch. A scale on the fixed sleeve counts the full turns. A circular scale on the thimble has $N$ divisions, and one division means $\dfrac{1}{N}$ of a turn. So

Here $N$ is the number of divisions on the circular scale.

A common gauge has a pitch of $0.5$ mm and $50$ divisions, so $\text{LC} = 0.5/50 = 0.01$ mm. Another has a pitch of $1$ mm and $100$ divisions. It also gives $0.01$ mm.

The picture shows a gauge with a pitch scale reading of $2.5$ mm and a circular scale reading of $39$. The reading is $2.5\ \text{mm} + 39 \times 0.01\ \text{mm}$ $= 2.89$ mm.

To read the gauge:

1. Read the pitch scale reading (PSR) on the sleeve: the last mark that the edge of the thimble has passed.
2. Read the circular scale reading (CSR): the division of the thimble that lies on the reference line.
3. Reading $= \text{PSR} + \text{CSR} \times \text{LC}$. Then correct for the zero error.

The zero error of a screw gauge. Turn the screw until the two studs touch. Use the ratchet, not the thimble, for the last part of the turn. If the circular scale reads $c$ divisions on the reference line, the error is:

• positive, $+c \times \text{LC}$, when $c$ is a small number, so that the closed gauge reads a little above zero;
• negative, $-(N - c) \times \text{LC}$, when $c$ is close to $N$, so that the closed gauge reads a little below zero.

A worked example. A screw gauge has a pitch of $0.5$ mm and $50$ divisions. With the studs touching, the circular scale reads $47$. With a wire between the studs, the pitch scale reading is $0.5$ mm and the circular scale reading is $39$. Find the diameter.

1. The least count. $\text{LC} = \dfrac{0.5\ \text{mm}}{50} = 0.01$ mm.
2. The zero error. $47$ is close to $50$, so the error is negative: $-(50 - 47) \times 0.01\ \text{mm}$ $= -0.03$ mm.
3. The observed reading. $0.5\ \text{mm} + 39 \times 0.01\ \text{mm}$ $= 0.89$ mm.
4. The corrected reading. $0.89\ \text{mm} - (-0.03\ \text{mm})$ $= 0.92$ mm.

The ratchet and the backlash. Turn the screw in one direction only when you close it on an object. If you turn it back, the thread can slip a little in its nut. This loss of motion is the backlash error. The ratchet at the end of the thimble slips at a fixed grip, so that every measurement presses the object with the same force.

From one reading to a result

One reading is never enough. Take the diameter of a wire at several places along its length. Take it in two directions at right angles at each place, because the wire may not be round. Then find the mean and the mean deviation.

Practise with new numbers

Try it in Python

The program works with the screw gauge of the worked example. It finds the zero error, corrects five readings of the diameter of a wire, and finds the mean and the mean deviation.

Common mistakes

• Adding the zero error. The correction is the observed reading minus the zero error. A negative zero error is added.
• Reading the wrong vernier division. Use the division that lines up best, not the one beside the zero.
• Counting vernier divisions from the wrong end for a negative error. The error is $-(N - k) \times \text{LC}$, not $-k \times \text{LC}$.
• Forgetting the pitch scale. The sleeve scale gives whole turns. In a gauge with a pitch of $0.5$ mm, the marks are $0.5$ mm apart.
• Using the thimble to close the screw. Use the ratchet. A hard grip changes the zero error.
• Writing too many digits. A reading of $0.92$ mm is worth two decimals, not five.

Remember these

• $\text{LC}$ (vernier) $= \dfrac{1\ \text{MSD}}{N}$, and $\text{LC}$ (screw gauge) $= \dfrac{\text{pitch}}{N}$.
• Vernier: reading $= \text{MSR} + n \times \text{LC}$. Screw gauge: reading $= \text{PSR} + \text{CSR} \times \text{LC}$.
• Corrected reading $=$ observed reading $-$ zero error.
• Positive zero error: $+k \times \text{LC}$. Negative zero error: $-(N - k) \times \text{LC}$.
• Take many readings, in two directions, and find the mean and the mean deviation.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A vernier has 20 divisions that fit into 19 divisions of a main scale of 1 mm each. What is its least count?

2. With the jaws closed, the 3rd vernier division lines up (10 divisions in 9 mm). What is the zero error, and what correction is made to every reading?

3. A screw gauge has a pitch of $1$ mm and $100$ circular divisions. With the studs touching, the reading is $96$. A sheet gives a pitch scale reading of $2$ mm and a circular reading of $17$. What is the thickness?

4. Why do we turn the ratchet, not the thimble, when we close the screw gauge on a wire?

References

• NCERT, Laboratory Manual: Physics, Class XI, the experiments on the vernier calipers and the screw gauge.
• OpenStax University Physics Volume 1, Chapter 1, the section on significant figures and measurement.