Vernier Calipers & Screw Gauge

In physics and engineering, measurements must go far beyond what a simple ruler can provide. When the diameter of a wire, the thickness of a coin, or the bore of a cylinder needs to be recorded to a fraction of a millimetre, scientists rely on precision instruments. The two most fundamental precision instruments encountered in a school laboratory are the Vernier calliper and the screw gauge (also called a micrometer screw gauge). Both devices exploit clever mechanical principles to extend human perception into the sub-millimetre world. Every measuring instrument has a property called its least count (LC) — the smallest division it can reliably resolve. An ordinary ruler has a least count of 1 mm. A Vernier calliper typically achieves 0.1 mm or 0.02 mm, while a screw gauge routinely measures to 0.01 mm (10 micrometres). Understanding how these instruments are constructed and read is essential for reporting measurements with appropriate precision and for correctly estimating experimental uncertainty. This article explains the operating principle of each instrument, derives the least count formula from first principles, demonstrates how to take a reading, and discusses the sources of error — particularly zero error — that every careful experimenter must account for.

Key takeaways

The Vernier Scale Principle
Reading a Vernier Calliper
The Screw Gauge Principle
Reading a Screw Gauge
Zero Error and Its Correction

The Vernier Scale Principle

A Vernier calliper has two scales: a fixed main scale (graduated in mm) and a sliding Vernier scale. The Vernier scale has $n$ divisions that together span $(n-1)$ main-scale divisions. Each Vernier division therefore equals $\frac{n-1}{n}$ mm. The difference between one main-scale division (1 mm) and one Vernier division is $1 - \frac{n-1}{n} = \frac{1}{n}$ mm. This difference is the least count: $\text{LC} = \frac{1 \text{ MSD}}{n}$ , where MSD stands for Main Scale Division. For a 10-division Vernier, LC = 0.1 mm; for a 50-division Vernier, LC = 0.02 mm.

Reading a Vernier Calliper

To read the calliper: (1) Record the main scale reading — the last main-scale graduation that is fully to the left of the Vernier zero. (2) Identify which Vernier division aligns most perfectly with any main-scale line; call this number $m$ . (3) The total reading is $\text{Main Scale Reading} + m \times \text{LC}$ . For example, if the main scale reads 23 mm and the 7th Vernier division coincides on a 10-division calliper (LC = 0.1 mm), the measurement is $23 + 7 \times 0.1 = 23.7$ mm.

The Screw Gauge Principle

A screw gauge uses a precision screw whose pitch $p$ is the linear distance the spindle advances per complete rotation. The thimble scale has $n$ equal divisions around its circumference. One division of the thimble therefore corresponds to a linear advance of $\frac{p}{n}$ , which is the least count. Most laboratory screw gauges have $p = 0.5$ mm and $n = 50$ divisions, giving $\text{LC} = \frac{0.5}{50} = 0.01$ mm. The instrument magnifies the tiny linear displacement into a large rotational arc, making sub-millimetre resolution easy to read.

Reading a Screw Gauge

To read the screw gauge: (1) Read the main scale (sleeve scale) — note the last fully visible millimetre mark and also whether the 0.5 mm mark beyond it is visible. (2) Read the thimble scale — find which thimble division aligns with the datum line on the sleeve. (3) Add: $\text{Total} = \text{sleeve reading} + (\text{thimble division} \times \text{LC})$ . If the sleeve reads 5.5 mm (the 0.5 mark is exposed) and the thimble reads 24, then $\text{Total} = 5.5 + 24 \times 0.01 = 5.74$ mm.

Zero Error and Its Correction

When the jaws of a calliper are fully closed (or the faces of a screw gauge touch), the zero of the Vernier or thimble should align with the datum line. If it does not, a zero error exists. A positive zero error means the instrument reads too high; subtract it from every reading. A negative zero error means the instrument reads too low; add its magnitude. Corrected reading $=$ Observed reading $-$ Zero error. For example, if a screw gauge has a zero error of $+0.03$ mm and reads 4.52 mm on a wire, the true diameter is $4.52 - 0.03 = 4.49$ mm.

The Key Equation

$$\text{Least Count} = \frac{\text{Pitch}}{\text{Number of divisions on thimble}} = \frac{p}{n}

- **p** — Pitch (mm): The linear distance the spindle advances in one complete revolution of the thimble. - **n** — Number of thimble divisions (dimensionless): The number of equal graduations marked around the circumference of the thimble. - **\text{LC}** — Least count (mm): The smallest length that can be resolved by the instrument; equal to one thimble division's linear equivalent. ## Worked Example **Problem.** A student uses a Vernier calliper with 20 Vernier divisions (spanning 19 mm on the main scale) to measure the outer diameter of a hollow cylinder. The main scale reads 31 mm and the 13th Vernier division coincides with a main-scale line. A zero error of −0.05 mm was noted before the measurement. Find the corrected diameter.

d = 31.70 \text{ mm}

**Answer.** The corrected outer diameter of the cylinder is 31.70 mm.

Figure

Diagram

Origins · 1 minPierre Vernier and how this was discovered

Pierre Vernier

1580–1637 · French

Pierre Vernier was a French mathematician and instrument maker who in 1631 published 'La Construction, l'usage, et les propriétés du quadrant nouveau de mathématiques', in which he described the auxiliary sliding scale that now bears his name. The Vernier scale allowed navigators and surveyors to read angles and lengths far more precisely than previously possible, representing a landmark in the history of precision measurement. Vernier originally developed his scale to improve the accuracy of a surveying quadrant used for measuring angles — it was only later adapted to linear measurement devices like the calliper we use in school labs today.

Significant Figures and Precision — Readings from Vernier callipers and screw gauges must be recorded with the correct number of significant figures to reflect the instrument's precision and avoid overstating accuracy.
Errors and Uncertainties in Measurement — Zero error, parallax, and systematic errors encountered when using precision instruments are all specific examples of measurement errors discussed in this broader topic.
Units and Dimensional Analysis — Correctly expressing measurements in SI units (metres, centimetres, millimetres) and converting between them is fundamental when recording calliper or screw gauge readings.
Systematic and Random Errors — Zero error in a screw gauge is a classic example of a systematic error, while reading the coinciding Vernier line involves an element of random error — both are key concepts in experimental physics.