Least Count: Vernier and Screw Gauge

The least count (LC) of a measuring instrument is the smallest length it can measure with certainty. Two NEET workhorses are the vernier callipers and the screw gauge (micrometer). Both rely on a coincidence trick to magnify the precision of a simple linear scale.

Concept

For vernier callipers, the main scale (MS) has 1 mm divisions and the vernier scale (VS) has $N$ divisions matching $(N-1)$ MS divisions. For a screw gauge, the pitch is the linear distance the spindle moves in one full rotation, and the circular scale typically has 50 or 100 divisions.

Formula Derivation

Vernier:

$\text{LC} = 1\,\text{MSD} - 1\,\text{VSD} = \frac{1\,\text{MSD}}{N}$

For a standard 10-division vernier with MSD = 1 mm:

$\text{LC} = \frac{1\,mm}{10} = 0.1\,mm = 0.01\,cm$

Screw gauge:

$\text{LC} = \frac{\text{Pitch}}{\text{Number of circular divisions}}$

Reading $=$ MS reading $+$ (coinciding division $\times$ LC), corrected for zero error if any.

NEET-style Worked Example

A screw gauge has pitch 1 mm and 100 circular divisions. The main scale reads 4 mm and the 47th circular division coincides with the reference line. There is no zero error. Find the diameter.

$\text{LC} = \frac{1\,mm}{100} = 0.01\,mm$

$D = 4\,mm + 47 \times 0.01\,mm = 4.47\,mm$

Common Confusions

• Zero error positive when the zero of the vernier (or circular scale) is to the right of the main scale zero — subtract it.
• Zero error negative when it is to the left — add its magnitude.
• Pitch is not the LC; pitch is divided by the number of circular divisions to get LC.
• The coinciding division on the vernier must be read in vernier division units, not millimetres.

Key Takeaways

• $\text{LC}_{vernier} = \text{MSD}/N$.
• $\text{LC}_{screw} = \text{pitch}/N$.
• Reading $=$ main + (coincidence $\times$ LC) $\pm$ zero error correction.
• Smaller LC means higher precision.