Least count of a vernier caliper

How do you find the least count?

The least count is the difference between one main scale division (MSD) and one vernier scale division (VSD). NCERT’s Class 11 lab manual also calls it the vernier constant (VC). On the usual vernier, n VSD = (n − 1) MSD, so one VSD is (n − 1)/n of an MSD and the difference is LC = 1 MSD ÷ n.

Example from the NCERT manual: 1 MSD = 1 mm, 10 vernier divisions = 9 main scale divisions, so 1 VSD = 0.9 mm and LC = 1 − 0.9 = 0.1 mm = 0.01 cm. The other form gives the same: 1 mm ÷ 10 = 0.1 mm.

Least count of common verniers

A main scale in millimetres with a 10-, 20- or 50-division vernier gives three least counts. More divisions give a smaller least count; the lines get closer together and are harder to read without a magnifier.

What if N vernier divisions equal M main scale divisions?

Work out the length of one VSD first, then compare it with the main scale length it nearly matches. When N VSD = (N − 1) MSD, LC = 1 MSD − 1 VSD = MSD ÷ N, whatever the size of the main scale division. A long vernier spreads N divisions over 2N − 1 main scale divisions; then one VSD is compared with 2 MSD: LC = 2 MSD − 1 VSD, which again equals MSD ÷ N.

• 20 VSD = 19 MSD, MSD = 1 mm (the vernier on a Fortin barometer, an NCERT exercise): 1 VSD = 0.95 mm, LC = 1 − 0.95 = 0.05 mm.
• 25 VSD = 24 MSD, MSD = 0.5 mm: 1 VSD = 0.48 mm, LC = 0.5 − 0.48 = 0.02 mm.
• 20 VSD = 39 MSD, MSD = 1 mm (a long vernier): 1 VSD = 1.95 mm, LC = 2 − 1.95 = 0.05 mm, the same as 1 mm ÷ 20.

How do you take a reading?

1. Read the main scale just to the left of the vernier zero (main scale reading, M).
2. Find the one vernier line that coincides with a main scale line and note its number N.
3. Vernier reading = N × LC. Put it in the same unit as M before adding.
4. Observed reading = M + N × LC. Do not estimate between lines.

Example with LC = 0.01 cm: the vernier zero is just past 2.4 cm and the 7th vernier line coincides. Observed reading = 2.4 + 7 × 0.01 = 2.47 cm (24.7 mm).

Positive and negative zero error

Close the jaws and look at the two zeros. If the vernier zero lies to the right of the main scale zero, the zero error is positive; to the left, it is negative. NCERT gives the correction for both: true reading = observed reading − zero error, with the sign of the zero error.

• Positive: count from the vernier zero. If the n-th vernier line coincides, zero error = +n × LC. With LC = 0.01 cm and n = 5: +0.05 cm.
• Negative: count back from the end of the vernier. If the n-th line of an N-division vernier coincides, zero error = −(N − n) × LC. With N = 10, LC = 0.01 cm and n = 7: −(10 − 7) × 0.01 = −0.03 cm.

Why the negative case counts from the end: if the vernier zero sits a distance x to the left of the main zero, its n-th line lies at n × (MSD − LC) − x and meets main scale line n − 1, so x = MSD − n × LC = (N − n) × LC, because N × LC = 1 MSD. In the NCERT example (N = 10, n = 5) both ways of counting give 0.05 cm; for any other line only N − n gives the right value.

Zero correction and the corrected reading

The zero correction is the zero error with its sign reversed, added to the reading. Subtracting a negative zero error and adding a positive correction are the same step.

Keep the zero in 2.50 cm: the least count is 0.01 cm, so the reading has two decimals in centimetres.

Recording the Class 11 practical

NCERT Experiment 1 measures the diameter of a sphere or cylinder, the sides of a rectangular block, and the internal diameter and depth of a beaker. Each table has the columns main scale reading M, coinciding vernier division N, vernier reading N × VC and measured value M + N × VC, with the zero error written under the table.

• Hold the body gently between the jaws, square to the diameter; no undue pressure.
• Look straight down at the coinciding line to avoid parallax.
• Take three readings at different positions, find the mean, then subtract the zero error.
• Write every value with its unit and to the least count.

Practising in KELIBRON

The lab’s vernier caliper has a 1 mm main scale and a 50-division vernier spread over 49 mm, so its least count is 0.02 mm. On the Ø25 shaft it reads 24.98 mm. Its zero-error demonstration closes at +0.06 mm and reads 25.04 mm on the same shaft: 25.04 − 0.06 = 24.98 mm. The lab has no 10- or 20-division vernier and does not show a negative zero error.