The main scale of a vernier callipers has n divisions/cm. n divisions of the vernier scale coincide with (n − 1) divisions of main scale. The least count of the vernier callipers is,

CONCEPT:

Vernier calipers are defined as a device which is used for the measurement of linear dimensions and also used for the measurement of diameters of round objects like thin wire. The least count for the Vernier caliper is 0.1 mm.

Least count (L.C.) = 1 MSD - 1 VSD    ----(1)

CALCULATION:

Given: n divisions = 1 cm

1 division = \(\frac{1}{n}\) cm

1 Main Scale Divison = \(\frac{1}{n}\) cm

n ( VSD) = (n -1) MSD

⇒ VSD = \(\frac{n-1}{n}\) MSD

⇒ 1 VSD = \(\frac{n-1}{n}\times \frac{1}{n}\)

⇒ 1 VSD = \(\frac{n-1}{n^2}\)

Now, using equation (1) we have;

Least count (L.C.) = 1 MSD - 1 VSD

⇒ Least count (L.C.) = 1 MSD - \(\frac{n-1}{n^2}\) MSD

⇒ Least count (L.C.) = 1 MSD - \(\frac{n-1}{n^2}\)MSD

⇒ Least count (L.C.) = \(\frac{1}{n} - \frac{n-1}{n^2}\) 

Therefore, Least count (L.C.) = \(\frac{1}{n^2}\)