In a vernier caliper, each cm on the main scale is divided into 20 equal parts. If the tenth vernier scale division coincides with the ninth main scale division. Then the value of the vernier constant will be ______ × 10^-2 mm.

Concept: 

In this, we have used the concept of the vernier caliper which is used as a measuring tool to take linear dimension measurements. With the aid of the measuring  Jaws, it is also used to determine the diameter of circular objects.

N vernier scale divisions (VSD) = N - 1 main scale divisions (MSD)

Then, 1 VSD = (N - 1/N) MSD   -----(1)

value of vernier constant = 1 MSD - 1 VSD   -----(2)

Or we can calculate using an alternate method which is,

Vernier constant (VC) = [1 - ((N - 1)/N)] × (1/ MSD)   -----(3)

which can be used when 'N' VSD coincides with 'N - 1' MSD.

Calculations:

Given: 1 cm in main scale = 20 divisions of the Main scale

The 10th vernier scale division matches with the 9th main scale division.

20 MSD = 1 cm ⇒ 1 MSD = 1/20 cm  -----(4)

According to equation (2), we get:

10 VSD = 9 MSD then 1 VSD = (9/10) MSD   -----(5)

Using equations (4) and (5) we get:

1VSD = (9/10) × (1/20) cm = (9/200) cm

using equation (2) we get: Vernier constant (VC) = (1/20) - (9/200) cm = (1/200) cm = 5 × 10^-3 cm = 5 × 10^-2 mm

hence the correct answer is 5.

Alternative solution: 

Using equation (3) we get vernier constant (VC) = [1 - ((N-1)/N)] × (1/ MSD) 

Here, the 10th vernier scale division matches with the 9th main scale division.

 So VC = [1-(9/10)] × (1/20) cm = 1/200 cm = 5 × 10-3 cm = 5 × 10-2 mm