how does vernier caliper works
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Answer:
The frame has one set of markings (the main scale), usually graduated with divisions of size 1mm. There is another set of markings on the movable slide (the vernier scale) that lines up with the fixed scale. The vernier scale has finer divisions; in a standard basic vernier like in Fig 2., 10 divisions of the vernier scale correspond to 9 divisions of the main. The vernier divisions are marked 0 through 9 and then 0 again. This means that the divisions on the vernier are separated by 0.9mm. This pair of vernier calipers has an accuracy of 0.1mm. A priori, it's not clear that we can measure 0.1mm with 0.9mm graduations.
When the 0 of the vernier is aligned with the 0 of the main, it's clear that the latexi\textsuperscriptth division on the vernier lags the main by 0.1mm. When the movable tooth is advanced by 0.1mm, the 1st division of the vernier aligns with the main scale; advance it by 0.2mm and the second division aligns.
Fig 2. A standard vernier caliper that has an accuracy of 0.1mm
This means that we can measure distances as follows: read the main scale by looking at or immediately to the left of the zero of the vernier. Suppose it reads 11mm. See which division of the vernier aligns with the main scale; suppose its the 6th. Then the measurement is 11.6mm.
It is easily to generalize this simple idea to achieve arbitrary accuracy: Suppose the main scale has divisions of size latexa, and you want to achieve an accuracy of latexa/m, where latexm is an integer. We have to match, then, latexp divisions of the main scale with latexq divisions of the vernier scale, where latexp and latexq are relatively prime integers. Then, latexpa/q is the size of each vernier division, and the distance between a main scale division and vernier scale division is latexa(q−p)/q. In other words,
latexaccuracy=aq−pq=a1m
Clearly, we must have latexq−p=1, q=m. The standard vernier in Fig 2. sets latexp=9, q=10, a=1 to achieve an accuracy of 0.1mm.
The vernier in Fig 1. has a main scale with divisions of size 1mm. The vernier scale has 50 divisions. They correspond to 49 divisions on the main scale. That is, we have latexp=49, q=50, a=1, which gives an accuracy of latex1/50=0.02mm.