I know how to use a caliper but on what principle does devices like caliper and screw gauge works. Can someone derive its formulae. I am more or less interested in its mathematical aspect.
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1I don't want to be a jerk here, but have you started by looking closely at the two scales that make up a vernier scale? Set the device to zero and try it. The most surprising thing about them is how uncomplicated they are. From there you can probably deduce how the screw micrometer works. – dmckee --- ex-moderator kitten Jun 09 '15 at 02:25
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If the scale is using decimal system, the marks on the vernier scale will be 9/10 the length of each spacing of the main scale. So, say you had a .01cm precision scale and were measuring something exactly 6.00cm, this means the vernier 0 line would match up with the 0 on the "rough" scale. If you now measured something 6.01cm, this is just enough for the vernier scale line labeled "1" to line up with "1" mark on the main scale. Similar things happen for 6.05cm or 6.08cm, the 5 or 8 will be the only ones lined up correctly, since each spacing on the vernier scale is 9/10 the "main scale" (which is .1cm in this case.) The reason it has 9/10 the size is only one can line up at a time and when it gets to 6.09cm, the very next .01cm movement will start the cycle again and line up the 0 on the vernier again, meaning you're measuring 6.10cm now. I think this image explains it pretty well. (Notice however that the top part is a measurement of 2.00 and the bottom is a measurement of 2.34)
