Tue, 11 Nov 2003

Vernier scale

I've been going through some of the instruments and gadgets that belonged to my father. A common feature on many instruments is a vernier scale.

I can remember Dad teaching me how to read a vernier scale, but I never gave much thought, until now, to how and why it works.

The vernier scale is based on some very simple mathematics that yields a surprisingly powerful and useful result.

Consider the following scale. The base scale is on top, the verier scale on the bottom.

Simple vernier scale

Notice that the vernier scale divides 9 units on the base scale into 10 equal parts. Therefore, each vernier unit is 0.9 base units.

1 on the vernier scale falls on a point that must be 0.9 on the base scale, even though the base scale is not marked at that point. If we slide the vernier scale 0.1 base units to the right, the 1 on both scales will align precisely, and the arrow at the origin of the vernier scale will be positioned precisely at 0.1 base units.

If we slide the vernier scale to the right until the 2 marks on both scales align, the arrow on the vernier scale will be positioned at 0.2 base units. It must be, because each vernier unit is 0.9 base units; 2 vernier units equal 1.8 base units; 2 – 1.8 = 0.2.

So, as we can see, when a mark on the vernier scale aligns with a mark on the base scale, that mark on the vernier scale indicates, in tenths, where the vernier arrow falls on the base scale.

Take the following setting, for example.

vernier scale set to 2.3

The 3 on the vernier scale aligns with a mark on the base scale, so the arrow falls at 2.3 on the base scale.

A common set of vernier calipers allows measurements in thousands of an inch. The base scale is marked inches, and tenths, with each tenth divided into 4 parts (0.025 inches). The vernier scale divides 24 of the smallest marked units on the base scale into 25 vernier units. Each unit on the vernier scale is, therefore 0.024 inches long, 0.001 inches shorter than a unit on the base scale. When a measurement is taken, the thousandths indicated on the vernier scale are added to the nearest 0.025 inch mark on the base scale, left of the vernier origin.

It would be impractical to mark a rule with thousandths of an inch and even more impractical to use, but by employing a vernier scale, measuring that accurately is simple.

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