PROBLEM STATEMENT

Suppose a ruler has markings every tenth of an inch but it is desired to read accurately to hundredths. Aseparate scale (fig. 6-3) is added to the ruler. It has 10 markings on it that take up the same distance as 9 markings on the ruler scale. Thus, each space on the vernier is 1/10 of 9/10 inch, or 9/100 inch. How much smaller is a space on the vernier than a space on the ruler? The ruler space is 1/10 inch, or 10/100 and the vernier space is 9/100 inch. The vernier space is smaller by the difference between these two numbers, as follows:

Figure 6-3.-Vernier scale.

Each vernier space is 1/100 inch smaller than a ruler space. As an example of the use of the vernier scale, suppose that we are measuring the steel bar shown in figure 6-4. The end of the bar almost reaches the 3-inch mark on the ruler, and we estimate that it is about halfway between 2.9 inches and 3.0 inches. The vernier marks help us to decide whether the exact measurement is 2.94 inches, 2.95 inches, or 2.96inches.

Figure 6-4.-Measuring with a vernier,

The 0 on the vernier scale is spaced the distance of exactly one ruler mark (in this case, one tenth- of an inch) from the left hand end of  the vernier. Therefore the 0 is at a position between ruler marks which is comparable to the position of the end of the bar. In other words, the 0 on the vernier is about halfwaybetween two adjacent marks on the ruler, just as the end of the bar is about halfway between two adjacent marks. The 1 on the vernier scale is a little closer to alignment with an adjacent ruler mark; in fact, it is one hundredth of an inch closer to alignment than the 0. This is because each space on the vernier is one hundredth of an inch shorter than each space on the ruler.

Each successive mark on the vernier scale is one hundredth of an inch closer to 0 on the vernier must be five hundredths of an inch from the nearest ruler mark, since five increments, each one hundredth of an inch in size, were used before a mark was found in alignment.

We conclude that the end of the bar is five hundredths of an inch from the 2.9 mark on the ruler, since its position between marks is exactly comparable to that of the 0 on the vernier scale. Thus the value of our measurement is 2.95 inches.