11.1 Vernier Callipers

  Basic Mathematics and Measurements

Vernier calipers, an instrument for making very accurate linear measurements was introduced in 1631 by Pierre Vernier of France.

Vernier calipers are used to measure

(i). The length of a rod or any object.
(ii). The diameter of a sphere.
(iii). The internal and external diameter of a hollow cylinder.
(iv). The depth of a small beaker.

The detailed description of every component of vernier scale is given below,

1. Main scale: The main scale runs along the beam of the caliper. On a metric vernier caliper, the main scale is graduated in centimetres ($cm$) and millimetres ($mm$).

On an imperial vernier caliper, the main scale is graduated in inches, with each inch divided into increments of a tenth of an inch ($0.1$ inch).

2. Vernier scale: A small, movable auxiliary graduated scale attached parallel to a main graduated scale and calibrated to indicate fractional parts of the subdivisions of the larger scale. Vernier scales are used on certain precision instruments to increase accuracy in measurement.

3. Inner jaws: The inner jaws of a caliper are used for taking inside measurements such as the internal diameter of a container or slot.

4. Outer jaws: The outer jaws are used for measuring outside dimensions such as width, length, and diameter.

5. Depth probe: The depth rod is used for measuring the holes.

6. Thumb screw: The thumb screw is used to precisely adjust the measuring faces of the caliper (both sets of jaws and the depth probe). It helps the user to get a tight grip on the object they are measuring.

7. Lock screw: The lock screw secures the jaws into place, so the object being measured can be removed, and reading can be taken.

(A). Least count $(L.C.):$ It is the smallest length that can be accurately measured with the vernier scale.

In other words, the difference between the values of one main scale division and one vernier scale division.

Mathematically,
$$V.C. = 1\,M.S.D. - 1\,V.S.D.$$
Let $n$ vernier scale divisions $(V.S.D.)$ coincide with $(n-1)$ main scale divisions $(M.S.D.)$ Then,$$n\,V.S.D. = \left( {n - 1} \right)M.S.D.$$$$1\,V.S.D. = \left( {\frac{{n - 1}}{n}} \right)M.S.D.$$
So, the least count is, $$V.C. = 1\,M.S.D. - \left( {\frac{{n - 1}}{n}} \right)M.S.D.$$$$V.C. = \frac{{M.S.D}}{n}$$

For the vernier calliper shown in the figure, $9\ M.S.D.$ concides with $10\ V.S.D.$ and $1\ M.S.D=1\ mm$. So, the value of $n=10$

Therefore, the least count becomes,
$$V.C. = \frac{{M.S.D}}{n} = \frac{1}{{10}}mm$$$$V.C. = 0.1\,mm$$

(B). Reading of a vernier calliper $(R)$

The reading of a vernier calliper is,
$$R = M.S.R + n \times L.C.$$
where,

$R:$ Total reading
$M.S.R.:$ Main scale reading before on the left of the zero of the vernier scale.
$n:$ Number of vernier divisions which just coincides with any of the main scale division.

Let us understand how to take the reading with vernier calliper better with the help of an illustration.

As we can see from the above diagram we can write,

$M.S.R.=1.8\ cm \ \text{or}\ 18\ mm$
$n=4$
$L.C.=0.1\ mm$

So the diametre of the ball or the reading of vernier calliper is given by,
$$R = M.S.R + n \times L.C.$$
$$R = 18\,mm + 4 \times \left( {0.1\,mm} \right)$$$$R = \left( {18 + 0.4} \right)mm$$$$R = 18.4\,mm\,{\text{or}}\,1.84\,cm$$