Evaluation of accuracy of direct measurements

Laboratory work 1. Evaluation of accuracy of direct measurements

          Objective: to learn how to make measurements with ruler, micrometer and  caliper; assess the accuracy of direct measurements.

          Devices and materials: ruler, caliper, micrometer, bodies to be measured.         

Questions students should know before they are allowed to work:

1. What are system error and random error?

2. How is a systematic error accounted for?

3. How do you account for random error that occurs in direct measurement?

4. what is the construction of a caliper, micrometer?

5. how do you measure with a caliper, micrometer?

6. How is the value of the nonius part determined?

        Nonius and micrometer screw. Imagine two rulers stacked together as shown in figure 1. Let the scale value (length of one division) of the upper ruler be l1, and the scale value of the lower ruler be l2. The rulers form a nonius if there exists such a number k, at which

                                                   Kl2=(k + 1)l1                                (1)

The rulers in figure 1 have k = 4. The upper sign in formula (1) refers to the case when the divisions of the lower ruler are longer than those of the upper ruler, i.e. when l2 > l1. In the opposite case the lower sign should be chosen. Let us assume for certainty that L2 > L1. The value (quantity)

                                                      δ= l2 – l1 = l1/k =l2/(k + 1)                     (2)

is called the accuracy of the nonius.

[pic 1]

fig. 1.

         In particular, if L1 = 1mm, k =10, then the accuracy of the nonius δ = 0,1mm .  As shown in fig. 1, when the zero graduations of the lower and upper scales coincide, the k-th graduation of the lower and (k+1)- degrees of the upper scale, the 2k-th graduation of the lower and 2(k+1) - degrees of the upper scale, etc. also coincide.

         Start gradually moving the upper ruler to the right. The zero divisions of the rulers will separate and the first divisions of the rulers will coincide from the beginning. This will happen at a shift l2 –l1 equal to the precision of the nonius δ. At a double shift the second graduations will coincide, etc. If the m-th graduations coincide, one can obviously state that their zero graduations have shifted by mδ.

       The above statements are true, in the case, if the shift of the upper ruler relative to the lower ruler does not exceed one graduation of the lower ruler. At a shift of exactly one division (or several divisions), the zero point of the upper scale coincides no longer with the zero, but with the first (or n-th) division of the lower ruler. With a slight additional shift, the lower scale graduation is no longer zero, but the first graduation, and so on.

         In technical noniuses, the upper ruler is usually made short, so that only one of the divisions of this ruler can coincide with the lower ones. From now on, we will always assume that the nonius ruler is short in this sense.

        Let's use a nonius to measure the length of body А (Fig. 2). As you can see from the figure, in our case the length L of body А is equal to

                        L = nl2 + mδ                                     (3)

(l2>l1). Here n is an integer number of divisions of the lower scale, lying to the left of the beginning of the upper ruler, and m is the number of division of the upper ruler, coinciding with one of the divisions of the lower scale (in case when none of the divisions of the upper ruler coincides exactly with the divisions of the lower scale, m is taken as the number of division which comes closest to one of the divisions of the lower scale).

[pic 2]

fig. 2.

          Often the movable part of the nonius (upper ruler in fig. 1) has larger divisions, i.e. l1>l2. The method of determining the body length in this case is recommended to be found independently.

     Not only linear, but also angular noniuses can be constructed in a similar way. Noniuses are supplied with calipers (Fig. 3), theodolites and many other instruments.

For precise distance measurements micrometric screws - screws with small and very precisely adjusted step are often used. These screws are used, for example, in micrometers (Fig. 4). One turn of the micrometer screw moves the shaft by 0.5 mm. The drum, connected with the rod, is divided into 50 divisions. One graduation corresponds to a shifting of the stick by 0.01 mm. This is the accuracy with which the micrometer is normally measured.

[pic 3]

Measuring with a caliper and processing the measurement results.

        Caliper consists of steel millimetre ruler, one side of which has fixed foot. The other leg has nonius and can be moved along the ruler. When the feet touch, the zero of the ruler and the zero of the nonius coincide. To measure length of an object, it is placed between the legs, which are moved until it touches the legs of the object (without pressing hard), and secured with the screw f. After that, the ruler and nonius are read off and the length of the object is calculated according to the formula (5).