Self-examination questions

1. Give the definition of measurement; explain the essence of direct and indirect measurement.

2. What types of errors are there? Give definition of these errors.

3. What factors does the Student’s coefficient value depend on?

4. Give the rules of error estimation of direct and indirect measurements.

5. What is an instrumental error? How is it possible to determine it?

6. Deduce the expression for calculation of cylinder density.

7. Formulate the round off rules.

Table 5.2

Exp. No.	Diameter di, mm	Random deviation Ddi, mm	Height hi, mm		Random deviation Dhi, mm	Mass m, g		Random deviation Dmi, g
1 2 3 4 5
Root-mean- square error		Sd, mm		Sh, mm			Sm, g
Random error		Ddr, mm		Dhr, mm			Dmr, g
Total error		Dd, mm		Dh, mm			Dm, g

Laboratory work 1.2

Random distribution study and determination of gravity acceleration using mathematical pendulum

The purpose of the work: to study the random values distribution law and distribution curve construction; to determine gravity acceleration.

Theoretical information

Random errors can take place during experimental measurement of some physical quantity. As a result of n experiments, the sequence of values x₁, x₂,…x_n is obtained. Let’s find out the smallest value x_min and the largest value x_max and divide the range into k equal intervals. The width of each interval is determined as follows

.

N ow it is possible to determine a number of measurements within every interval n₁,n₂…n_k and calculate the rate of reappearance of the measured values in each interval n₁/n, n₂/n,… n_k/n.

To find out the distribution of random values, we are to draw a diagram. The measured magnitude is put on the abscissa axis and the rate of reappearance within the corresponding interval is put on the ordinate axis. Such a diagram is termed a histogram (Fig. 5.2). As it is evident from the histogram, some values appear more often than others.

If the number of values approaches infinity and interval magnitude tends to zero, the upper sides of rectangles form continuous curve. This curve and its function are termed distribution curve and distribution function. In practice, normal distribution law is mostly applied. A typical curve of normal random distribution is shown in Fig. 5.3. Analytic expression of this distribution has been received by German mathematician Gauss. Depending on a standard deviation , the distribution curve can vary (see Fig. 5.3.).

Distribution function has the peak value (absolute error is equal to zero) when the measured value is equal to the true one.