Execution of the laboratory work

1. Study the rules of DL division.

2. Create a subroutine of DL division.

3. Input the subroutine into your computer.

4. Get results based on numerical data.

5. Compile your Experimentation Paper.

Contents of the Experimentation Paper

1. Principle of performing division in the binary number system and DL representation.

2. An algorithm of the DL-division subroutine.

3. Examples of calculations using the subroutine.

Questions for the Self-Testing

1. What is the difference between division in the binary system and that in DL representation?

2. What are the causes of ending the operation of division?

Laboratory work 4.3

PROCESSING OF MEASUREMENT RESULTS

Aim of the work: The aim of the work is to learn to process experimental data of physical quantity measurements.

Basic Principles

A physical quantity is a characteristic of one of the properties of a physical object (system, phenomenon, or process). The value of a physical magnitude is its estimate in terms of a certain number of units accepted for this quantity and expressed by the product of its numerical value and the chosen unit.

There is a true value of a physical quantity which ideally represents the object’s properties and its measured which is found experimentally.

Measurement is a set of operations performed with the help of technical aids which allow relating the quantity being measured to its unit and getting a value of this quantity.

There are two basic types of measurements: direct and indirect.

Direct measurement is measurement of a physical quantity when its value is found directly from experimental data.

Indirect measurement is measurement of a physical quantity when its value is found on the basis of a known relationship between this quantity and quantities whose values are found by direct measurements.

In measuring a quantity it is impossible to find its true value due to imperfections of measuring devices and methods of measurement, the influence that measuring conditions may have on measurements, and some random causes. To find out how close the being measured is to the true value, it is necessary to take into account the measurement error.

Numerically, the measurement error of a quantity x is characterized by its absolute Δx and relative value ε_x:

or ,

where Δx is an absolute error, ε_x is a relative error, x is a measured value, and x₀ is a true value.

In the general case the error of direct measurement is found from the formula:

,

where Δx_sys is a systematic error and Δx_ran is a random error.

,

where Δx_i is an instrumental error, Δx_m is a method error, and Δx_o is an error due to variability of the object properties.

For a measuring device of a given accuracy class, the instrumental error is calculated by the following formula:

Δx_i = (ε_c.a.∙X_r)/100%,

where ε_c.a. is the accuracy class index and X_r is the upper limit of the instrument range.

If the instrumental error is unknown, the absolute error of measurement is defined either by the scale-division value of an analog device or by the unit of the least significant digit of a digital device.

Methodical errors are defined individually for each method.

Errors due to variability of the object properties are reduced to random ones by multiple measurements.

To calculate a random error, we must know its distribution law. Most often it is the Gaussian distribution law whose curve is shown in Fig. 4.3. Since it is symmetrical about X_max, errors equal in absolute values but opposite in sign are equiprobable.

The probability of the measured results being in the interval from x to x+Δx is:

.

The measured result is defined as the most probable value x_prob which is calculated as the mean arithmetic value:

,

where x_i is the result of i-th measurement, n is the number of measurements.

If the number of measurements tends to infinity, the confidence interval is defined as a root-mean-square error:

.

If the number of measurements is limited, the confidence interval is extended to Δx_lim = t_αS_n, where t_α is the Student factor which depends on the number of measurements and the confidence probability α.

For 10 measurements t_α=0.9 =1.83, t_α=0.95 =2.26, and t_α=0.99 =3.25.

Date: 2016-03-03; view: 576