Metrology. Direct and indirect measurements. General and differences between indirect, cumulative and joint measurements Which measurements of physical quantities are direct indirect
RMG 29 -99 introduces the concept of measurement domain - a set of measurements of physical quantities characteristic of any field of science or technology and distinguished by its specificity. In accordance with the definition, a number of measurement areas are distinguished: mechanical measurements, magnetic, acoustic, measurements of ionizing radiation, etc.
A type of measurement is a part of the measurement area that has its own characteristics and is characterized by the homogeneity of the measured values. As examples of types of measurements, measurements of electrical resistance, electromotive force, electrical voltage, magnetic induction, related to the field of electrical and magnetic measurements, are given. Additionally, subtypes of measurements are identified - part of the type of measurement, distinguished by the peculiarities of measurements of a homogeneous quantity (by range, by size of quantity, etc.) and examples of subtypes (measurements of large lengths, having the order of tens, hundreds, thousands of kilometers or measurements of ultra-short lengths - film thicknesses as subtypes of measurements length).
This interpretation of the types and especially subtypes of measurements is ineffective and not very correct - the subtypes of measurements are not actually defined, and unsuccessful examples confirm this.
A broader interpretation of the types of measurements (using various classification bases) allows us to include among them also the measurements given in the same document, but not formed into classification groups, characterized by the following alternative pairs of terms:
- direct and indirect measurements,
- aggregate and joint measurements,
- absolute and relative measurements,
- single and multiple measurements,
- static and dynamic measurements,
- equal and unequal measurements.
Direct and indirect measurements are distinguished depending on the method of obtaining the measurement result. Direct measurement is a measurement in which the desired value of a physical quantity is obtained directly. The note notes that with a strict approach, only direct measurements exist and it is proposed to use the term direct measurement method. This proposal cannot be called successful (see below for the classification of measurement methods). Examples of direct measurements are given: measuring the length of a part with a micrometer, current strength with an ammeter, mass on a scale.
During direct measurements, the desired value of a quantity is determined directly from the device for displaying measurement information of the measuring instrument used. Formally, without taking into account the measurement error, they can be described by the expression
where Q is the measured quantity,
x is the measurement result.
Indirect measurement - determination of the desired value of a physical quantity based on the results of direct measurements of other physical quantities that are functionally related to the desired quantity. It is further said that instead of the term indirect measurement, the term indirect measurement method is often used. It is preferable not to use this option as it is clearly unsuccessful.
In indirect measurements, the desired value of a quantity is calculated based on the known relationship between this quantity and the quantities subjected to direct measurements. Formal notation for such a measurement
Q = F (X, Y, Z,…),
where X, Y, Z,… are the results of direct measurements.
The fundamental feature of indirect measurements is the need to process (convert) the results outside the device (on paper, using a calculator or computer), as opposed to direct measurements, in which the device produces a finished result. Classic examples of indirect measurements include finding the angle of a triangle from the measured lengths of the sides, determining the area of a triangle or other geometric figure, etc. One of the most common cases of using indirect measurements is determining the density of a solid material. For example, the density ρ of a cylindrical body is determined from the results of direct measurements of mass m, height h and cylinder diameter d, related to the density by the equation
ρ = t/0.25π d2 h
Discussions and a number of misunderstandings are associated with the distinction between direct and indirect measurements. For example, there are disputes about whether measurements of radial runout (b = Rmax - Rmin) or the height of the part are indirect when setting the device to a division other than zero. Some metrologists refuse to recognize indirect measurements as such (“there are only direct measurements, and everything else is mathematical processing of the results”). A compromise solution can be proposed: to recognize the right to exist for indirect measurements, since the specifics of mathematical processing of the results of such measurements and the assessment of their errors are not disputed by anyone.
Direct and indirect measurements characterize the measurements of some specific single physical quantity. The measurement of any set of physical quantities is classified according to the homogeneity (or heterogeneity) of the measured quantities. This is the basis for the distinction between cumulative and joint measurements.
Cumulative measurements are measurements of several quantities of the same name carried out simultaneously, in which the desired values of the quantities are determined by solving a system of equations obtained by measuring these quantities in various combinations. The given example is the determination of the mass values of individual weights of a set from the known value of the mass of one of the weights and from the measurement results (comparisons) of the masses of various combinations of weights confirms that the definition corresponds not to measurements, but to special studies aimed at finding errors in a number of mass measures.
In reality, cumulative measurements should include those in which several quantities of the same name are measured, for example, lengths L1, L2, L3, etc. Such measurements are performed on special devices (measuring installations) for simultaneous measurement of a number of geometric parameters of shafts.
Joint measurements are measurements of two or more different quantities carried out simultaneously to determine the relationship between them. As an example, we can consider simultaneous measurements of lengths and temperatures to find the temperature coefficient of linear expansion. In a narrower interpretation, joint measurements imply the measurement of several different quantities (X, Y, Z, etc.). Examples of such measurements can be complex measurements of electrical, power and thermodynamic parameters of an electric motor, as well as measurements of movement parameters and vehicle condition (speed, fuel reserve, engine temperature, etc.).
To display the results obtained from measurements, different rating scales can be used, including those graduated in units of the physical quantity being measured, or in some relative units, including unnamed ones. In accordance with this, it is customary to distinguish between absolute and relative measurements.
Absolute measurement - a measurement based on direct measurements of one or more basic quantities and (or) the use of the values of physical constants. This extremely unfortunate definition is accompanied by an example (the measurement of force F = mg is based on the measurement of the basic quantity - mass m and the use of the physical constant g at the point of mass measurement), which confirms the absurdity of the proposed interpretation. The note says that the concept of absolute measurement is used as the opposite of the concept of relative measurement and is considered as the measurement of a quantity in its units, and that it is precisely this understanding that is finding more and more application in metrology. It is this interpretation that makes sense to use for these alternative types of measurements.
Relative measurement is a measurement of the ratio of a quantity to a quantity of the same name, which plays the role of a unit, or a measurement of a change in a quantity in relation to a quantity of the same name, taken as the initial one.
Example - Measurement of the activity of a radionuclide in a source in relation to the activity of a radionuclide in a similar source certified as a reference measure of activity.
Based on the number of repeated measurements of the same quantity, single and multiple measurements are distinguished. Single measurement - a measurement performed once.
Note - In many cases, in practice, only single measurements are performed. For example, measuring a specific point in time using a clock is usually done once. (The example does not stand up to criticism, since repeated measurements of one period of time are impossible).
Multiple measurement - a measurement of a physical quantity of the same size, the result of which is obtained from several successive measurements, i.e., consisting of a number of single measurements.
Depending on the goal, the number of repeated measurements can vary widely (from two measurements to several tens and even hundreds). Multiple measurements are carried out either to insure against gross errors (in this case, three to five measurements are sufficient) or for subsequent mathematical processing of the results (often more than fifteen measurements with subsequent calculations of average values, statistical assessment of deviations, etc.). Multiple measurements are also called “measurements with multiple observations.”
Static measurement is a measurement of a physical quantity that is taken, in accordance with a specific measurement task, to be unchanged throughout the measurement time. The examples given (measuring the length of a part at normal temperature and measuring the size of a plot of land) are more likely to confuse than to clarify the situation.
Dynamic measurement is the measurement of a physical quantity that changes in size.
Notes
1 The term element “dynamic” refers to the measured quantity.
2 Strictly speaking, all physical quantities are subject to certain changes in time. This is confirmed by the use of more and more sensitive measuring instruments, which make it possible to detect changes in quantities previously considered constant, therefore the division of measurements into dynamic and static is conditional.
The interpretation of static and dynamic measurements as measurements of a constant or variable physical quantity is primitive and philosophically always ambiguous (“everything flows, everything changes”). There are almost no “unchangeable” physical quantities other than physical constants in measurement practice; all quantities differ only in accordance with the rate of change.
Instead of abstract reasoning, definitions based on a pragmatic approach are desirable. It is most logical to consider static and dynamic measurements depending on the mode in which the measuring instrument receives the input signal of measuring information. When measuring in a static mode (or quasi-static mode), the rate of change of the input signal is disproportionately lower than the speed of its conversion in the measuring circuit, and the results are recorded without dynamic distortion.
When measuring in dynamic mode, additional dynamic errors appear due to too rapid changes in either the measured physical quantity itself or the input signal of measuring information coming from a constant measured quantity. For example, measuring the diameters of rolling elements (constant physical quantities) in the bearing industry is carried out using inspection and sorting machines. In this case, the rate of change of measurement information at the input may be comparable to the rate of measurement transformations in the device circuit. Measuring temperature with a mercury thermometer is disproportionately slower than measurements with electronic thermometers; therefore, the measuring instruments used can largely determine the measurement mode.
Based on the realized accuracy and the degree of dispersion of the results during multiple repetitions of measurements of the same quantity, they distinguish between equally accurate and unequally accurate, as well as equally scattered and unequally scattered measurements.
Equal-precision measurements are a series of measurements of any quantity made by measuring instruments of equal accuracy under the same conditions with the same care.
Unequal measurements are a series of measurements of any quantity made by measuring instruments that differ in accuracy and (or) under different conditions.
The notes to the last two definitions suggest that before processing a series of measurements, make sure that all measurements are equally accurate, and process unequal measurements taking into account the weight of the individual measurements included in the series.
The assessment of equal accuracy and non-equivalence, as well as equidispersion and non-equidispersion of measurement results depends on the selected values of the limiting measures of accuracy discrepancy or scattering estimates. Acceptable discrepancies between estimates are established depending on the measurement task. Measurement series 1 and 2 are called equivalent, for which the error estimates Δi and Δj can be considered almost identical
and unequal accuracy includes measurements with differing errors
Measurements in two series are considered equally scattered (Δ1 ≈ Δ2), or at (Δ1 ≠ Δ2)
unequally scattered (depending on the coincidence or difference in the estimates of the random components of the measurement errors of the compared series 1 and 2).
Depending on the planned accuracy, measurements are divided into technical and metrological. Technical measurements should include those measurements that are performed with a predetermined accuracy. In other words, in technical measurements, the measurement error Δ should not exceed a predetermined value [Δ]:
where [Δ] is the permissible measurement error.
It is these measurements that are most often carried out in production, which is where their name comes from.
Metrological measurements are performed with the maximum achievable accuracy, achieving a minimum (with existing limitations) measurement error Δ, which can be written as
Such measurements take place when standardizing units and when performing unique studies.
In cases where the accuracy of the measurement result is not of fundamental importance, and the purpose of the measurements is to approximate an estimate of an unknown physical quantity, they resort to approximate measurements, the error of which can fluctuate within a fairly wide range, since any error Δ realized during the measurement process is taken as acceptable [Δ ]
The commonality of the metrological approach to all these types of measurements is that for any measurements the values Δ of the realized errors are determined, without which a reliable assessment of the results is impossible.
Indirect measurements differ from direct ones in that the desired value of a quantity is determined based on the results of direct measurements of other physical objects. quantities that are functionally related to the desired quantity. In other words, the desired PV value is determined based on the results of direct measurements of such quantities that are associated with the desired specific relationship. Indirect measurement equation: y = f(x 1, x 2,...,x n), where x i - i is the th result of direct measurement. Examples: In modern microprocessor-based measuring instruments, calculations of the desired measured value are very often carried out “inside” the device. In this case, the measurement result is determined in a manner characteristic of direct measurements, and there is no need or possibility of separately taking into account the methodological error of the calculation. It is included in the error of the measuring device. Measurements carried out by measuring instruments of this kind are classified as direct. Indirect measurements include only those measurements in which the calculation is carried out manually or automatically, but after receiving the results of direct measurements. In this case, the calculation error can be taken into account separately. An example of such a case is measuring systems for which the metrological characteristics of their components are standardized separately. The total measurement error is calculated based on the standardized metrological characteristics of all system components. Aggregate measurements involve solving a system of equations compiled from the results of simultaneous measurements of several homogeneous quantities. Solving the system of equations makes it possible to calculate the desired value.
In cumulative measurements, the values of a set of quantities of the same name Q 1 ...... Q k ., as a rule, are determined by measuring the sums or differences of these quantities in various combinations:
where coefficients c ij take values ±1 or 0.
Thus, we are talking about measurements of several quantities of the same name carried out simultaneously, in which the desired values of the quantities are determined by solving a system of equations obtained by measuring various combinations of these quantities.
Joint measurements- these are simultaneous (direct or indirect) measurements of two or more heterogeneous (not identical) physical. quantities to determine the functional relationship between them. In essence, cumulative measurements are no different from joint measurements, except that in the first case the measurements refer to quantities of the same name, and in the second - to non-identical ones. Indirect, cumulative and joint measurements are united by one fundamentally important common property: their results are determined by calculation based on known functional relationships between the measured quantities and the quantities subjected to direct measurements.
Thus, we emphasize once again that the difference between indirect, cumulative and joint measurements lies only in the form of the functional dependence used in the calculations. With indirect measurements, it is expressed by one equation in explicit form, with joint and cumulative measurements - by a system of implicit equations.
Indirect measurement
Direct measurement
Direct measurement- this is a measurement in which the desired value of a physical quantity is found directly from experimental data as a result of comparison of the measured quantity with standards.
- measuring length with a ruler.
- measuring electrical voltage with a voltmeter.
Indirect measurement
Indirect measurement- a measurement in which the desired value of a quantity is found on the basis of a known relationship between this quantity and the quantities subjected to direct measurements.
- We find the resistance of the resistor based on Ohm's law by substituting the values of current and voltage obtained as a result of direct measurements.
Joint measurement
Joint measurement- simultaneous measurement of several different quantities to find the relationship between them. In this case, a system of equations is solved.
- determination of the dependence of resistance on temperature. In this case, different quantities are measured, and the dependence is determined based on the measurement results.
Aggregate Measurement
Aggregate Measurement- simultaneous measurement of several quantities of the same name, in which the desired values of the quantities are found by solving a system of equations consisting of the resulting direct measurements of various combinations of these quantities.
- measuring the resistance of resistors connected in a triangle. In this case, the resistance value between the vertices is measured. Based on the results, the resistor resistances are determined.
Wikimedia Foundation. 2010.
See what “Indirect measurement” is in other dictionaries:
indirect measurement- Determination of the desired value of a physical quantity based on the results of direct measurements of other physical quantities that are functionally related to the desired quantity. Example. Determination of the density D of a cylindrical body based on the results of straight lines... ... Technical Translator's Guide
indirect measurement- 3.6 indirect measurement: Measurement by which individual components and/or groups of components that are not present in the working reference gas mixture are determined using relative coefficients... ...
indirect measurement- netiesioginis matavimas statusas T sritis automatika atitikmenys: engl. indirect measurement vok. indirekte Messung, f; mittelbare Messung, f rus. indirect measurement, n pranc. mesurage indirect, m; mesure indirecte, f … Automatikos terminų žodynas
indirect measurement- netiesioginis matavimas statusas T sritis Standartizacija ir metrologija apibrėžtis Dydžio vertės radimas netiesioginiu būdu, kai ieškomoji vertė randama naudojant kitų dydžių tiesioginių matavimų rezultatus. pavyzdys(iai) Vienalytės medžiagos… … Penkiakalbis aiškinamasis metrologijos terminų žodynas
indirect measurement- netiesioginis matavimas statusas T sritis fizika atitikmenys: engl. indirect measurement vok. indirekte Messung, f rus. indirect measurement, n pranc. mesure indirecte, f … Fizikos terminų žodynas
Indirect measurement- 1. Measurement in which the desired value of a quantity is determined based on the results of direct measurements of other quantities associated with the desired quantity by a known functional relationship. Used in the document: OST 45.159 2000 Industry... ... Telecommunications dictionary
Indirect measurement (calculation) of individual complex indicators of TOU functioning- Indirect automatic measurement (calculation) is performed by converting a set of partial measured values into a resulting (complex) measured value using functional transformations and subsequent direct measurement... ... Dictionary-reference book of terms of normative and technical documentation
Indirect measurement (calculation) of individual complex indicators of TOU functioning- Kos in cm os automatic measurement (calculation) is performed by converting a set of private measured quantities into a resultrucctsuk (complex) value using functional transformations and subsequent direct... ... Dictionary-reference book of terms of normative and technical documentation
Measurement is a set of operations to determine the ratio of one (measured) quantity to another homogeneous quantity, taken as a unit stored in a technical device (measuring instrument). The resulting value is called a numerical value... ... Wikipedia
This term has other meanings, see Measurement (meanings). Measurement is a set of operations to determine the ratio of one (measured) quantity to another homogeneous quantity, taken as a unit stored in technical... ... Wikipedia
Classification of types of measurements can be carried out according to various classification criteria, which include the following:
A method for finding the numerical value of a physical quantity,
Number of observations
The nature of the dependence of the measured value on time,
The number of measured instantaneous values in a given time interval,
Conditions that determine the accuracy of the results
Method of expressing measurement results.
By method of finding the numerical value of a physical quantity measurements are divided into the following types: direct, indirect,cumulative and joint.
Direct measurement called a measurement in which the value of the measured quantity is found directly from experimental data. Direct measurements are performed using tools designed to measure these quantities. The numerical value of the measured quantity is calculated directly from the reading of the measuring device. Examples of direct measurements: current measurement with an ammeter; voltage - with a voltmeter; mass - on lever scales, etc.
The relationship between the measured value X and the measurement result Y during direct measurement is characterized by the equation:
those. the value of the measured quantity is assumed to be equal to the result obtained.
Unfortunately, direct measurement is not always possible. Sometimes the appropriate measuring instrument is not at hand, or it is unsatisfactory in accuracy, or has not even been created yet. In this case, you have to resort to indirect measurement.
Indirect measurements These are measurements in which the value of the desired quantity is found on the basis of a known relationship between this quantity and the quantities subjected to direct measurements.
In indirect measurements, it is not the actual quantity being determined that is measured, but other quantities that are functionally related to it. The value of the quantity measured indirectly X found by calculation using the formula
X = F(Y 1 , Y 2 , … , Y n),
Where Y 1 , Y 2 , … Y n– values of quantities obtained by direct measurements.
An example of an indirect measurement is the determination of electrical resistance using an ammeter and a voltmeter. Here, by direct measurements, the voltage drop values are found U on resistance R and current I through it, and the desired resistance R is found by the formula
R = U/I.
The operation of calculating the measured value can be performed by both a person and a computing device placed in the device.
Direct and indirect measurements are currently widely used in practice and are the most common types of measurements.
Aggregate Measurements – these are measurements of several quantities of the same name made simultaneously, in which the desired values of the quantities are found by solving a system of equations obtained by direct measurements of various combinations of these quantities.
For example, to determine the resistance values of resistors connected by a triangle (Fig. 3.1), the resistances at each pair of vertices of the triangle are measured and a system of equations is obtained:
 |
From the solution of this system of equations the resistance values are obtained
, 
, 
,
Joint measurements– these are measurements of two or more quantities of the same name that are made simultaneously X 1, X 2,…,X n, whose values are found by solving the system of equations
F i(X 1, X 2, …, X n; Y i1 , Y i2 , … ,Y im) = 0,
Where i = 1, 2, …, m > n; Y i1 , Y i2 , … ,Y im– results of direct or indirect measurements; X 1, X 2, …, X n– values of the required quantities.
For example, the inductance of the coil
L = L 0 ×(1 + w 2 × C × L 0),
Where L 0– inductance at frequency w =2×p×f tending to zero; WITH– interturn capacitance. Values L 0 And WITH cannot be found by direct or indirect measurements. Therefore, in the simplest case we measure L 1 at w 1, and then L 2 at w 2 and form a system of equations:
L 1 = L 0 ×(1 + w 1 2 × C× L 0);
L 2 = L 0 ×(1 + w 2 2 × C× L 0),
solving which, the required inductance values are found L 0 and containers WITH
; 
.
Cumulative and joint measurements are a generalization of indirect measurements to the case of several quantities.
To increase the accuracy of aggregate and joint measurements, the condition m ³ n is provided, i.e. the number of equations must be greater than or equal to the number of required quantities. The resulting inconsistent system of equations is solved by the least squares method.
By number of measurement observations are divided:
On ordinary measurements – measurements performed with a single observation;
- statistical measurements – measurements with multiple observations.
Observation during measurement - an experimental operation performed during the measurement process, as a result of which one value is obtained from a group of values of quantities that are subject to joint processing to obtain measurement results.
Observation result– the result of a quantity obtained from a separate observation.
By the nature of the dependence of the measured quantity on time dimensions are divided:
On static , in which the measured quantity remains constant over time during the measurement process;
- dynamic , in which the measured quantity changes during the measurement process and is not constant over time.
In dynamic measurements, this change must be taken into account to obtain the measurement result. And to assess the accuracy of the results of dynamic measurements, knowledge of the dynamic properties of measuring instruments is necessary.
According to the number of measured instantaneous values in a given time interval, measurements are divided into discrete And continuous(analog).
Discrete measurements are measurements in which, over a given time interval, the number of measured instantaneous values is finite.
Continuous (analog) measurements – measurements in which, over a given time interval, the number of measured instantaneous values is infinite.
According to the conditions determining the accuracy of the results, measurements are:
- highest possible accuracy, achieved with the existing level of technology;
- control and verification, the error of which should not exceed a certain specified value;
- technical measurements, in which the error of the result is determined by the characteristics of the measuring instruments.
By way of expressing results distinguish between absolute and relative measurements.
Absolute measurements – measurements based on direct measurements of one or more basic quantities and (or) use of the values of physical constants.
Relative measurements – measuring the ratio of a quantity to a quantity of the same name, which plays the role of a unit, or measuring a quantity in relation to a quantity of the same name, taken as the initial one.
Measurement methods and their classification
All measurements can be made using various methods. There are two main measurement methods: direct assessment method And methods of comparison with a measure.
Direct assessment method characterized by the fact that the value of the measured quantity is determined directly from the reading device of the measuring device, previously calibrated in units of the measured quantity. This method is the simplest and therefore is widely used in measuring various quantities, for example: measuring body weight on a spring scale, electric current with a dial ammeter, phase difference with a digital phase meter, etc.
The functional diagram of measurement using the direct assessment method is shown in Fig. 3.2.
The measure in direct assessment instruments is the division of the scale of the reading device. They are not placed arbitrarily, but based on the calibration of the device. Thus, the divisions of the scale of the reading device are, as it were, a substitute (a “fingerprint”) of the value of a real physical quantity and therefore can be used directly to find the values of the quantities measured by the device. Consequently, all direct assessment devices actually implement the principle of comparison with physical quantities. But this comparison is multi-temporal and is carried out indirectly, using an intermediate means - divisions of the scale of the reading device.
Methods for comparison with a measure – measurement methods in which the measured value is compared with the value reproduced by the measure. These methods are more accurate than the direct assessment method, but a little more complicated. The group of methods for comparison with a measure includes the following methods: contrast method, zero method, differential method, coincidence method and substitution method.
Defining characteristic comparison methods is that in the process of measurement there is a comparison of two homogeneous quantities - a known (reproducible measure) and a measured one. When measuring by comparison methods, real physical measures are used, and not their “fingerprints”.
Comparison can be simultaneous and multi-simultaneous. With simultaneous comparison, the measure and the measured quantity act on the measuring device simultaneously, and with multi-temporal– the impact of the measured quantity and measure on the measuring device is separated in time. In addition, comparison can be direct And indirect.
In direct comparison, the measured quantity and measure directly affect the comparison device, and in indirect comparison, through other quantities that are uniquely related to the known and measured quantities.
Simultaneous comparison is usually carried out using methods oppositions, zero, differential And coincidences, and multi-temporal - by substitution method.
LECTURE 4
MEASUREMENT METHODS
Direct measurements These are measurements that are obtained directly using a measuring device. Direct measurements include measuring length with a ruler, calipers, measuring voltage with a voltmeter, measuring temperature with a thermometer, etc. The results of direct measurements can be influenced by various factors. Therefore, the measurement error has a different form, i.e. There are instrument errors, systematic and random errors, rounding errors when taking readings from the instrument scale, and misses. In this regard, it is important to identify in each specific experiment which of the measurement errors is the largest, and if it turns out that one of them is an order of magnitude greater than all the others, then the latter errors can be neglected.
If all the errors taken into account are the same order of magnitude, then it is necessary to evaluate the combined effect of several different errors. In general, the total error is calculated using the formula:
Where – random error, – instrument error, – rounding error.
In most experimental studies, a physical quantity is measured not directly, but through other quantities, which in turn are determined by direct measurements. In these cases, the measured physical quantity is determined through directly measured quantities using formulas. Such measurements are called indirect. In the language of mathematics, this means that the desired physical quantity f related to other quantities X 1, X 2, X 3, … ,. X n functional dependence, i.e.
F= f(x 1 , x 2 ,….,X n )
An example of such dependencies is the volume of a sphere
.
In this case, the indirectly measured quantity is V- the ball, which is determined by direct measurement of the radius of the ball R. This measured value V is a function of one variable.
Another example would be the density of a solid
.
(8)
Here – is an indirectly measured quantity, which is determined by direct measurement of body weight m and indirect value V. This measured value is a function of two variables, i.e.
= (m, V)
Error theory shows that the error of a function is estimated by the sum of the errors of all arguments. The smaller the errors of its arguments, the smaller the error of a function.
4. Plotting graphs based on experimental measurements.
An essential point of experimental research is the construction of graphs. When constructing graphs, first of all you need to select a coordinate system. The most common is a rectangular coordinate system with a coordinate grid formed by equally spaced parallel lines (for example, graph paper). On the coordinate axes, divisions are marked at certain intervals on a certain scale for the function and argument.
In laboratory work, when studying physical phenomena, it is necessary to take into account changes in some quantities depending on changes in others. For example: when considering the movement of a body, a functional dependence of the distance traveled on time is established; when studying the electrical resistance of a conductor as a function of temperature. Many more examples can be given.
Variable value U called a function of another variable X(argument) if each has a value U will correspond to a very specific value of the quantity X, then we can write the dependence of the function in the form Y = Y(X).
From the definition of the function it follows that to specify it it is necessary to specify two sets of numbers (argument values X and functions U), as well as the law of interdependence and correspondence between them ( X and Y). Experimentally, the function can be specified in four ways:
Table; 2. Analytically, in the form of a formula; 3. Graphically; 4. Verbally.
For example: 1. Tabular method of specifying the function - dependence of the magnitude of direct current I on the voltage value U, i.e. I= f(U) .
table 2
2.The analytical method of specifying a function is established by a formula, with the help of which the corresponding values of the function can be determined from the given (known) values of the argument. For example, the functional dependence shown in Table 2 can be written as:
(9)
3. Graphical method of specifying a function.
Function graph I= f(U) in the Cartesian coordinate system is the geometric locus of points constructed from the numerical values of the coordinate point of the argument and function.
In Fig. 1 plotted dependence I= f(U) , specified by the table.
Points found experimentally and plotted on a graph are clearly marked as circles and crosses. On the graph, for each plotted point, it is necessary to indicate errors in the form of “hammers” (see Fig. 1). The size of these “hammers” should be equal to twice the absolute errors of the function and argument.
The scales of the graphs must be chosen so that the smallest distance measured from the graph is not less than the largest absolute measurement error. However, this choice of scale is not always convenient. In some cases, it is more convenient to take a slightly larger or smaller scale along one of the axes.
If the studied interval of values of an argument or function is distant from the origin of coordinates by an amount comparable to the value of the interval itself, then it is advisable to move the origin of coordinates to a point close to the beginning of the studied interval, both along the abscissa and ordinate axis.
Fitting a curve (i.e., connecting experimental points) through points is usually done in accordance with the ideas of the method of least squares. In probability theory, it is shown that the best approximation to experimental points will be a curve (or straight line) for which the sum of the least squares of vertical deviations from the point to the curve will be minimal.
The points marked on the coordinate paper are connected by a smooth curve, and the curve should pass as close as possible to all experimental points. The curve should be drawn so that it lies as close as possible to the points where the errors are not exceeded and so that there are approximately equal numbers of them on both sides of the curve (see Fig. 2).
If, when constructing a curve, one or more points fall outside the range of permissible values (see Fig. 2, points A And IN), then the curve is drawn along the remaining points, and the dropped points A And IN how misses are not taken into account. Then repeated measurements are taken in this area (points A And IN) and the reason for such a deviation is established (either it is a mistake or a legal violation of the found dependence).
If the studied, experimentally constructed function detects “special” points (for example, points of extremum, inflection, discontinuity, etc.). Then the number of experiments increases at small values of the step (argument) in the region of singular points.