Physical quantities are characterized by the concept of "accuracy of the error." There is a saying that by taking measurements you can come to knowledge. So you can find out what is the height of the house or the length of the street, like many others.
Introduction
We will understand the meaning of the concept of "measure quantity." The measurement process consists in comparing it with homogeneous quantities, which are taken as units.
Liters are used to determine volume, and grams are used to calculate mass. To make it more convenient to make calculations, we introduced the SI system of international classification of units.
For measuring the length of a stuck meter, mass - kilograms, volume - cubic liters, time - seconds, speed - meters per second.
When calculating physical quantities, it is not always necessary to use the traditional method; it is enough to apply the calculation using the formula. For example, to calculate indicators such as average speed, you need to divide the distance traveled by the time spent on the journey. So the average speed is calculated.
Using units of measurement that are ten, one hundred, a thousand times higher than the indicators of the accepted measuring units, they are called multiples.
The name of each prefix corresponds to its number of multiplier:
- Soundboard.
- Hecto.
- Kilo.
- Mega.
- Giga.
- Thera.
In physical science, the power of 10 is used to record these factors. For example, a million is denoted as 10 6 .
In a simple ruler, length has a unit of measure - a centimeter. It is 100 times less than a meter. The 15-centimeter ruler has a length of 0.15 m.
The ruler is the simplest kind of measuring instrument in order to measure length indicators. More complex devices are represented by a thermometer - to measure temperature, a hygrometer - to determine humidity, an ammeter - to measure the level of force with which an electric current propagates.
How accurate will the measurements be?
Take a ruler and a simple pencil. Our task is to measure the length of this stationery.
First you need to determine what the division price is indicated on the scale of the measuring device. On two divisions, which are the nearest strokes of the scale, numbers are written, for example, β1β and β2β.
It is necessary to calculate how many divisions are in the range of these numbers. With the correct calculation, you get "10". Subtract from the number that is large, the number that will be smaller, and divide by the number that make up the division between the digits:
(2-1) / 10 = 0.1 (cm)
So we determine that the price determining the division of office supplies is the number 0.1 cm or 1 mm. It is clearly shown how the price indicator for division is determined using any measuring device.
By measuring a pencil with a length that is slightly less than 10 cm, we use the knowledge we have gained. If there was no shallow division on the ruler, the conclusion would follow that the object has a length of 10 cm. This approximate value is called the measurement error. It indicates the level of inaccuracy that may be tolerated during the measurement.
By determining the parameters of the length of the pencil with a higher level of accuracy, a greater measuring accuracy is achieved at a higher division price, which provides less error.
In this case, absolutely accurate measurements cannot be. And indicators should not exceed the size of the price of division.
It was established that the dimensions of the measuring error are Β½ of the price, which is indicated on the divisions of the device, which is used to determine the dimensions.
After performing measurements of the pencil in 9.7 cm, we determine the indicators of its error. This is a gap of 9.65 - 9.85 cm.
The formula that measures such an error is the calculation of:
A = a Β± D (a)
A - as a value for measuring processes;
a is the value of the measurement result;
D is the designation of absolute error.
If you debug or subtract the values, taking into account the error, this number will be the sum of the digits that indicate the error and are available for each individual value.
When subtracting or adding values ββwith an error, the result will be equal to the sum of the error indicators, which is each individual value.
Acquaintance with the concept
If we consider the classification of errors depending on the method of its expression, we can distinguish the following varieties:
The absolute measurement error is indicated by the letter "Delta" in capital. This concept is defined as the difference between the measured and actual values ββof the physical quantity that is being measured.
The absolute expression of the measurement error is the units of the quantity that must be measured.
When measuring mass, it will be expressed, for example, in kilograms. This is not a standard for measurement accuracy.
How to calculate the error of direct measurements?
There are ways to display measurement errors and their calculation. For this it is important to be able to determine the physical quantity with the necessary accuracy, to know what the absolute measurement error is, that no one can ever find it. Only its boundary value can be calculated.
Even if this term is conventionally used, it indicates precisely the boundary data. The absolute and relative measurement error are indicated by the same letters, the difference in their spelling.
When measuring length, the absolute error will be measured in those units in which the length is calculated. And the relative error is calculated without dimensions, since it is the ratio of the absolute error to the measurement result. This value is often expressed as a percentage or in fractions.
The absolute and relative measurement error have several different calculation methods, depending on which method of measuring physical quantities.
The concept of direct measurement
The absolute and relative error of direct measurements depend on the accuracy class of the device and the ability to determine the weighing error.
Before talking about how the error is calculated, it is necessary to clarify the definitions. Direct is the measurement in which the result is directly read from the dashboard.
When we use a thermometer, ruler, voltmeter or ammeter, we always carry out direct measurements, since we directly use the device with a scale.
There are two factors that affect the effectiveness of indications:
- Instrument error.
- The error of the reference system.
The boundary of the absolute error in direct measurements will be equal to the sum of the error that the device shows and the error that occurs during the countdown.
D = D (ave.) + D (exhaust)
Medical Thermometer Example
Error indicators are indicated on the device itself. An error of 0.1 degrees Celsius is recorded on a medical thermometer. The error of reference is half the price of division.
D out. = C / 2
If the division price is 0.1 degrees, then for a medical thermometer, you can calculate:
D = 0.1 o WITH + 0.1 o C / 2 = 0.15 o C
On the back of the scale of another thermometer there is a TU and it is indicated that for the correct measurement it is necessary to immerse the thermometer with the entire back. Measurement accuracy not specified. All that remains is the reading error.
If the scale division value of this thermometer is 2 o , then the temperature can be measured with an accuracy of 1 o . These are the limits of the permissible absolute measurement error and the calculation of the absolute measurement error.
A special system for calculating accuracy is used in electrical meters.
Electrical Instrument Accuracy
To set the accuracy of such devices, a quantity called the accuracy class is used. For its designation, the letter "Gamma" is used. In order to accurately determine the absolute and relative measurement error, you need to know the accuracy class of the device, which is indicated on the scale.
Take, for example, an ammeter. On its scale, an accuracy class is indicated, which shows the number 0.5. It is suitable for measurements on direct and alternating current, relates to devices of the electromagnetic system.
This is a fairly accurate device. If you compare it with a school voltmeter, it can be seen that it has an accuracy class of 4. This value must be known for further calculations.
Knowledge application
Thus, D c = c (max) X Ξ³ / 100
We will use this formula for specific examples. We will use a voltmeter and find the error in measuring the voltage that the battery gives.
We connect the battery directly to the voltmeter, after checking if the arrow is at zero. When connecting the device, the arrow deviated by 4.2 divisions. This condition can be described as follows:
- It can be seen that the maximum value of U for this subject is 6.
- Accuracy class - (Ξ³) = 4.
- U (o) = 4.2 V.
- C = 0.2 V
Using this formula data, the absolute and relative measurement error is calculated as follows:
DU = DU (ave.) + C / 2
DU (ave.) = U (max) X Ξ³ / 100
DU (ave.) = 6 V X 4/100 = 0.24 V
This is the error of the device.
The calculation of the absolute measurement error in this case will be performed as follows:
DU = 0.24 V + 0.1 V = 0.34 V
Using the above formula, you can easily find out how to calculate the absolute measurement error.
There is a rule for rounding errors. It allows you to find the average indicator between the border of the absolute error and the relative.
Learning to determine the weighing error
This is one example of direct measurements. In a special place is weighing. After all, lever scales do not have a scale. We will learn to determine the error of such a process. The accuracy of mass measurement is affected by the accuracy of weights and the perfection of the scales themselves.
We use lever scales with a set of weights, which must be put on the right side of the scales. To weigh, take a ruler.
Before starting the experiment, you need to balance the balance. We put the ruler on the left bowl.
The mass will be equal to the sum of the installed weights. Define the measurement error of this quantity.
D m = D m (weights) + D m (weights)
The error in measuring the mass consists of two terms related to weights and weights. To find out each of these values, the plants for the production of scales and weights are equipped with special documents that allow you to calculate the accuracy.
Application Tables
We will use the standard table. The error of the balance depends on how much weight was put on the balance. The larger it is, the correspondingly greater is the error.
Even if you put a very light body, the error will be. This is due to the friction process occurring in the axes.
The second table relates to a set of weights. It indicates that each of them has its own mass error. 10 grams has an accuracy of 1 mg, like 20 grams. We calculate the sum of the errors of each of these weights, taken from the table.
It is convenient to write mass and mass error in two lines, which are located one below the other. The lower the weight, the more accurate the measurement.
Summary
In the course of the considered material it was found that it is impossible to determine the absolute error. You can only establish its boundary indicators. For this, the formulas described above in the calculations are used. This material is offered for study at a school for students in grades 8-9. Based on the knowledge gained, it is possible to solve the problem of determining the absolute and relative error.