PHYSICS UNIT - 1 Error Analysis Complete knowledge class 11th complete Solution
Physics
UNIT - I
CHAPTER - 5
Error Analysis
SIGNIFICANT FIGURES
The significant figures express the accuracy with which a physical quantity may be expressed
The digits whose values are accurately known in a particular measurement, are called is significant figures
Greater. the number significant figures obtained when making a measurement, more Conversely, a measurement made to only a few significant accurate is the measurement. figures is not. a very occur a one. For example, a recorded figure of 1-21 means the relied on as accurate to three significant figures and a figure of 1-212 is quantity can bei qua he accurate to four significant figures. Thus, significant figures in a measured an quantity indicate the In the example given above, one can easily find the number of significant the number of digits in which we have confidence.
the example given above, one can easily find the number of significant Tt is because, each digit used to express the magnitude of the measurement was significant. u significant. However, when we use the different systems of notation, this is not always the always the case. Suppose that we write the average distance S from the earth to the moon as S 377,000 km = 3-77 x 105 km. Does this number have six significant figures ? No, this measurement is given to a accuracy of only three significant figures. The 3, the 7 and the 7are significant. The last three zeros merely indicate he correct location of the decimal point. Similarly, 0-000123 cm contains only three significant figures. The zeros which merely locate a decimal point are not significant figures. The following rules have been set up for determining the number of significant figures:
All non-zero digits are significant. = 152,48 contains five significant figures.
All zeros occurring between two non-zero digits are significant. = 108-006 contains six significant figures.
All zeros to the right of a decimal point and to the left of a non-zero digit re never significant.
=0-00967 contains three significant figures. The single zero conventionally placed to the left of the decimal point in such an expression is also ever significant.
All zeros to the right of a decimal point are significant, if they are not lowed by a non-zero digit. = 30-00 contains four significant figures.
All zeros to the right of the last non-zero digit after the decimal point are significant. = 0-07800 contains four significant figures.
All zeros to the right of the last (right most) non-zero digit are nothing significant. = 2020 contains three significant figures.
All zeros to the right of the last non-zero digit are significant, if they come from a measurement.= Suppose that the distance between two objects is measured to be 2020 m (to the nearest metre). Then, 2020 m contains four significant figures. This distance can be expressed as 202000 cm or 2-02 km. So that significant figures still remain 4 (irrespective of the way of expressing the measurement), the distance should be expressed as 2-020 x 10 m or 2-020 x 10" cm or 2-020 km.
SIGNIFICANT FIGURES IN CALCULATIONS
In an experiment, usually a number of measurements are made and to obtain the result, they are, then, compounded ie. added, subtracted, multiplied or divided. If all the observations have been made with great accuracy except one observation, then the inaccuracy in the single observation is going to mark the result adversely. To understand this point, consider a strong iron chain of which onel ink is weak.Obviously, the chain cannot be stronger than its weakest link. Actually, it forms the basis of compounding the measurements and then to know the number of significant figures.
SIGNIFICANT FIGURES IN ADDITION AND SUBTRACTION
The accuracy of a sum or a difference is limited to the accuracy of the least accurate observation in the addition and subtraction.
Rule = Do not retain a greater number of decimal places in a result computed from addition and/lor subtraction than in the observation, which has the fewest decimal places.
We have,
428.5 428.5
17.23 17.23
Sum 445.73 Difference 411.27
But in physics,the sum and difference taken in this manner are discouraged. In fact, in the data 428-5, we have assumed zero to be in second place after decimal.The data 428-5 might have been written to first decimal only, because of the inability of the instrument to measure it to the further accuracy. Therefore, the choice of zero only in the second decimal place of the data is not justified.
i) By rounding off the answer = The data 428-5 is the weakest link as its value is know upto first decimal only. Therefore, the answer should also be retained only up to first decimal place i.e.
428.5 428.5
17.23 17.23
Sum 445.73 Difference 411.27
As said earlier,in case the second decimal is occupied by 5 or more than 5, the number in first decimal is increased by 1. On the other hand, if the second decimal is occupied by a number less than 5, it is ignored. Rounding off the results of the above sum and difference to the first decimal, we have
Correct = 445.7 correct difference = 411.3
ii) By rounding off the other data. = The result can also be obtained by rounding off the other data in accordance with the data, which is the weakest link. The data 17-23 should be rounded off to 17-2 (3 in second decimal place is ignored) and then added to or subtracted from 428-5. Thus, we have
428.5
17.2
Correct sum 445.7
428.5
17.2
Correct difference 411.3
SIGNIFICANT FIGURES IN MULTIPLICATION AND DIVISION
When the values of different observations are multiplied or divided, the number of digits to be retained in the answer depends upon the number of significant figures in the weakest link.
Rule Donot retain a greater number of significant figures ina result computed from multiplication and/or division than the least number of significant figures in the data from which the result is computed.
THEORY OF ERRORS (PRECISION OF MEASUREMENT)
Foundation of a science, particularly of physics, is experiment. In physics, the discovery of a new law or principle is acceptable only, when the experiment approves it. In order to get as close to the truth as possible, physicists have not only been trying to design more and more perfect instruments but have also developed a theory of errors, which helps in eliminating possible errors in the observations.
The theory of error originates from the following two assumptions
1. That every observer performing an experiment is careless to some extent and is thus liable to commit mistake.
2. That every instrument used in an experiment is defective to some extent. Hence, the observation made with an instrument has got some inherent error.
in fact, errors and mistakes are two separate things. The term mistake is used to denote a gross error. It can be avoided by care on the part of the experimenter and should be avoided as the result cannot be corrected for them. On the other hand, uncertainty in a measurement is called an error. It tells the limits, within which the true value may lie.
ERROR OF MEASUREMENTS
There is only a limit up to which measurements can be made with a given measuring instrument. This limit is called the least count of the instrument. For example, a metre rod can measure length accurately up to 0-1 cm, whereas a vernier callipers can measure length accurately up to 0-01 cm. However, when we make use of various measuring instruments, various types of errors creep into the observations.
There can be the following possible types of errors in measurements
1. Constant error = When the result of a series of observations are in error by the same amount, the error is said to be a constant one. For example, in measuring the length of a cylinder by a vernier callipers whose graduations are faulty, say one centimetre on the scale is actually 0.9 centimetre, the measured length will always be greater than the true value by a constant amount.
2. Systematic error. = A systematic error is one that always produces an error off the same sign. These are due to known causes. This type of error is eliminated by detecting the source of the error and the rule governing this error. Systematic error may be subdivided into the following types :
i) Instrumental errors. = These are inherent errors of the apparatus and the measuring instruments used. A simple example is the zero-error of a measuring instrument. All instrumental errors come under this category. The instrumental error, r any, can be detected by interchanging two similar instruments or by using different methods for measuring the same physical quantity.
ii) Observational or personal errors. = The errors due to the personal peculiarities of the experimenter are known as personal errors. For example, parallax error While reading the positions of uprights on the optical bench. The more experience in a tne experimenter is, the lesser it is. hese errors can be minimized by obtaining several readings carefully and then taking their arithmetical mean.
iii) Errors due to external causes. = These errors are caused by external Conditions (pressure, temperature, wind, etc). For example, expansion of a scale due to increase in temperature. These errors can be taken care of by applying suitable corrections.
iv) Errors due to imperfection = Sometimes, even when we know the nature of the error, it cannot be eliminated due to imperfection in experimental arrangement. For example, in calorimetry, loss of heat due to radiation, the effect on weighing due to buoyancy of air, etc. These errors will always exist but observations can be corrected for them.
3. Random errors These errors are due to unknown causes and are sometimes termed as chance errors. In an experiment, even the same person repeating an observation may get different reading every time. For example, measuring diameter of a wire with a screw gauge, one may get different readings in different observations. If may happen due to many reasons. For example, due to non-uniform area of cross- section of the wire at different places, the screw might have been tightened unevenly in the different observations, etc. ln such a case, it may not be possible to indicate. which observation is most accurate. However, if we repeat the observation a number of times, the arithmetical mean of all the readings is found to be most accurate or very close to the most accurate reading for that observation. That is why, in an experiment, it is recommended to repeat an observation a number of times and then to take their arithmetical mean.
GROSS ERRORS (MISTAKES)
These are the results of sheer carelessness on the part of experimenter. No correction can be applied for them and hence must be avoided by exercising due care.
They are of the following types
1.Neglect of the sources of error. = This type of gross error results due to negligence towards sources of errors. For example, for plotting the field of a magnet, improper setting of the magnet along NS-line, presence of magnetic material in the vicinity of the magnet, etc.
2. Reading the instrument incorrectly. = Sometimes, over a metre scale, one cm is divided in 20 parts instead of 10 parts or in a voltameter or ammeter, 1 volt or 1 ampere might have been divided in 20 or 5 parts instead of 10. The experimenter may read the instrument without paying due attention to the value of 1 division.
3.Improper recording of the reading. = This type of mistake is committed by the experimenter, when he records the reading wrongly. For example, he may record 21-3in place of 23.1. It happens in undue haste or when the experimenter mentally carries the reading for a long time.
ABSOLUTE RELATIVE AND PERCENTAGE ERROR
Absolute error = The difference of the true value (standard value) and the experimental value (observed value) of a physical quantity is called absolute error
It is expressed in the units of measured quantity. Therefore,
Absolute error = true value - experimental value
Suppose in an experiment on determining the value of acceleration due to gravity, value comes out to be 945 cm s4. We know that the value of the acceleration due to gravity is generally taken as 980 cm s-2. Therefore,
absolute error in the value of acceleration due to gravity
= 980-945 = 35cm s²
Relative error = The knowledge of the relative error is more important than that of the absolute error in a measurement.
The relative error is defined as the ratio of the absolute error to the true value.
Therefore, relative error= absolute error / true value
It has no units. Therefore, in the above example,
the relative error in the value of acceleration due to gravity = 35/980 = 0.036
Percentage error = The relative error expressed as percentage is known as the percentage error.
Therefore, percentage error = absolute/ true value × 100
Therefore, in the above example, the percentage error in the value of acceleration due to gravity
= 35/980×100 = 3.6%
PERMISSIBLE ERROR IN A RESULT
Even when an experimenter has managed to avoid gross error by exercising due care and the instrumental errors have been avoided by selecting a perfect apparatus for the experiment, still another type of error may creep in the result of an experiment.It is due to the limits put on the measuring abilities of various kinds of instruments used, while performing the experiment owing to their least counts. Such an error is called the permissible error.
For example, suppose that the temperature of a liquid is read at 25-4°C on a the thermometer calibrated in degrees. If the reading is estimated to the nearest 0-1°C, the temperature should be recorded as 25.4 +0.1°C. This is the scientific way of recording a reading with the limits of error. In the present case, the temperature of liquid will be in the range 25-3°C to 25-5°C. Similarly, if the length of an object is read as 34-7 cm with a metre rod calibrated in cm and correct to the nearest 0-1 cm, then the length of the object should be recorded as 34.7+0:1 cm.
PROPAGATION OF ERROR
The final result of an experiment is calculated from a formula from the number of Observations for the various quantities taken with the help of different instruments. to H Shall see, each of these observed quantities does not influence the result to the same extent Some observation may influence the result more than what the other does.
We shall now calculate the maximum permissible error in different cases.
1.when the result involves the sum of two observed quantities.
2. When the result involves the difference of two observed quantities
3. When the result involves the product of two observed quantities
4. When the result involves the quotient of two observed quantities
5. When the result involves the product of the powers of observed quantities
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