CBSE Class 11 Physics notes on Chapter 2, Units and Measurement (Part 3) are available here. Important topics covered in this part are Accuracy, Precision, Errors of Measurements, Types of Errors, Absolute Error, Relative Error and Percentage Error, Combination of Errors, Significant Figures, Rules for Arithmetic Operations with Significant Figures. Other important topics such as Units, Types of Units (Fundamental Units & Supplementary Units), Measurements of Length, Mass & Time etc are already covered in Part 1 and Part 2. These notes are important for the preparation of CBSE Class 11 exams and competitive exams like JEE Main, NEET etc.
The notes are given below:
Accuracy is the extent to which a reported measurement approaches the true value of the quantity measured.
Problems with accuracy are generally due to human error (while reading an instrument), instrument error (error is design), lack of calibration etc.
As, a person reduce errors, measurement accuracy increases.
Precision is the degree of exactness or refinement of a measurement. Basically it describes the limitations of the measuring instrument.
Example: A measurement of 5.382m is more accurate than measurement of 5.3 m.
Lack of precision is typically due to limitations of the measuring instrument and is not the result of human error or lack of calibration.
Note: Measurement’s precision is determined by least count of the measuring instrument. Smaller the least count, greater is the precision.
Measurement is the foundation of all experimental science and technology. The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error.
Errors of Measurements
It is the difference between true value and the measured of value of quantity is known as error of measurement.
Types of Errors:
In general, the errors in measurement can be broadly classified as (a) systematic errors and (b) random errors (c) Least count error
The systematic errors are those errors that tend to be in one direction, either positive or negative. Basically, these are the errors whose causes are known.
(a) Instrumental errors:
These errors arise from the errors due to imperfect design or calibration of the measuring instrument, zero error in the instrument, etc.
Temperature graduations of a thermometer may be inadequately calibrated [it may read 103 °C at the boiling point of water at STP (Standard Temperature & Pressure) whereas it should read 100 °C).
In a vernier calliper the zero mark of vernier scale may not coincide with the zero mark of the main scale, or simply an ordinary metre scale may be worn off at one end.
(b) Imperfection in experimental technique or procedure
To determine the temperature of a human body, a thermometer placed under the armpit will always give a temperature lower than the actual value of the body temperature.
Other external conditions (such as changes in humidity, temperature, wind velocity, etc.) during the experiment may systematically affect the measurement.
(c) Personal errors
Such errors arise due to an individual’s bias, lack of proper setting of the apparatus or individual’s carelessness in taking observations without observing proper precautions, etc. For example, if you, by habit, always hold your head a bit too far to the right while reading the position of a needle on the scale, you will introduce an error due to parallax.
The random errors are those errors, which occur irregularly and hence are random with respect to sign and size. These can arise due to random and unpredictable fluctuations in experimental conditions (e.g. unpredictable fluctuations in temperature, voltage supply, mechanical vibrations of experimental set-ups, etc), personal (unbiased) errors by the observer taking readings, etc. For example, when the same person repeats the same observation, it is very likely that he may get different readings everytime.
Least count error
Least count is the smallest value that can be measured by the measuring instrument.
Least count error is the error associated with the resolution of the instrument.
For example, a vernier callipers has the least count as 0.01 cm; a spherometer may have a least count of 0.001 cm.
Least count error belongs to the category of random errors but within a limited size; it occurs with both systematic &random errors.
If we use a metre scale for measurement of length, it may have graduations at 1 mm division scale spacing or interval.
Absolute Error, Relative Error and Percentage Error
Let the values obtained in several measurements are a1, a2, a3...., an.
The arithmetic mean of these values is taken as the best possible value of the quantity under the given conditions of measurement as,
amean = (a1 + a2 + a3 … + an)/n
The magnitude of the difference between the true value of the quantity and the individual measurement value is called the absolute error of the measurement.
This is denoted by | Δa |.
Note: In the absence of any other method of knowing true value, we considered arithmatic mean as the true value.
Then the errors in the individual measurement values are
Δa1 = amean – a1,
Δa2 = amean – a2,
.... .... ....
.... .... ....
Δan = amean – an
The Δa calculated above may be positive in certain cases and negative in some other cases. But absolute error |Δa| will always be positive.
Mean absolute error
It is the arithmetic mean of the magnitude of absolute errors in all the measurement of the quantity. It is generally represented by Δamean
If we do a single measurement, the value we get may be in the range amean ± Δamean
i.e., a = amean ± Δamean
The relative error is the ratio of the mean absolute error Δamean to the mean value amean of the quantity measured.
i.e., Relative error = Δamean/ amean
Percentage error, δa = (Δamean/amean) × 100%
Combination of Errors
(a) Error of a sum or a difference
When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.
Z = A + B
We have by addition, Z ± ΔZ = (A ± ΔA) + (B ± ΔB).
The maximum possible error in Z
ΔZ = ΔA + ΔB
For the difference Z = A – B, we have
Z ± Δ Z = (A ± ΔA) – (B ± ΔB) = (A – B) ± ΔA ± ΔB
or, ± ΔZ = ± ΔA ± ΔB
The maximum value of the error ΔZ is again ΔA + ΔB.
(b) Error of a product or a quotient
When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.
Suppose Z = AB and the measured values of A and B are A ± ΔA and B ± ΔB. Then
Z ± ΔZ = (A ± ΔA) (B ± ΔB) = AB ± B ΔA ± A ΔB ± ΔA ΔB.
Dividing LHS by Z and RHS by AB we have,
1 ± (ΔZ/Z) = 1 ± (ΔA/A) ± (ΔB/B) ± (ΔA/A)(ΔB/B).
Since ΔA and ΔB are small, we shall ignore their product.
Hence the maximum relative error
ΔZ/ Z = (ΔA/A) + (ΔB/B).
(c) Error in case of a measured quantity raised to a power
The relative error in a physical quantity raised to the power k is the k times the relative error in the individual quantity.
Suppose Z = A2,
ΔZ/Z = (ΔA/A) + (ΔA/A) = 2 (ΔA/A).
Hence, the relative error in A2 is two times the error in A.
In general, if Z = (Ap Bq)/Cr
ΔZ/Z = p (ΔA/A) + q (ΔB/B) + r (ΔC/C).
Significant figures in the measured value of a physical quantity tell the number of digits in which we have confidence.
Larger the number of significant figures obtained in a measurement, greater is the accuracy of the measurement. The reverse is also true.
Important Rules of counting significant figures
• All the non-zero digits are significant.
• All the zeros between two non-zero digits are significant, no matter where the decimal point is, if at all.
• If the number is less than 1, the zero(s) on the right of decimal point but to the left of the first non-zero digit are not significant.
[In 0.00 2308, the underlined zeroes are not significant].
• The terminal or trailing zero(s) in a number without a decimal point are not significant.
[Thus 123 m = 12300 cm = 123000 mm has three significant figures, the trailing zero(s) being not significant.] However, you can also see the next observation.
• The trailing zero(s) in a number with a decimal point are significant.
[The numbers 3.500 or 0.06900 have four significant figures each.]
• For a number greater than 1, without any decimal, the trailing zero(s) are not significant.
• For a number with a decimal, the trailing zero(s) are significant.
Rules for Arithmetic Operations with Significant Figures
(1) In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures.
(2) In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.
Rounding off the Uncertain Digits
(i) If the digit dropped is less than 5, then the preceding digit is left unchanged.
(ii) If the digit to be dropped is more than 5, then the preceding digit is raised by one.
(iii) If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit is raised by one.
(iv) If the digit to be dropped is 5 or 5 followed by zeroes, then the preceding digit isleft unchanged, if it is even.
(v) If the digit to be dropped is 5 or 5 followed by zeroes then the preceding digit is raised by one, if it is odd.
Rules for Determining the Uncertainty in the Results of Arithmetic Calculations
Rules for determining the uncertainty in the results of arithmetic calculations can be understood with the following examples:
(i) Suppose, l = 16.2 ± 0.1 cm = 16.2 cm ± 0.6 % and b = 10.1 ± 0.1 cm
= 10.1 cm ± 1 %
Then, the error of the product of two (or more) experimental values, using the combination of errors rule, will be
l b = 163.62 cm2 + 1.6% = 163.62 + 2.6 cm2
This leads us to quote the final result as
l b = 164 + 3 cm2
Here, 3 cm2 is the uncertainty or error in the estimation of area of rectangular sheet.
(ii) If a set of experimental data is specified to n significant figures, a result obtained by combining the data will also be valid to n significant figures.
Example: 12.9 g – 7.06 g, both specified to three significant figures, can’t properly be evaluated as 5.84 g but only as 5.8 g, as uncertainties in subtraction or addition combine in a different fashion (smallest number of decimal places rather than the number of significant figures in any of the number added or subtracted).
(iii) The relative error of a value of number specified to significant figures depends not only on n but also on the number itself.
Example: the accuracy in measurement of mass 1.02 g is ± 0.01 g whereas another measurement 9.89 g is also accurate to ± 0.01 g.
The relative error in 1.02 g is = (± 0.01/1.02) × 100 % = ± 1%
Similarly, the relative error in 9.89 g is = (± 0.01/9.89) × 100 % = ± 0.1 %
Note: Intermediate results in a multi-step computation should be calculated to one more significant figure in every measurement than the number of digits in the least precise measurement.