This article is on the Units and Measurements Class 11 Notes of Physics. The notes on Units and Measurements of class 11 physics have been prepared with great care keeping in mind the effectiveness of it for the students. This article provides the revision notes of the Units and Measurements chapter of Class 11 for the students so that they can give a quick glance of the chapter.
Units and Measurements
To measure a physical quantity we need some standard unit of that quantity. The measurement of the quantity is mentioned in two parts, the first part gives magnitude and the second part gives the name of the unit. Thus, suppose I say that the length of this wire is 5 meters. The numeric part 5 says that it is 5 times the unit of length and the second part meter says that the unit chosen here is the meter.
The basic physical quantities, which are independent of other quantities, are known as the fundamental quantities. For example, mass, length and time are considered to be the fundamental quantities. In the same manner, the units which can be derived from the fundamental units are known as derived units. In mechanics, virtually all quantities can be expressed in terms of mass, length and time. The main systems of units are given as follows:
(a) CGS or Centimetre, Gram, Second System
(b) FPS or Foot, Pound, Second System
(c) MKS or Metre, Kilogram, Second System
(d) SI system: Totally, there are seven basic or fundamental quantities in the international system of units called the SI system which can express all physical quantities including heat, optics and electricity and magnetism. We now provide these basic seven quantities with their units and symbols:
There are also two supplementary units used as radian (rad) for plane angle and steradian (sr) for solid angle. The above mentioned International System of Units (SI) is now extensively used in scientific measurements. However, the following practical units of length are also conveniently used and are expressed in terms of SI system of units.
(a) Micron is a small unit for measurement of length. 1 micron =1 µ = m 10-6 m
(b) Angstrom is a unit of length in which the size of an atom is measured and is used in atomic physics. 1 Angstrom=1Å=10–10 m.
(c) Light year is a unit of distance travelled by light in 1 year free space and is used in astrophysics.
1 Light year = 3 x 108 m / s 365 x 24 x 60 x 60 = × 9.5 x 1015 meters
(d) Fermi is a unit of distance in which the size of a nucleus is measured. 1 Fermi = 10-15 m
(e) Atomic mass unit: It is a unit of mass equal to 1/12th of the mass of carbon-12 atom.
1 atomic mass unit ≅ 1.67×10-27 kg
Note: There are only seven fundamental units. Apart from these, there are two supplementary units—plane angle (radian) and solid angle (steradian). By using these units, all other units can be derived. However, we need to know the fact that both radian and steradian have no dimensions.
All the physical quantities of interest can be derived from the base quantities. Thus, when a quantity is expressed in terms of the base quantities, it is written as a product of diﬀerent powers of the base quantities. Further, the exponent of a base quantity that enters into the expression is called the dimension of the quantity in that base. To make it clear, consider the physical quantity ”force.” As we shall learn later, force is equal to mass times acceleration. We know that acceleration is the change in velocity divided by time interval but velocity is length divided by time interval. Thus,
Force = Mass X Acceleration = Mass X Velocity/Time = Mass x (Length / Time)/Time = Mass x Length x (Time)-2
Thus, the dimensions of force are 1 in mass, 1 in length and –2 in time. The dimensions in all other base quantities are zero. Note, however, that in this type of calculation, the magnitudes are not considered. This is because only equality of the type of quantity is what that matters. Thus, change in velocity, average velocity, or final velocity all are equivalent in this discussion, as each one is expressed in terms of length/time.
USES OF DIMENSIONS
The major uses of dimensions are listed hereunder:
(a) Conversion from one system of units to another.
(b) To test and validate the correctness of a physical equation or formula.
(c) To derive a relationship between diﬀerent physical quantities in any physical phenomenon.
(d) Conversion from one system of units to another: If we consider n1 as numerical value of a physical quantity with dimensions a, b and c for units of mass, length and time as M1, L1, and T1, then the numerical value of the same quantity, n2 can be calculated for diﬀerent units of mass, length and time as M2, L2 and T2, respectively.
n2 = n1 [M1/M2]a [L1/L2]b [T1/T2]c
(e) To test and validate the correctness of a physical equation or formula: The principle of homogeneity requires that the dimensions of all the terms on both sides of physical equation or formula should be equal if the physical equation of any derived formula is correct.
(f) To derive a relationship between diﬀerent physical quantities in any physical phenomenon: Suppose that if a physical quantity depends upon a number of parameters whose dimensions are not known, then the principle of homogeneity of dimensions can be used. As we know that the dimensions of a correct dimensional equation are equal on both sides, it can be used to find the unknown dimensions of these parameters on which the physical quantity depends. Further, it can be used to derive the relationships between any physical quantity and its dependent parameters.
Sn = u + a/2(2n – 1)
Sn = Sn + Sn-1 = un + ½ (an2) – (n(n-1) + ½ a(n-1)2)
Sn = u(1) + ½ a(1)(2n – 1) = [u + a/2 (2n – 1)](1)
(We ignore ‘1’ in formula but it carries dimension of time.)
Where, n – dimension of time; u – dimension of velocity; s – dimension of displacement; and a – dimension of acceleration.
Significant figures in the measured value of a physical quantity provide information regarding the number of digits in which we have confidence. Thus, the larger the number of significant figures obtained in a measurement, the greater is the precision of the measurement. “All accurately known digits in a measurement plus the first uncertain digit together form significant figures.”
Rules for Counting Significant Figures
For counting significant figures, we make use of the rules listed hereunder:
(a) All non-zero digits are significant. For example, x = 2567 has clearly four significant figures.
(b) The zeroes appearing between two non-zero digits are counted in significant figures. For example, 6.028 has 4 significant figures.
(c) The zeroes located to the left of the last non-zero digit are not significant. For example, 0.0042 has two significant figures.
(d) In a number without decimal, zeroes located to the right of the non-zero digit are not significant. However, when some value is assigned on the basis of actual measurement, then the zeroes to the right non-zero digit become significant. For example, L = 20 m has two significant figures but x = 200 has only one significant figure.
(e) In a number with a decimal, zeroes located to the right of last non-zero digit are significant. For example, x = 1.400 has four significant figures.
(f) The power of ten is not counted as a significant digit(s). For example, 1.4 ×10-7 has only two significant figures, i.e., 1 and 4.
(g) Change in the units of measurement of a quantity, however, does not change the number of significant figures. For example, suppose the distance between two stations is 4067 m. It has four significant figures. The same distance can be expressed as 4.067 km or 4.067 ×105 cm. In all these expressions, however, the number of significant figures continues to be four.
Rounding Oﬀ a Digit
The rules for rounding oﬀ a measurement are listed hereunder:
(a) If the number lying to the right of the cut oﬀ digit is less than 5, then the cut oﬀ digit is retained as such. However, if it is more than 5, then the cut oﬀ digit is increased by 1. For example, x = 6.24 is rounded oﬀ to 6.2 (two significant digits) and x = 5.328 is rounded oﬀ to 5.33 (three
(b) If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit is increased by 1. For example, x = 14.252 is rounded oﬀ to x = 14.3 to three significant digits.
(c) If the digit to be dropped is simply 5 or 5 followed by zeroes, then the preceding digit is left unchanged if it is an even number. For example, x = 6.250 or x = 6.25 becomes x = 6.2 after rounding oﬀ to two significant digits.
(d) If the digit to be dropped is 5 or 5 followed by zeroes, then the preceding digit is raised by one if it is an odd number. For example, x = 6.350 or 6.35 becomes x = 6.4 after rounding oﬀ to two significant digits.
Algebraic Operations with Significant Figures
(a) Addition and subtraction: Suppose in the measured values to be added or subtracted, the least number of significant digits after the decimal is n. Then, in the sum or diﬀerence also, the number of significant digits after the decimal should be n. Example: Suppose that we have to find the sum of number 420.42 m, 420.4m and 0.402m by arithmetic addition
(b) Multiplication or division:
Suppose in the measured values to be multiplied or divided, the least number of significant digits is n; then, in the product or quotient, the number of significant digits should also be n. Example: 1.2 x 36.72 = 44.064 ≈ 44.
In the example shown, the least number of significant digits in the measured values is two. Hence, the result when rounded oﬀ to two significant digits becomes 44. Therefore, the answer is 44.
Example: 1100ms-1/10.2ms-1 =107.8431373 ≈ 108
We define the uncertainty in a measurement as an ‘error’. By this we mean the diﬀerence between the measured and the true values of a physical quantity under investigation. There are three possible ways of calculating an error as listed hereunder:
(i) Absolute error (ii) Relative error (iii) Percentage error
Let us consider a physical quantity measured by taking a repeated number of observations say x1, x2, x3, x4,….. if <x> be the average value of the measurement, then the error in the respective measurement is
∆x1 = x1 – <x>, ∆x2 = x2 – <x>……; ∆x = |xexperimental value – xtrue value|
However, if we take the arithmetic mean of all absolute errors, then we obtain the final absolute error ∆xmean. When arithmetic mean alone is considered, then only the magnitudes of the absolute errors are taken into account.
∆xmean = |∆x1| + |∆x2| +……. +|∆xn|/n = 1/n ∑ |∆xi|
It then follows clearly from the above discussion that any single measurement of x has to be such that
xmean – ∆xmean ≤ x ≤ xmean + ∆xmean
Relative error = ∆xmean/xmean; percentage error = (∆xmean/xmean) x 100
PROPAGATION OF ERRORS
Addition and Subtraction
If x = A ± B ; then ∆x = ∆A + ∆B
i.e., for both addition and subtraction, the absolute errors are to be added up. The percentage error, then, in the value of x is
Percentage error in the value of x = (∆A + ∆B/ A ± B) x 100%
Multiplication and Division
If y = AB or y = A/B then, ∆y/y = ∆A/A + ∆B/B => (∆y/y) x 100% = (∆A/A) x 100% + (∆B/B) x 100%
⇒Percentage error in the value of y= percentage error in the value of A + percentage error in the value of B
If y = k Al Bm/Cn then ∆y/y = l (∆A/A) + m (∆B/B) + n (∆C/C)
(Percentage error in the value of y) = l (Percentage error in the value of A) + m (Percentage error in the value of B) + n (Percentage error in the value of C)
This article has tried to highlight all the important points of Units and Measurements in the form of notes for class 11 students in order to understand the basic concepts of the chapter. The notes on Units and Measurements have not only been prepared for class 11 but also for the different competitive exams such as iit jee, neet, etc.