NCERT 11th Class : Units & Measurement - II

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Significant Figures : In case of a measured value of a physical quantity the reliable digits plus the first uncertain digit are known as significant digits or significant figures. Significant figures indicates the precision of measurement. A choice of change of different units does not change the number of significant digits or figures in a measurement.

Rules for Finding Significant digits

1. All the non - zero digits are significant.
2. All the zeros between two non-zero digits are significant, no matter where the decimal point is, if at all. eg. 2003.25 has got 6 significant figures.
3. 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 0.002308, the underlined zeroes are not significant.
4. The terminal or trailing zero(s) in a number without a decimal point are not significant. eg. 123m = 12300cm = 123000mm has only 3 significant figures.
5. The trailing zero(s) in a number with a decimal point are significant. eg. 0.3500 and 0.006400 has got 4 significant figures.

Rules for arithmetic operations

1. In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.
2. 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.

Rounding off uncertain digits

1. The rule by convention is that the preceding digit is raised by 1 if the insignificant digit to be dropped is more than 5, and is left unchanged if the latter is less than 5. For eg. 1.24 becomes 1.2 and 1.27 becomes 1.3.
2. If the insignificant digit to be dropped is 5 then if the preceding digit is even, the insignificant digit is simply dropped and, if it is odd, the preceding digit is raised by 1. For eg. 1.245 becomes 1.24 and 1.235 becomes 1.24.

To calculate the uncertainty, below process should be used. 

1. Add a lowest amount of uncertainty in the original numbers. Example uncertainty for 3.2 will be ± 0.1 and for 3.22 will be ± 0.01. 
2. Calculate these in percentage also. After the calculations, the uncertainties get multiplied/divided/added/subtracted. 
3. Round off the decimal place in the uncertainty to get the final uncertainty result. 

Example, for a rectangle, if length l = 16.2 cm and breadth b = 10.1 cm 
Then, take l = 16.2 ± 0.1 cm or 16.2 cm ± 0.6% and breadth = 10.1 ± 0.1 cm or 10.1 cm ± 1%. 
On Multiplication, area = length x breadth = 163.62 cm² ± 1.6% or 163.62 ± 2.6 cm². 
Therefore after rounding off, area = 164 ± 3 cm². Hence 3 cm2 is the uncertainty or the error in estimation.

Dimensions of Physical Quantities : The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity. All derived units can be expressed as a combination of seven fundamental or base quantities. (M L T A K cd mol).

For example dimension for force F = mass x acceleration
Mass - M,  Acceleration - LT^-2
Dimension of force = M L T^-2

Dimensional Analysis : A thorough understanding of dimensional analysis helps us in deducing certain relations among different physical quantities and checking the derivation, accuracy and dimensional consistency or homogeneity of various mathematical expressions. Physical quantities represented by symbols on both sides of a mathematical equation must have the same dimensions. Only physical quantities having same dimension can be added and subtracted. This is called the principle of homogeneity of dimensions.

Applications of Dimensional Analysis

1. Checking Dimensional Consistency : A dimensionally correct equation must have same dimensions on both sides of the equation. A dimensionally correct equation need not be a correct equation but a dimensionally incorrect equation is always wrong. It can test dimensional validity but not find exact relationship between the physical quantities.
   For example, S = ut + 0.5at²
   Dimensionally, [L] = [L] + [LT^-1][T] + [LT^-2][T²]
   Dimensions on both sides will be [L] as [T] gets cancelled out. Hence this is dimensionally correct equation.
2. Deducing Relations Among Physical Quantities : To deduce relation among physical quantities, we should know the dependence of one quantity over others (or independent variables) and consider it as product type of dependence. Dimensionless constants cannot be obtained using this method.
   Example consider equation for time period of spring T = k l^xg^ym^z
   Dimensionally, [L⁰M⁰T¹] = [L¹]^x  [L¹T^-2]^y [M¹]^z= [L^x+yT^-2y M^z]
   Equating x + y = 0, -2y = 1, z= 0
   i.e. x = 1/2, y = -1/2, z = 0
   T = k l^1/2g^-1/2
   T = k√l/g