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Session Objective Standard and Units Dimensions Errors Significant Figures Accuracy and Precision Dimensional Analysis
Standards and Units Laws of physics : expressed in terms of physical quantities Physical quantities : expressed in terms of fundamental quantities. Fundamental quantities : defined by measurements and expressed by standards. Measurements : comparison with a standard. Standards are defined and universally accepted by competent authority.
Standards and Units Physical quantity (q) given by a number and a unit. q = n . u n : pure number. u : unit of the standard. Because q is the same whatever be the standard
SI (International system) units mol Mole Amount of substance 7. cd Candela Luminous intensity 6. K Kelvin Temperature 5. A Ampere Electric current 4. s Second Time 3. kg Kilogram Mass 2. m Meter Length 1. Symbol of SI unit Name of SI unit Basic physical quantity
Dimensions of physical quantities Number of times a fundamental quantity is repeated in physical quantity q Volume is 3 dimensional in length Area is 2 dimensional in length a b c
Dimension • Quantities with same dimensions only can be added • Power of dimension on both sides of an equation must match
Class Exercise - 8 Show dimensionally which of the following physical quantities have an influence on the time period of a simple pendulum? (i) Mass of the bob (ii) Length of the string (l) (iii) Acceleration due to gravity (g) and (iv) Angular displacement ( )
Solution - 8 Time period = T Then Relation with cannot be found dimensionally.
Class Exercise - 9 What is the value of a force of 10 N in a system with fundamental units of centimetre, gram and hour?
Class Exercise - 10 Check dimensionally if the relation is correct.
Solution - 10 Dimension of left-hand side (s) = [M 0 L 1 T 0 ] On right-hand side: ut = Velocity × Time = [M 0 L 1 T –1 ][T] = [M 0 L 1 T 0 ] same as LHS = [M 0 L 1 T 0 ] [Same as LHS] Equation is dimensionally correct.
Errors An observation is limited by the least count of instrument Measured value q m = q real q Exact value of q real is not known Only mean value of q can be found
Errors Random errors are expected when several observations (q i ) are made
Errors In products or quotients, RELATIVE ERRORS are added In sums and differences, ABSOLUTE ERRORS are added A B = C C C = A B ( A + B)
Class Exercise - 3 The percentage errors of X, Y, X are x, y and z respectively. The total percentage error in the product XYZ is (a) xyz (b) x + y + z Percentage errors are added in a product. Solution :- b
Class Exercise - 6 The least count of a stop watch is 0.2 s. The time of 20 oscillations of a pendulum is measured to be 25 s. The percentage error in the measurement of time is
8% (b) 1.8%
(c) 0.8% (d) 0.1%
Solution Total time measured is important and not time period. So percentage error
Accuracy and Precision (i) Accuracy Sign has to be retained while expressing accuracy. Accuracy : degree of agreement of a measurement with the true (accepted) value. (ii) Precision Precision is expressed without any sign. Precision : degree of agreement between two or more measurements done in an identical manner.
Significant figures Significant figures in 1.007, 12.012 and 10.070 are 4, 5 and 5 respectively. Significant figures are the meaningful digits in a measured or calculated quantity.
Class Exercise - 2 Which of the following, in the measurement of length, is most accurate? (a) 2 × 10 2 m (b) 200.0 m (c) 20 × 10 2 m (d) 200 m 200.0 has four significant figures, which is maximum in the group. Solution :-
Class Exercise - 5 Which of the following measurements is most precise? (a) 2345 m (b) 234.5 m (c) 23.45 m (d) 2.345 m Solution - d 2.345 m measures till the smallest fraction of a meter.
Class Exercise - 4 With regard to the significant figures, (12.5) 2 is equal to (a) 156.250 (b) 156.25 (c) 156.2 (d) 156 (12.5) 2 = 156.25. But as only three significant figures are to be considered, 156 is the right answer. Solution
Class Exercise - 1 Which of the following statements is false among the statements given below? (a) All non-zero digits are significant. (b) Zeroes in the middle of a numerical expression are significant, while those immediately following a decimal point are not. (c) While counting the number of significant figures, the powers of 10 are to be considered. (d) Greater the number of significant figures in a measurement, smaller is the percentage error.
Solution - 1 In powers of 10 placed as: 212.2 = 2.122 × 10 2 , , 10 2 is not significant. Ans. c