# Units and Measurement

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Session of Units and Dimensions for Arc-IITJEE study program

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### Units and Measurement

1. 1. Physics
2. 2. Phys - 1 Units And Measurements
3. 3. Session Opener We know an elephant is heavier than a feather Physics wants to know by how many time, by what standards and with what accuracy.
4. 4. Session Objectives
5. 5. Session Objective Standard and Units Dimensions Errors Significant Figures Accuracy and Precision Dimensional Analysis
6. 6. 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.
7. 7. Standards and Units Physical quantity (q) given by a number and a unit. q = n . u n : pure number. u : unit of the standard. 1 n u ∝ Because q is the same whatever be the standard
8. 8. Basic physical quantity Name of SI unit Symbol of SI unit 1. Length Meter m 2. Mass Kilogram kg 3. Time Second s 4. Electric current Ampere A 5. Temperature Kelvin K 6. Luminous intensity Candela cd 7. Amount of substance Mole mol SI (International system) units
9. 9. Dimensions of physical quantities Number of times a fundamental quantity is repeated in physical quantity q q M L T .............. α β γ =               Volume is 3 dimensional in length 3 V L=       Area is 2 dimensional in length 2 A L=       a b c
10. 10. Dimension • Quantities with same dimensions only can be added • Power of dimension on both sides of an equation must match
11. 11. Questions
12. 12. Class Exercise - 7 Dimensionally, specific heat is proportional to dimension of mass as (a)[M0 ] (b) [M1 ] (c) [M–1 ] (d) [M2 ]
13. 13. Solution - 7 Specific heat is (dimensionally) Heat (energy) per unit mass per unit temperature (q) Then, = ×θ [Energy] [C] [M ] = = θ θ –2 0 2 –2 –1[MLT ] [M L T ] [M ]
14. 14. Dimensional Analysis To test whether a relation is wrong. For interconversion of units To justify /derive interrelation of quantities. Dimensional analysis is a powerful method
15. 15. Questions
16. 16. 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 (θ)
17. 17. Solution - 8 Time period = T Then α β γ =   l[T] m g Relation with θ cannot be found dimensionally. ( ) γ α β α β+γ γ   = × =     –2 –2M L LT M L Tα β γ ⇒ =   l0 0 1M L T m g ∴α = γ = β = γ0, –2 1, – γ = β = 1 1 or – , 2 2 = l So T Constant g
18. 18. Class Exercise - 9 What is the value of a force of 10 N in a system with fundamental units of centimetre, gram and hour?
19. 19. Solution - 9 q = n1u1 = n2u2 10 N = n2 new unit ∴ =2 Newton n 10 New unit × × ⇒ = × × –2 2 –2 kg m s n 10 g cm hr     =  ÷ ÷ ÷ ÷ ÷    23 210 g 10 cm hr 10 1 g 1 cm s = 106 × 60 × 60 = 3.6 × 109
20. 20. Class Exercise - 10 Check dimensionally if the relation is correct.= + 21 s ut at 2
21. 21. Solution - 10 Dimension of left-hand side (s) = [M0 L1 T0 ] On right-hand side: ut = Velocity × Time = [M0 L1 T–1 ][T] = [M0 L1 T0 ] same as LHS    = = × =     2 2 0 1 –2 21 1 at Acceleration Time M L T T 2 2    ÷   1 has no dimension 2 = [M0 L1 T0 ] [Same as LHS] Equation is dimensionally correct.
22. 22. Errors An observation is limited by the least count of instrument Measured value qm = qreal ± ∆q Exact value of qreal is not known Only mean value of q can be found
23. 23. Errors Random errors are expected when several observations (qi) are made =σ = − ∑ n 2 i i 1 x RMS error : n(n 1) = = → = − ∑ r r n i i 1 i i i 1 Meanq q n Error in q x q q
24. 24. Errors In sums and differences, ABSOLUTE ERRORS are added A ± B = C C ± ∆C = A ± B ± (∆ A + ∆B) In products or quotients,RELATIVE ERRORS are added A B C C A B C A B × = ∆ ∆ ∆  ± = ± + ÷   ∆ = × o o C percentage error 100 C = ∆ ∆ ∆  ± = ± + ÷   A C B C A B C A B
25. 25. Questions
26. 26. 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 + + 1 1 1 (c) x y z ( )+ +xy yz xz (d) x y z Percentage errors are added in a product. Solution :- b
27. 27. 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 (a)8% (b) 1.8% (c) 0.8% (d) 0.1% Solution Total time measured is important and not time period. So percentage error = × = 0.2 100% 0.8% 25
28. 28. (i) Accuracy Sign has to be retained while expressing accuracy. Accuracy : degree of agreement of a measurement with the true (accepted) value. Accuracy and Precision (ii) Precision Precision is expressed without any sign. Precision : degree of agreement between two or more measurements done in an identical manner.
29. 29. 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.
30. 30. i. All non-zero digits are significant. Rules to determine significant figures iv Zeroes to the right of the decimal point are significant. iii. Zeroes between non-zero digits are significant. ii. Zeroes to the left of the first non-zero digit are not significant.
31. 31. Questions
32. 32. Class Exercise - 2 Which of the following, in the measurement of length, is most accurate? (a) 2 × 102 m (b) 200.0 m (c) 20 × 102 m (d) 200 m 200.0 has four significant figures, which is maximum in the group. Solution :-
33. 33. 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.
34. 34. 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
35. 35. 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.
36. 36. Solution - 1 In powers of 10 placed as: 212.2 = 2.122 × 102 ,,102 is not significant. Ans. c
37. 37. Thank you