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Session of Units and Dimensions for Arc-IITJEE study program

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- 1. Physics
- 2. Phys - 1 Units And Measurements
- 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. Session Objectives
- 5. Session Objective Standard and Units Dimensions Errors Significant Figures Accuracy and Precision Dimensional Analysis
- 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. 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. 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. 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. Dimension • Quantities with same dimensions only can be added • Power of dimension on both sides of an equation must match
- 11. Questions
- 12. Class Exercise - 7 Dimensionally, specific heat is proportional to dimension of mass as (a)[M0 ] (b) [M1 ] (c) [M–1 ] (d) [M2 ]
- 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. 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. Questions
- 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. 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. 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. 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. Class Exercise - 10 Check dimensionally if the relation is correct.= + 21 s ut at 2
- 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. 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. 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. 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. Questions
- 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. 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. (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. 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. 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. Questions
- 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. 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. 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. 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. Solution - 1 In powers of 10 placed as: 212.2 = 2.122 × 102 ,,102 is not significant. Ans. c
- 37. Thank you

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