Lecture 1 BEng24_5.pdf

LECTURE 1 Engineering Physics 1 PHS1005 TOPIC: MEASUREMENT AND ERROR ANALYSIS  2 Course Overview Assessments Test 1: 12.5% Week 7 Test 2: 12.5% Week 13 Tutorial Assignment 5% Weeks 2 -13 Project/Presentation 10% Week 13 Laboratory...

LECTURE 1 Engineering Physics 1 PHS1005 TOPIC: MEASUREMENT AND ERROR ANALYSIS  2 Course Overview Assessments Test 1: 12.5% Week 7 Test 2: 12.5% Week 13 Tutorial Assignment 5% Weeks 2 -13 Project/Presentation 10% Week 13 Laboratory Experiments: Units 1-7 20% Final Assessments: Units 1-7 40% **All assessments will be face to face 3 Major units of Syllabus Unit 1: Measurement & Errors Unit 2: Kinematics & Newton’s Laws Unit 3: Harmonic Motion Unit 4: Waves Unit 5: Mechanical Properties of Matter Unit 6: Conduction Unit 7: Thermodynamics 4 Important Information  TEXTBOOK AND REFERENCES: Giancoli, Douglas C. – Physics for Scientist and Engineers; Prentice Hall; Englewood Cliffs, New Jersey; (Latest Edition) Azzopardi F. & Stewart B., - ACCESSIBLE PHYSICS; Macmillan Press LTD Nelkon M. & Parker P.,-ADVANCE LEVEL PHYSICS; Heinemann Educational Publishers  Tutors for lab sessions and tutorials will contact each group. 5 What is an Error? Error (or uncertainty) is defined as the difference between a measured or estimated value of a quantity and its true value. Error is inherent in all measurements. Knowledge of the type and degree of error likely to be present is essential if data are to be used wisely, 6 Types of Errors in Measurements  Systematic Errors  Gross Errors  Random Errors 7 Systematic Errors Systematic errors are predictable and expected. Systematic errors will result in all readings being either above or below the accepted value. They affect each reading in the same way. 8 Systematic Errors Causes of Systematic Errors Include: ✓Non-zeroing of instruments. ✓Incorrect calibration of measuring instrument. ✓Instrument is wrongly used by the experimenter. 9 Minimization of Systemic Errors Systematic errors can be difficult to identify and correct. Given a particular experimental procedure and setup, it doesn't matter how many times you repeat and average your measurements; the error remains unchanged. No statistical analysis of the data set will eliminate a systematic error, or even alert you to its presence. Systematic error can be located and minimized with by comparing your results to other results obtained independently, using different equipment or techniques; or by trying out an experimental procedure on a known reference value, and adjusting the procedure until the desired result is obtained. 10 Random Errors  These are unpredictable variations between successive measurements made under apparently identical experimental conditions.  These errors affect the measurement in an unpredictable manner and fluctuate from one reading to another. 11 Random Errors Causes of Random Errors Include: ✓Imperfect observation. ✓Inability to reproduce some measurements due to a sudden change. 12 Minimization of Random Errors Random errors can usually be estimated and minimized through statistical analysis of repeated measurements. 13 Gross (Personal) Errors Gross errors can be defined as physical errors/mistakes in analysis or in calculating and recording measurements. They can be minimized by: Careful reading as well as a recording of information. Taking numerous readings of the instrument by different operators. 14 Precision and Accuracy Precision describes the degree to which several measurements of a single quantity provide results very close to each other. It also refers to the reproducibility of a measurement Accuracy describes the nearness of a measurement to the standard or true value. It also refers to the correctness of a measurement. 15 Precision and Accuracy  Both Precise and Accurate  Precise not Accurate  Neither Precise nor Accurate 16 What is an Error Analysis ? This is the process of evaluating the uncertainty (error) associated with a measured result and its subsequent use and propagation in calculations. The term error excludes gross errors/mistakes which can be put down to the individual. Mistakes can arise through:  Misreading scales.  Faulty arithmetic.  Trying to apply a theory where it does not apply.  Faulty transcription. 17 Sources of Error in Experiments Physical variations Parallax Instrument Drift Lag time Instrument Resolution Failure to calibrate or check for zero error Environmental Factors Personal errors 18 Least Count Errors This is a number that expresses the tolerance or range within which the exact (true) value lies. An estimate of the precision of the instrument used. Least count error depends upon the instrument. 19 Least Count Error for Measurements  For this module the least count error for a single measurement is taken as the smallest division on the scale of the instrument that is being used to take the measurement. 20 Example The length of a box could be expressed as: ℓ = (5.00 ± 0.20) cm OR ℓ = (4.90 ± 0.10) cm OR ℓ = (4.90 ±0.05) cm 21 Absolute Error for Repeated Measurements The absolute error in repeated measurements is taken as the mean deviation from the average. The following three (3) measurements were obtained: ℓ1 = 1.60 ± 0.05 m ℓ2 = 1.65 ± 0.05 m ℓ3 = 1.85 ± 0.05 m Calculate the absolute error in the measurements.  Absolute Error for Repeated Measurements 22 ℓ1 = (1.60 ± 0.05) m ℓ2 = (1.65 ± 0.05) m ℓ3 = (1.85 ± 0.05) m 1.60+1.65+1.85 Mean length 𝑙= = 1.70 𝑚 3 Deviations are: 1.60 – 1.70 = - 0.10 m 1.65 – 1.70 = - 0.05 m 1.85 – 1.70 = + 0.15 m 𝑠𝑢𝑚 𝑜𝑓 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛𝑠 𝑟𝑒𝑔𝑎𝑟𝑑𝑙𝑒𝑠𝑠 𝑜𝑓 𝑠𝑖𝑔𝑛𝑠 𝑚𝑒𝑎𝑛 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑒𝑎𝑑𝑖𝑛𝑔𝑠 0.10+0.05+0.15 = = 0.10 𝑚 3 Therefore ℓ = (1.70 ± 0.10) m 23 Fractional & Percentage Errors Fractional error = Absolute error. Measured Value Percentage error = Fractional error * 100 24 Relative Error  Relative error is an estimate of how far off a measurement is from the accepted or true value as a percentage of the measurement.  Accuracy is often reported quantitatively by using the relative error. Relative error = Measured Value – Expected Value Expected Value 25 Propagation of Errors Rules: 1) When quantities are added or subtracted their absolute errors add. 2) When quantities are multiplied or divided their fractional (and percentage) errors add. 26 Propagation of Errors Rule 1 y =a+b−c Absolute error in y is : y =  ( a + b + c )  Propagation of Errors 27 Given y = a + b – c, Calculate y, the absolute error in y and the fractional error in y a = (2.1 ± 0.2) mm b = (1.6 ±0.1) mm c = (0.50 ±0.05) mm Hence, y = 2.1 + 1.6 - 0.5 = 3.2 mm Absolute error in y, ∆y = 0.2 + 0.1 + 0.05 = ± 0.35mm The result is then y = (3.20 ± 0.35) mm 28 Propagation of Errors Rule 2 𝑎𝑏 𝑦= 𝑐 Fractional error in y is : y  a b c  =  + +  y  a b c   Propagation of Errors 29 𝑝𝑞 In calculating a quantity, z, using the formula 𝑧= 𝑠 one measures p = (7.5 ± 0.5) kg q = (4.0± 0.2) m s = (7.0 ±0.3) m Hence, Fractional error in z = ±(fractional error p + fractional error in q + fractional error in s) ∆𝑧 ∆𝑝 ∆𝑞 ∆𝑠 =± + + 𝑧 𝑝 𝑞 𝑠 ∆𝑧 0.5 0.2 0.3 =± + + 𝑧 7.5 4.0 7.0  Propagation of Errors 30 ∆𝑧 = ± (0.067 + 0.05 + 0.043) = ± 0.16 𝑧 Absolute error in z, ∆z = ± 0.16 × z ∆z = ±(0.16 × 4.3) = ± 0.7 kg The result is then z = 4.3 ± 0.7 kg  Propagation of Errors 31 In calculating a quantity, z, using the formula 𝑝𝑞 2 𝑧= 𝑠 one measures p = (7.5 ± 0.5) kg q = (4.0 ± 0.2) m s = (7.0 ± 0.3) m Hence, Fractional error in z = ± (fractional error p + 2(fractional error in q) + fractional error in s)  Propagation of Errors 32 In symbols ∆𝑧 ∆𝑝 2∆𝑞 ∆𝑠 =± + + 𝑧 𝑝 𝑞 𝑠 ∆𝑧 0.5 2 × 0.2 0.3 =± + + 𝑧 7.5 4.0 7.0 ∆𝑧 = ± [0.07 + 2(0.05) + 0.04] = ± 0.21 𝑧 Absolute error in z, ∆z = ± 0.21 × z ∆z = ± (0.21 × 17) = ± 3.6 kgm The result is then z = (17 ± 3.6) kgm  Propagation of Errors 33 Problem 1 Force = mass x acceleration Mass = 10 kg ± 1 kg Acceleration = 2.00 ms-2 ± 0.05 ms-2 1. What is the force? 2. What is the fractional error of the mass? 3. What is the fractional error of the acceleration? 4. What is the percentage error of the force? 5. What is the absolute error of force? 34 Propagation of Errors Rules 1 and 2 combined: a+b T= c Fractional error in T is : T  a + b c  =  +  T  a+b c  35 Propagation of Errors Absolute error in T is: 36 Propagation of Errors Rules 1 and 2 combined: x= ( a + b)c d Fractional error in x is : x  ( a + b ) c d  =  + +  x  ( a + b) c d  37 Problem 2 Given that a body has an initial velocity u = (200  10) ms-1, an acceleration a = (12  2) ms-2 and in covering a certain distance (s) it takes a time (t) = (6.0  0.2) s. Calculate (i) distance travelled if s = ut + at2/2 (ii) error in the distance travelled. 38 Solution to Problem 2 (i) s = ut + at2/2 s = (200 × 6.0) + (12) × (6.0)2/2 s = 1200 + 216 = 1416 m 39 (ii) ) s = ut + at2/2 let B = ut and C = at2/2 then s = (B + C) If B= ut then B/B = [ (u/u) + (t/t)] & B = 1200 m B = 1200[ (10/200) + (0.2/6.0)] B =  1200[ 0.05 + 0.033] B =  99.6 m 40 If C = ½ at2 then C/C =  [ (a/a) + 2(t/t)] & C = 216 m C =  216[ (2/12) + 2(0.2/6.0)] C =  216[ 0.17 + 0.07] =  51.84 m And because s = (B + C) s =  [ 99.6 + 51.84] =  151.44 m Calculation done without consideration of sig. figs 41 Significant Figures & Decimal Places Rules for significant figures All non-zero digits are significant When a zero is in between non-zero digits it is significant. The numbers 1001 and 3.007 both have 4 sig. figs. When a decimal point is in a number the first non- zero number and those after it are significant. The number 0.0052 has 2 sig. figs. 42 Significant Figures Examples: 5.000 L Count all the digits starting at the first non-zero digit on the left. 4 significant figures 0.005 m Count all the digits starting at the first non-zero digit on the left. 1 significant figure 43 Significant Figures When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant: 190 m may be 2 or 3 significant figures 50,600 m may be 3, 4, or 5 significant figures. To eliminate the uncertainty about the number of significant figures scientific notation is often used. In Scientific Notation all numbers before the multiplication sign are significant. 44 Significant Figures 7.100 × 103 has 4 s.f. 7.10 × 103 has 3 s.f. 7.1 × 103 has 2 s.f. 45 Use of significant figures for calculations For multiplication and division, the number of significant digits in an answer should equal the least number of significant digits in any one of the numbers being multiplied, divided etc. For addition and subtraction the number of decimal places in the answer should be the same as the least number of decimal places in any of the numbers being added or subtracted. 46 Significant Figures (S.F.): Rules for Multiplication and Division  For example, consider John's measurement of the rectangular piece of board, the area is:  Area = Length x Width (11.63 cm) x (5.74 cm) The input number with the smallest number of S.F. is the width measurement, which has 3 S.F. Therefore the answer must also have 3 S.F. Calculator result is 66.7562, rounded off to the correct number of S.F, Area = 66.8 cm2 47 Significant Figures (S.F.): Rules for Multiplication and Division If the board thickness is 0.42 cm, Find Volume. Volume = Length x Width x Thickness = (11.63 cm) x (5.74 cm) x (0.42 cm) The calculator gives 28.037604, but because the thickness has only 2 S.F., the result must have the same 2 S.F. Therefore, Volume = 28 cm3 48 Significant Figures: Rules for Addition and Subtraction For addition and subtraction, the accuracy with which you quote an answer does not depend directly on the number of S.F. in the input numbers as above. It is determined by the position of the least significant digit in any of the input numbers. 49 Significant Figures: Rules for Addition and Subtraction See examples below 2.34 2.34 2.34 2.34 2.341234 +0.18 +2.8 −2.32 +8.43 −2.020000 2.52 5.1 0.02 10.77 0.3003 50 The Two Greatest Mistakes Regarding Significant Digits Writing more digits in an answer (intermediate or final) than justified by the number of digits in the data. Rounding-off, say, to two digits in an intermediate answer, and then writing three digits in the final answer. 51 Calculators Calculators will not give you the right number of significant figures; they usually give too many but sometimes give too few (especially if there are trailing zeroes after a decimal point). The calculator shows the result of 2.0 / 3.0 = (0.6666) Use your physics sense. We recommend you not to round- off too early. 52 Calculators The calculator shows the result of 2.5 x 3.2 = (8) Use your physics sense. We recommend you not to round-off too early. 53 Experimental Results Reported experimental results should always include a realistic estimate of their error, either explicitly, as plus/minus an error value, or implicitly, using the appropriate number of significant figures. Furthermore, you need to include the reasoning and calculations that went into your error estimate, if it is to be plausible to others. 54 Experimental Results An explicit estimate of the error may be given either as a measurement plus/minus an absolute error, in the units of the measurement; fractional or relative error, expressed as plus/minus a fraction or percentage of the measurement. The advantage of the fractional error format is that it gives an idea of the relative importance of the error. A 10-gram error is a tiny 0.0125% of the weight of an 80- kg man, but is 33.3% of the weight of a 30-g mouse. 55 References  https://www.e- ucation.psu.edu/files/geog482/image/quality_sys_random.gif  http://www.physics.umd.edu/courses/Phys276/Hill/Information/Notes/Err orAnalysis.html  http://www.newport.com/servicesupport/Tutorials/default.aspx?id=86  http://www.wellesley.edu/Astronomy/kmcleod/Toolkit/sigfigs.html  http://www.ruf.rice.edu/~kekule/SignificantFigureRules1.pdf  https://courses.cit.cornell.edu/.../Minimizing_Systematic_Error.shtml  https://www.elprocus.com/what-are-errors-in-measurement-types-of- errors-with-calculation/