Part-III Treatment of Data 1
OVERVIEW (1) Units of measurement (a) must be indicated in tables/graphs. (b) use scientific notation Examples µ 2.2x10 -6 2.2 V V 7.1x10 -2 71 m m m 8.34x10 4 83.4 kJ J 2.1x10 5 0.21 MW W 7.3x10 -11 73 pF F
Tabulation of Data Whether one is measuring the distance to the moon using laser interferometry or measuring the soil strength using a penetrometer or bond angles/lengths using XRD, carefully made (and recorded) observations are the cornerstone of a good experimental work. (i) Repeat measurements These are made even when the quantity (or the value) is NOT varying. Timing the fall of an object through a given distance or measuring the wavelength of light emitted from a lamp containing helium gas. (ii) Relationship between two variables : so assign y = f(x) different values to as , and measure , y y ...... ...... x x x 2 1 2 1
Table: Fall times (t) for an object to fall in air through 25 m (ambient temperature 298 K, 3 pm, 17-09-2013). t(s) 2.2 2.0 2.6 2.1 1.9 2.2 2.4 2.0 2.3 2.3 (3) Use Scientific Notation ≅ 2x10 -6 For example, 2 2x10 -3 mg g kg Pressure values (Pa) : 1.01x10 5 1.05x10 5 1.03x10 5 1.01x10 5 9.9x10 4 9.83x10 4 Table : Measured values of pressure. Pressure(x10 5 ) 1.03 1.01 0.99 0.983 1.01 1.05 Pa or Pressure( ) 103 101 99 98.3 101 105 kPa
(4) Uncertainties in measurements Despite our best efforts and/or the quality of instruments “Variability”/”uncertainty” in experimental data. graduation of a thermometer ∴ there is nothing called “Exact” measurement. (One can only do the best possible !) Fluctuating nature of the mercury in a thermometer ∼ ± 0.5 . o C
There is an uncertainty of 20 ± 0.5 o C What does it mean ? What you mean is that the temperature could be anything from 19.5 to 20.5 , with the o C o C possible average value of 20 ! o C 6
One must always include uncertainty estimates. Table : Dependence of electrical resistance on temperature of a Copper wire (Kirkup : Experimental Methods, Wiley 1994). T Electrical Resistance (K) ± 0.5K Ω ±0.001 Ω 281 0.208 289.5 0.213 296.5 0.222 305.0 0.229 313.5 0.232 327.5 0.243
OTHER FACTORS � Compounding of uncertainties? � Does uncertainty depend upon the magnitude of the measurement ? 8
(5) Significant Figures If a value is recorded as 5.1 2 your experiment is able to distinguish between 5.1 1 and 5.1 3 Similarly, 5.123 5.122 & 5.124 5.12 3 significant digits 5.123 4 significant digits Resist the temptation to record all the figures given by an instrument display or a calculator or a computer. Let us look at the value of 0.0020409
how many significant figures are there ? between the first non-zero and the last digit inclusive five significant figures Exercise 2.564 four 0.00489 three 64000 ? 1.20 three 1.2 two 0.20000878 eight
Rounding off Numbers � In calculations of derived variables, rounding off is needed. � How to do it ? 11
Reduce to 1.8671132 1.8 or 1.9 2 SF Reduce to 1.87 3 SF Reduce to 1.867 4 SF CALCULATION & SIGNIFICANT FIGURES Let us say, for a cylinder, D = 7.9 mm L = 1.5 mm π π 2 2 ( ) D 7.9 X- Sectional area, A = = 4 4 2 A = 49.016699mm Is this sensible ? Obviously not. Why ?
Well, if D is known to 2 significant figures, area can not be calculated to 8 significant figures !! 2 A = 49mm π 2 D Volume of cylinder = L 4 π 2 ( ) = × 7.9 ×1.5 4 3 = 73.525049mm ∴ V = 73 3 mm
Guidelines for rounding off Rule # 1 :Addition/Subtraction Round off the result to the same number of decimal places = least number of decimal places of the constituents Ex. 11.39 – 7.897 + 12.3538 = 15.8468 Correct answer is 15.85 14
Rule # 2 : Multiplication /Division The number of significant digits in the answer = least number of significant digits in the primary numbers. 3.17×3.393×3.3937 = 36.501992 3 significant 8 significant figures figures 36.5 CAVEAT Volume = Area×L 3 = 49×1.5 = 73.5mm One might be tempted to round it up to 74 ! 3 mm Do not round off the intermediate results
PROBLEM In an experiment, the density ( ρ ) of a metallic sphere is to be measured. Density ρ is defined as mass per unit volume , i.e., m/v. π 3 d V = 6 Lab notebook has the following entries Mass of sphere = 0.44 g Diameter of sphere = 4.76 mm π π 3 3 ( ) d 4.76 3 V = = = 451.761761m m 6 6 0.44 ρ ∴ -4 = = 9.739647 ×10 451.761761 Does it make sense ?
PRESENTATION OF DATA (Graphs, bar/ pie charts) 1. Purpose Graphs: our visual ability is very strong to detect trends as opposed to tabulated information. x y 9.3 19.0 7.3 15.0 9.8 20 1.8 4 5.3 11
Graph/plot can be used to show the: a) Range of data b) Uncertainty in each measurement c) Trend or the absence of a trend d) Data which do not fall in line with the majority of data points.
DEPENDENT/INDEPENDENT VARIABLES Horizontal axis (Independent variable) 2-D plots Vertical axis (Dependent variable) Independent variables : controlled or which is varied systematically P P can be varied through a high pressure gas line P Independent variable Q Dependent variable Q
Effect of temperature on solubility of salt in water Amount of water = 1 m 3 Mass of salt which can be dissolved = M kg ∴ solubility, S = M/1 (kg/m 3 ) T (K) S (kg/m 3 ) S 1 298 S 2 315 S 3 325 Independent variable Dependent variable
Example: Effect of temperature on the length of a wire T (K) L (m) 273 1.1155 298 1.1164 323 1.1170 348 1.1172 373 1.1180 398 1.1190 423 1.1199 448 1.1210 473 1.1213 498 1.1223 523 1.1223
(3) ORIGINS Is it necessary to have a (0,0) point on both axes ? Re-plot the T-L data This is not a good graph ! Why ?
(4) Error bars These show the uncertainties in both variables: For the independent variable: For the dependent variable: for any ( x, y) data point Cooling of an object Time (s) ± 5 s Temperature Uncertainty in y- value ( ° C) ± 4 ° C 10 125 Uncertainty in x- value 70 116 125 104 190 94 260 87 320 76 370 72
NOTE 1: If uncertainties are not constant, the size of the error bars must also vary. NOTE 2: % error is not same for each data point.
(5) Types of Graphs Linear x-y graphs • Semi-log graphs (one scale is linear and one is logarithmic) • Log- Log graphs (both scales are logarithmic) • (6) Linear x-y plots Dependence of relative density on sugar concentration at 298 K. Concentration Relative (kg/m 3 ) density (-) 0 1.005 50 1.034 100 1.066 150 1.095 200 1.122 250 1.150
Draw the best line by naked eye ? For RD = 1.053, calculate the sugar concentration
CAVEATS (1) Do not ignore outliers - investigate them further (2) Interpolation of results is justified when there is a sufficient number of data points ? ? (3) Extrapolation should always be avoided.
What are the two characteristics of a linear graph/plot? y = mx + C intercept slope For two data points, one can only draw one line and this is also the best line : (x 1 ,y 1 ), (x 2 ,y 2 ) − y y = m and C = value of y at x = 0 2 1 − x x 2 1 But when we have a large number of data points (x i ,y i ) (i=20 say) which pair of data points should be used to calculate the values of m and C? BEST FIT What are the uncertainties in the best fit values of m and C?
Example T (h) Size (mm) 0.5 1.5 ± 0.3 1 2.3 ± 0.3 1.5 3.3 ± 0.3 2 4.3 ± 0.3 2.5 5.4 ± 0.3
Linearization of Equations � Prior knowledge about the expected form of dependence. � Original data might exhibit a non-linear relationship . � Transform one of the variables in such a fashion that transformed variable leads to a linear relationship. Period of oscillation (T) ~ mass of the body (M) M
M (kg) T (s) 0.02 0.7 0.05 1.11 0.10 1.6 0.20 2.25 0.30 2.76 0.40 3.18 0.50 3.58 0.60 3.97 0.70 4.16 0.80 4.60 Dependence is not linear Let us recall our high school physics: M (k : spring constant) M π T = ∝ 2 k ∴ M T ∝
π 2 = T M k y = m x + C We expect C = 0 Linear graphs great advantage
EXAMPLE - 1 Constant acceleration (a) Initial Velocity @ t = 0 1 u = u = + s ut at 2 s : distance travelled in time t 2 a = 9.81 m/s 2 Relationship between u = 1 m/s “s” and “t” is quadratic s 1 t = + u at 2 mx y C
s/t m = a/2 C = u 0 t 5 Linearization has come about at a price? We should now estimate uncertainty in (s/t) NOT in s.
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