Errors & Uncertainties

Metric Review Metric Base Units meter (m) Length  Mass  Volume  Time  gram (g) Liter (L) second (s) Note: In physics the kilogram (kg) is used as the fundamental unit for mass not the gram.

Largersmaller 1 kilo (k) =___________ base 1 mega (M)=___________ base 1 giga (G)=___________ base 1 base= ___________ deci (d) 1 base= ___________ centi (c) 1 base =___________ milli (m) 1 base=___________ micro (μ) 1 base=___________ nano (n) Notice that the 1 always goes with the larger unit!! There are always Lots of small units in a single large one! ,000,000 1,000,000, ,000,000 1,000,000,000

Scales Object Length (m) Distance to the edge of the observable universe Diameter of the Milky Way galaxy Distance to the nearest star10 16 Diameter of the solar system10 13 Distance to the sun10 11 Radius of the earth10 7 Size of a cell10 -5 Size of a hydrogen atom Size of a nucleus Size of a proton Planck length Object Mass (kg) The Universe10 53 The Milky Way galaxy10 41 The Sun10 30 The Earth10 24 Boeing 747 (empty)10 5 An apple.25 A raindrop10 -6 A bacterium Mass of smallest virus A hydrogen atom An electron Order of magnitude  The difference between exponents. Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005

Order of Magnitude ► Give an order of magnitude estimate for the mass (kg) of  An egg  The earth  The difference between the mass of an egg and the earth. ► The ratio to the nearest order of magnitude is

Fundamental vs. Derived Units: Fundamental Units ► Basic quantities that can be measured directly ► Examples: length, time, mass, etc… Derived Units ► Calculated quantities from fundamental units ► Examples: speed, acceleration, area, etc… Volume can be measured in liters (fundamental units), or calculated by multiplying length x width x height to give derived units in meters 3

IB Fundamental Units ► Length – meter (m)  Defined as the distance travelled by light in a vacuum in a time of 1/299,792,458 seconds ► Mass – kilogram (kg)  Standard is a certified quantity of a platinum-iridium alloy stored at the Bureau International des Poides et Measures (France) ► Time – second (s)  Defined as the duration 9,192,631,770 full oscillations of the electromagnetic radiation emitted in a transition between the two hyperfine energy levels in the ground state of a cesium-133 (Cs) atom

IB Fundamental Units ► Temperature – Kelvin (K)  Defined as 1/ of the thermodynamic temperature of the triple point of water. ► Molecules – mole (mol)  One mole contains as many molecules as there are atoms in 12 g of carbon 12. (6.02 x molecules – Avogadro’s number) ► Current – Ampere (A)  Defined as the current which when flowing in two parallel conductors 1m apart, produces a force of 2 x N on a length of 1m of the conductors. ► Light Intensity – candela (cd)  The intensity of a source of frequency 5.40 x Hz emitting 1/685 W per steradian. Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005

Errors ► Systematic Errors – error that arises for all measurements taken.  incorrectly calibrated instrument (not zeroed) ► Reading Errors – impreciseness of measurement due to limitations of reading the instrument. ► ► Digital scale   Safe to estimate the reading error (uncertainty) as the smallest division (Ex. Digital stopwatch – smallest division is.01 s so the uncertainty is ±0.01 s) ► ► Analog scale   Safe to estimate the uncertainty as half the smallest scale division (Ex. Ruler - smallest division is.001 m so the reading error is ± m) x Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005

Errors Source: Kirk, 2007, p. 3

Errors ► Random Errors – shown by fluctuations both high and low in the data.  Reduced by averaging repeated measurements (¯)  Error calculated with the standard deviation. whereMeasurement is whereMeasurement is  Estimating random error ► Calculate the average ► Find the highest deviation in the data above and below the average. ► The largest of these deviations becomes the uncertainty. x Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005

Estimating Uncertainty ► Suppose a ruler was used to make the following measurements with the observer noting the reading error to be ±0.05 cm. ► Calculate the average, standard deviation, uncertainty. ► Estimate the uncertainty Length (±0.05 cm) Deviation Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005 Excel Length (±0.05 cm)

Estimating Uncertainty ► Average ( ► Average (¯) = cm ► ► Standard deviation = ► ► Since the random error is larger than the reading error it must be included. ► ► Thus, the measurement is ± 0.16 cm.   Note: IB rounds uncertainty to one significant digit and you match the SD of measurement to the uncertainty ±0.2 cm ► ► Estimation of uncertainty   Largest deviations above/below 0.23 &   Estimated uncertainty ± 0.23 w/ IB rounding 14.8 ± 0.2 cm Length (±0.05 cm) Deviation x Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005

Errors in Measurements ► ► Best estimate ± uncertainty (x best ± δx) standard error notation ► ► Rule for Stating Uncertainties – experimental uncertainties should almost always be rounded to one significant digit. ► ► Rule for Stating Answers – The last significant figure in any stated answer should be of the same order of magnitude as the uncertainty (same decimal position) ► ► Number of decimals places reflect the precision of the measuring instrument ► ► For clarity in graphing we need to convert all data into standard form (scientific notation). ► ► If calculations are made the uncertainties are propagated. ► ►

Relative and Absolute Uncertainty ► Absolute uncertainty is the uncertainty of the measurement.  Ex ±0.02 s  Ex.=

Absolute and Relative (%) Error: ► Useful when comparing to an established value. ► Absolute Error: E a =  O – A  Where  O = observed value  A = accepted value ► Relative or % Error: or E a /A x 100

Sample Problem: ► In a lab experiment, a student obtained the following values for the acceleration due to gravity by timing a swinging pendulum: m/s m/s m/s m/s 2 The accepted value for g at the location of the lab is m/s 2. ► Give the absolute error for each value. ► Find the relative error for each value.

Rules for the Propagation of Error

► To state our answer we now choose the number half-way between these two extremes and for the uncertainty we take half of the difference between them. or S = 4000mm² ± 130mm² or S = 4000mm² ± 130mm² 3. If two (or more) measured quantities are multiplied or divided then their relative uncertainties are added. ► Relative uncertainties: x is 1/50 or 0.02mm and y is 1/80 or mm. So, the relative uncertainty in the final result should be ( ) = ► Checking, the relative uncertainty in final result for S is 130/4000 =

Rules for the Propagation of Error

As previously we now state the final result as 4. If a measured quantity is raised to a power then the relative uncertainty is multiplied by that power. ► Relative uncertainty in r is 0.05/25 = ► Relative uncertainty in V is 393/65451 = ► x 3 = so, again the theory is verified V = 65451mm 3 ± 393mm 3

Summary

Propagation Step by Step ► For more complicated calculations, we break them down into a sequence of steps each involving one of these operations  Sums and differences  Products and quotients  Computation of a function of one variable (x n ) We then apply the propagation rule for each step and total the uncertainty.

Error Propagation

Extension ► For other functions use the ’worst’ value ► Example: uncertainty of sin θ if θ = 60 o ± 5 o ► Sin 60 o = 0.87, sin 65 o = 0.91, sin 55 o = 0.82 ► Since 0.91 is and 0.82 is -0.05, select the worst value ►  sin θ = 0.87 ± 0.05

Error Bars ► Lines plotted to represent the uncertainty in the measurements. ► If we plot both vertical and horizontal bars we have what might be called "error rectangles” ► The best-fit line could be any line which passes through all of the rectangles. x was measured to ±0·5s y was measured to ±0·3m Error Bars

Best Fit Line Source: Kirk, 2007, p. 3

Min & Max Slopes Source: Kirk, 2007, p. 9

Min & Max Y-Intercepts Source: Kirk, 2007, p. 9

Sources ► Kirk, T. (2007) Physics for the IB diploma: Standard and higher level. (2nd ed.). Oxford, UK: Oxford University Press ► Taylor, J. R. (1997) An introduction to error analysis: The study of uncertainties in physical measurements. (2nd ed.). Sausalito, CA: University Science Books ► Tsokos, K. A. (2009) Physics for the IB diploma: Standard and higher level. (5th ed.). Cambridge, UK: Cambridge University Press