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## Physics and Physical Measurement

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**Physics and Physical Measurement**Topic 1.2 Measurement and Uncertainties**Standards of Measurement**• SI units are those of the Système International d’Unités adopted in 1960 • Used for general measurement in most countries**Fundamental Quantities**• Some quantities cannot be measured in a simpler form and for convenience they have been selected as the basic quanitities • They are termed Fundamental Quantities, Units and Symbols**The Fundamentals**• Length metre m • Mass kilogram kg • Time second s • Electric current ampere A • Thermodynamic temp Kelvin K • Amount of a substance mole mol**Derived Quantities**• When a quantity involves the measurement of 2 or more fundamental quantities it is called a Derived Quantity • The units of these are called Derived Units**The Derived Units**• Acceleration ms-2 • Angular acceleration rad s-2 • Momentum kgms-1 or Ns • Others have specific names and symbols • Force kg ms-2 or N**Standards of Measurement**• Scientists and engineers need to make accurate measurements so that they can exchange information • To be useful a standard of measurement must be • Invariant, Accessible and Reproducible**3 Standards (for information)**• The Metre :- the distance traveled by a beam of light in a vacuum over a defined time interval ( 1/299 792 458 seconds) • The Kilogram :- a particular platinum-iridium cylinder kept in Sevres, France • The Second :- the time interval between the vibrations in the caesium atom (1 sec = time for 9 192 631 770 vibrations)**Conversions**• You will need to be able to convert from one unit to another for the same quanitity • J to kWh • J to eV • Years to seconds • And between other systems and SI**KWh to J**• 1 kWh = 1kW x 1 h • = 1000W x 60 x 60 s • = 1000 Js-1 x 3600 s • = 3600000 J • = 3.6 x 106 J**J to eV**• 1 eV = 1.6 x 10-19 J**SI Format**• The accepted SI format is • ms-1 not m/s • ms-2 not m/s/s • i.e. we use the suffix not dashes**Errors**• Errors can be divided into 2 main classes • Random errors • Systematic errors**Mistakes**• Mistakes on the part of an individual such as • misreading scales • poor arithmetic and computational skills • wrongly transferring raw data to the final report • using the wrong theory and equations • These are a source of error but are not considered as an experimental error**Systematic Errors**• Cause a random set of measurements to be spread about a value rather than being spread about the accepted value • It is a system or instrument value**Systematic Errors result from**• Badly made instruments • Poorly calibrated instruments • An instrument having a zero error, a form of calibration • Poorly timed actions • Instrument parallax error • Note that systematic errors are not reduced by multiple readings**Random Errors**• Are due to variations in performance of the instrument and the operator • Even when systematic errors have been allowed for, there exists error.**Random Errors result from**• Vibrations and air convection • Misreading • Variation in thickness of surface being measured • Using less sensitive instrument when a more sensitive instrument is available • Human parallax error**Reducing Random Errors**• Random errors can be reduced by • taking multiple readings, and eliminating obviously erroneous result • or by averaging the range of results.**Accuracy**• Accuracy is an indication of how close a measurement is to the accepted value indicated by the relative or percentage error in the measurement • An accurate experiment has a low systematic error**Precision**• Precision is an indication of the agreement among a number of measurements made in the same way indicated by the absolute error • A precise experiment has a low random error**Limit of Reading and Uncertainty**• The Limit of Reading of a measurement is equal to the smallest graduation of the scale of an instrument • The Degree of Uncertainty of a measurement is equal to half the limit of reading • e.g. If the limit of reading is 0.1cm then the uncertainty range is 0.05cm • This is the absolute uncertainty**Reducing the Effects of Random Uncertainties**• Take multiple readings • When a series of readings are taken for a measurement, then the arithmetic mean of the reading is taken as the most probable answer • The greatest deviation or residual from the mean is taken as the absolute error**Absolute/fractional errors and percentage errors**• We use ± to show an error in a measurement • (208 ± 1) mm is a fairly accurate measurement • (2 ± 1) mm is highly inaccurate**In order to compare uncertainties, use is made of absolute,**fractional and percentage uncertainties. • 1 mm is the absolute uncertainty • 1/208 is the fractional uncertainty (0.0048) • 0.48 % is the percentage uncertainity**Combining uncertainties**• For addition and subtraction, add absolute uncertainities • y = b-c then y ± dy = (b-c) ± (db + dc)**Combining uncertainties**• For multiplication and division add percentage uncertainities • x = b x c then dx = db + dc x b c**Combining uncertainties**• When using powers, multiply the percentage uncertainty by the power • z = bn then dz = n db z b**Combining uncertainties**• If one uncertainty is much larger than others, the approximate uncertainty in the calculated result may be taken as due to that quantity alone**Plotting Uncertainties on Graphs**• Points are plotted with a fine pencil cross • Uncertainty or error bars are required • These are short lines drawn from the plotted points parallel to the axes indicating the absolute error of measurement**y**x Uncertainties on a Graph**Significant Figures**• The number of significant figures should reflect the precision of the value or of the input data to be calculated • Simple rule: • For multiplication and division, the number of significant figures in a result should not exceed that of the least precise value upon which it depends**Estimation**• You need to be able to estimate values of everyday objects to one or two significant figures • And/or to the nearest order of magnitude • e.g. • Dimensions of a brick • Mass of an apple • Duration of a heartbeat • Room temperature • Swimming Pool**You also need to estimate the result of calculations**• e.g. • 6.3 x 7.6/4.9 • = 6 x 8/5 • = 48/5 • =50/5 • =10 • (Actual answer = 9.77)**Approaching and Solving Problems**• You need to be able to state and explain any simplifying assumptions that you make solving problems • e.g. Reasonable assumptions as to why certain quantities may be neglected or ignored • i.e. Heat loss, internal resistance • Or that behaviour is approximately linear**Graphical Techniques**• Graphs are very useful for analysing the data that is collected during investigations • Graphing is one of the most valuable tools used because**Why Graph**• it gives a visual display of the relationship between two or more variables • shows which data points do not obey the relationship • gives an indication at which point a relationship ceases to be true • used to determine the constants in an equation relating two variables**You need to be able to give a qualitative physical**interpretation of a particular graph • e.g. as the potential difference increases, the ionization current also increases until it reaches a maximum at…..**Plotting Graphs**• Independent variables are plotted on the x-axis • Dependent variables are plotted on the y-axis • Most graphs occur in the 1st quadrant however some may appear in all 4**Plotting Graphs - Choice of Axis**• When you are asked to plot a graph of a against b, the first variable mentioned is plotted on the y axis • Graphs should be plotted by hand**Plotting Graphs - Scales**• Size of graph should be large, to fill as much space as possible • choose a convenient scale that is easily subdivided**Plotting Graphs - Labels**• Each axis is labeled with the name and symbol, as well as the relevant unit used • The graph should also be given a descriptive title**Plotting Graphs - Line of Best Fit**• When choosing the line or curve it is best to use a transparent ruler • Position the ruler until it lies along an ideal line • The line or curve does not have to pass through every point • Do not assume that all lines should pass through the origin • Do not do dot to dot!**y**x**Analysing the Graph**• Often a relationship between variables will first produce a parabola, hyperbole or an exponential growth or decay. These can be transformed to a straight line relationship • General equation for a straight line is • y = mx + c • y is the dependent variable, x is the independent variable, m is the gradient and c is the y-intercept**The parameters of a function can also be obtained from the**slope (m) and the intercept (c) of a straight line graph**Gradients**• Gradient = vertical run / horizontal run • or gradient = y / x • uphill slope is positive and downhill slope is negative • Don´t forget to give the units of the gradient