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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 worldwide**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 7 Fundamentals**• Length metre m • Mass kilogram kg • Time second s • Electric current ampere A • Thermodynamic temp Kelvin K • Luminous Intensity candela cd • 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**Derived Units**Examples… • Acceleration ms-2 • Momentum kgms-1 or Ns Some derived units have been given their own specific names and symbols… • Force N = kg ms-2 • Joule J = kgm2s-2**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 (FYI – not tested)**• The Meter :- 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 (energy) • J to eV (energy) • Years to seconds (time) • And between other systems and SI ****Note: youshouldbeableto do basicconversionsnow and otherswillbedevelopedthroughouttheyear**SI Format**• The accepted SI format is • ms-1 not m/s • ms-2 not m/s/s The IB will recognize work reported with “/”, but will only use the SI format when providing info.**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 affected in the same way • It is a system or instrument issue**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 unpredictable variations in performance of the instrument and the operator**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**• Randomerrors can bereducedbytakingmultiplereadings, and eliminatingobviouslyerroneousresultorbyaveragingtherange 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**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 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**Absolute, fractional, and relative uncertainty**Assume we measure something to be 208 ± 1 mm in length... • 1 mm is the absolute uncertainty • 1/208 is the fractional uncertainty (0.0048) • 0.48 % is the relative (percent) uncertainty**Combining uncertainties**To determine the uncertainty of a calculated value... • For addition and subtraction, add absolute uncertainities • For multiplication and division add percentage uncertainities • When using exponents, multiply the percentage uncertainty by the exponent**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**Significant Figures**• Thenumber of significant figures shouldreflecttheprecision of thevaluesused as input data in a calculation Simple rule: • Formultiplication and division, thenumber of significant figures in a resultshouldnotexceedthat of theleast precise valueuponwhichitdepends**Graphical Techniques**• Graphing is one of the most valuable tools in data analysis because • 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**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**• Experimentally speaking, the dependent variable is plotted on the y axis and the independent variable is plotted on the x axis. • When you are asked to plot a graph of a against b, the first variable mentioned is plotted on the y axis.**Plotting Graphs - Scales**• Size of graph should be large, to fill as much space as possible…3/4 rule • choose a convenient scale that is easily subdivided**Plotting Graphs - Labels**• Each axis is labeled with the name of the quantity, as well as the relevant unit used… Temperature/K speed/ms-1 • The graph should also be given a descriptive title**Plotting Uncertainties on Graphs**• Error bars showing uncertainty are required - short lines drawn from the plotted points parallel to the axes indicating the absolute error of measurement**Plotting Graphs - Line of Best Fit**• When choosing the best fit line or curve it is easiest 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 play connect the dots!**y**x Uncertainties on a Graph Notice that the best fitting line or curve is one that passes through the error bars of the plotted points. A straight line could not accomplish that with this data set**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**Gradients**• Gradient = vertical run / horizontal run gradient = y / x • Don´t forget to give the units of the gradient • In lab work, always report the maximum and minimum gradient**Areas under Graphs**• The area under a graph is a useful tool. For example… • on a force vs. displacement graph the area is work (N x m = J) • on a speed time graph the area is distance (ms-1 x s = m) • Again, don´t forget the units of the area**y**x Standard Graphs - linear graphs • A straight line passing through the origin shows proportionality y x y = k x k = rise/run Where k is the constant of proportionality**y**y x x2 Standard Graphs - parabola • A parabola shows that y is directly proportional to x2 • i.e. y x2 or y = kx2 • where k is the constant of proportionality**y**y x 1/x Standard Graphs - hyperbola • A hyperbola shows that y is inversely proportional to x • i.e. y 1/xor y = k/x • where k is the constant of proportionality**y**y x 1/x2 Standard Graphs - hyperbola again • An inverse square law graph is also a hyperbola • i.e. y 1/x2 or y = k/x2 • where k is the constant of proportionality