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Prefixes and Uncertainties Prefixes and Uncertainties The use of standard form and / or prefixes such as milli will be encouraged throughout the course. Care must be taken when dealing with significant figures. Data should be carried through in calculations and the final rounding carried out at the end. Remember that you should only quote your final answer to the same significance as the least significant piece of data. SQA however will allow you to quote 2 too many or 1 too few significant figures. 1 Units, prefixes and scientific notation Units and prefixes SI units should be used with all the physical quantities. Prefixes should be used where appropriate. These include pico (p), nano(n), micro( giga(G) and tera(T). Significant figures When carrying out calculations and using relationships to solve problems, it is important to give answers to an appropriate number of significant figures. This means that the final answer can have no more significant figures than the value with least number of significant figures used in the calculation. Scientific notation Candidates should be familiar with the use of scientific notation and this may be used as appropriate when large and small numbers are used in calculations. 2 Uncertainties Random and systematic uncertainty All measurements of physical quantities are liable to uncertainty, which should be expressed in absolute or percentage form. Random uncertainties occur when an experiment is repeated and slight variations occur. Scale reading uncertainty is a measure of how well an instrument scale can be read. Random uncertainties can be reduced by taking repeated measurements. Prefixes and Uncertainties Systematic uncertainties occur when readings taken are either all too small or all too large. They can arise due to measurement techniques or experimental design. Uncertainties and data analysis The mean of a set of readings is the best estimate of a ‗true‘ value of the quantity being measured. When systematic uncertainties are present, the mean value of measurements will be offset. When mean values are used, the approximate random uncertainty should be calculated. When an experiment is being undertaken and more than one physical quantity is measured, the quantity with the largest percentage uncertainty should be identified and this may often be used as a good estimate of the percentage uncertainty in the final numerical result of an experiment. The numerical result of an experiment should be expressed in the form final value ± uncertainty These learning outcomes will be covered throughout the course when doing experimental work. Prefixes and Uncertainties Uncertainties The measurement of any Physical Quantity is subject to an uncertainty. There are a number of factors which may contribute to this uncertainty and we need to be able to identify the effects of each. Reading Uncertainty Whatever instrument we use to take a measurement, we can estimate the uncertainty in the reading. Analogue Scales (rulers, protractors etc.) can be read to ±½ of the smallest division. E.g. a metre stick graduated in centimetres can be read to ±½ cm. Digital Scales (digital multimeter, stop watch etc.) Can be read to the value of the last digit on the display. For example if a digital voltmeter reads 5.76V then the uncertainty in the reading is ± 0.01V. We can say with confidence the actual value is between 5.75V and 5.77V. Systematic Effects These are caused by faults in the apparatus or in poor experimental technique. For example, if a voltmeter reads 0.2V when it should read 0V then all readings taken with that instrument will be too high by 0.2V. Random Uncertainties When we do experiments it is good practice to take a number of readings for each measurement. This often results in a scatter of values. We can deal with these by calculating the mean value and the random uncertainty for the mean. Calculation of Mean -Simply add up all the values you obtain and divide by the number of readings taken. Mean Sum of Values Number of Readings Calculation of Random Uncertainty (RUncert) –Subtract the minimum value from the maximum value and divide by the number of readings taken. RUncert Maximum Value - Minimum Value Number of Measuremen ts Made Prefixes and Uncertainties Example –A group of students measure the width of a sheet of A4 paper; Person Making Reading Value Obtained (mm) Alice 209 Bob 210 Claire 210 Damien 211 Ethel 212 Fred 210 Gemma 209 Horace 208 Irene 210 Jeremy 211 Calculation of Mean Mean 209 210 210 211 212 210 209 208 210 211 10 = 210mm Calculation of Random Uncertainty RUncert Maximum Value - Minimum Value Number of Measuremen ts Made Uncert 212 208 10 = 0.4mm So we quote our final answer as 210 0.4mm Percentage Uncertainty It is sometimes useful to quote an uncertainty as a percentage of the mean value. We calculate percentage uncertainties using; %Uncertainty Random Uncertainty 100 Mean So in the example above; 0.4 %Uncertaint y 100 0.2% 210 Prefixes and Uncertainties So we can now quote the value as; 210 0.4mm or as 210mm 0.2% Percentage Uncertainties are particularly useful when we need to combine uncertainties. We may measure two physical quantities and calculate a third. As a rule, we take the largest % uncertainty in the data and simply quote this as the % uncertainty in the calculated value. Example; The voltage across a resistor is measured as 12V ± 0.5V The current through the resistor is measured as 2A ± 0.5A. Calculate the value of the resistor and quote the uncertainty in this value. 0.5 100 4.2% 12 0.5 %Uncertaint y in Current 100 25% 2 %Uncertaint y in Voltage Resistance is found using Ohm’s Law; R V 12 24 I 0.5 So we quote the final answer as; 24 ± 6 Ω (25% is the largest of the %uncertainties in the data, 25% of 24 is 6Ω)