# vce physics: dealing with numerical measurments

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• 1. Dealing with Numbers A guide to Numerical & Graphical Methods

2. 1.0 The Importance of Experiments

• Scientists and engineers spend a lot of time performing experiments.Why ?
• They form the basis for scientific and technical advances.
• They allow theory to be put to the test.
• They may reveal new, unexpected effects leading to new or modified theoreticalmodels or explanations.
• In the case of students in a VCE class, experiments are unlikely to break new ground, but they do provide you with the opportunity to acquire:
• Knowledge
• Skills
• Understanding
• through investigating the real world

3. 1.1 Experimental Results The Data OK youve done an experiment and collected some results. What are the important features of the data you have collected ? Measurements made or taken during an experiment generate raw data. This data must be recorded then presented and analysed. All data will have some uncertainty attached. It doesnt matter how good the experimenter, how well designed the experiment or how sophisticated the measuring device, ALL collected data has some uncertainty. (27.50.5) 0 C This statement of temperature indicates both its measured value and the uncertainty. The temperature could be anywhere between 27.5 0.5 = 27.0 0and 27.5 + 0.5 = 28.0 0 4. 1.2 UncertaintyThe uncertainty of the measurement is determined by the scale of the measuring device. The uncertainty quantifies (gives a number to) the amount of variation that has been found in a measured value. An alternative term to that of uncertainty is to use the term EXPERIMENTAL ERROR. This does NOT imply a mistake in your results, but simply the natural spread in the values of a repeatedly measured quantity. Uncertainty generally comes in three forms: Resolution Uncertainty how fine is the scale on the measuringdevice ? Calibration Uncertainty how well does the measuring device conformto the standard ? Reading Uncertainty- howwell did the operator use the device ? 5. 1.3 Systematic and Random Uncertainty

• Each form of uncertainty can have 2 categories:
• Systematic Uncertainty can exist without the experimenters knowledge.
• Can skew all readings or values one way.
• Mostly due to instruments rather than humans.
• RandomUncertainty produces scatter in measurements.
• Environmental factors often cause this type of error.
• Mostly due to humans rather than instruments.

Elimination of these experimental errors is the holy grail of experimental scientists and engineers. Systematic uncertainties can be reduced or eliminated from the measuring device by calibrating (comparing to a known standard) known to a high degree of both accuracy and precision. Random uncertainties can be controlled (but not eliminated) by taking multiple readings and using statistical analysis on the collected results. 6. 1.4 Precision Precision is a measure of how closely a group of measurements agreewith one another. Close agreement translates to a small uncertainty. However, precision DOES NOT mean that the measurements are close to the true value.An example here should explain: The true value on a dart board is the bullseye. This player is precise - all darts fall within a small area (small uncertainty) but he is certainly not accurate A player throws 5 darts 7. 1.5 Accuracy Accuracy is how closely the measurements agree with the true value. Again using the darts analogy:This player is BOTH accurate AND precise. What can you say about the following measurements ? Each dot represents one persons attempt to measure the length of a piece of string Inaccurate Imprecise Precise Imprecise Precise Inaccurate Accurate Accurate True Value 8. 1.6 Significant Figures Significant Figures can be regarded as another method of indicating the uncertainty in a measured quantity. Significant Figures THE RULES: 1. All NON ZERO integers are significant. 2. Zeros (a) Captive Zeros they fall between two non zero numbersthey always count as significant figures. (b) Decimal Point Zeros Zeros used to place a decimalpoint are NOT significant. (c) Trailing Zeros any zeros following a decimal point aresignificant. Number 12.5 0.003002 49,000 0.000234 123.00 Significant Figures 3 4 2 3 5 9. 1.7 Significant Figure Manipulation

• When adding and subtracting numbers, round the result of the calculation to the same number of decimal places as the number with the fewest decimal places used in the calculation.

46.379 Rounding to the least number of decimal places of those numbers added (21.1 with 1 decimal place).

• 2.MULTIPLICATION AND DIVISION:
• Identify the number in thecalculationwith the leastnumber of significant figures.Give your answer to the samenumber of sig figs.