physics prac

• Measurement and Uncertainty:

  • Accurate and precise measurements are essential for drawing appropriate conclusions
  • Definitions:
  • Accuracy: closeness of a measurement to the true value
  • Measurement error: the difference between a measured value and its true value
  • Precision: how close a set of measurements are to each other
  • Uncertainty: characterizes the range of values within which the true value is expected to lie
• When taking measurements, multiple readings are usually taken to account for slight variations in results
• Variation in results is due to the inherent uncertainty in all measurements
• Precision, or reliability of a measurement, describes how close a set of measurements are to each other
• Accuracy and precision are illustrated using the dartboard analogy
• Expressing measurements requires two numbers: the best estimate and the uncertainty
• Sources of experimental error and uncertainty:

  • It is impossible to measure with absolute accuracy and precision
  • Even with sophisticated instruments and careful readings, there will always be a range of uncertainty
• Three main limitations to measurement accuracy and/or precision:

  1. The precision of the measuring instrument limits the precision of the measurement.

  2. Systematic errors affect the accuracy of measurements and are consistently present in all readings.

  3. Random errors are caused by unpredictable fluctuations in measurement technique and the experimental environment
• Examples of random and systematic errors can be found at the following site: https://www.matrix.edu.au/systematic-vs-random-errors/
• Significant figures are used to reflect the precision of a measuring device
• Measurement instruments are either analogue (e.g. metre rule, thermometer, micrometer) or digital (e.g. stopwatch, weighing scale, multimeter)
• When a measurement is taken using an analogue device, it can be accurately read to the smallest scale division; thereafter an estimation is made to a fraction of this scale
• When a measurement is taken with a digital device, no estimation is involved, and the significant figures are directly read off the device
• Identifying significant figures:

  • All nonzero digits are significant
  • All zeros between significant digits (imbedded zeros) are significant
  • Trailing zeros are significant only if the decimal point is specified
  • Leading zeros are never significant
  • In scientific notation, the decimal point indicates the number of significant digits
• Carrying significant figures in calculations:

  • For addition and subtraction, round off the result to the same number of decimal places as the term with the least number of decimal places
  • For multiplication and division, round the result to the same number of significant figures as the factor with the least number of significant figures
• When dealing with significant figures in calculations:

  • For logarithms, retain as many digits as there are digits to the right of the decimal point in the original number
  • For trigonometrical functions and their inverses, retain as many digits in the result as there are significant figures in the argument
• Rounding off in calculations involving combinations of different operations should only be performed on the final result
• When rounding off a calculated result:

  • If the figure next beyond the last one to be retained is less than 5, retain the last figure unchanged
  • If the figure next beyond the last one to be retained is greater than 5, increase the last figure by 1
  • If the figure next beyond the last one to be retained is 5 and there are no figures or only zeros beyond it, increase the last figure by 1 if it is odd, and leave it unchanged if it is even
  • If the figure next beyond the last one to be retained is 5 and there are figures other than zero beyond it, increase the last figure by 1
• Exact numbers, such as defined conversions and counted numbers, are treated as if they have infinite significant figures and are free of errors
• When calculating random errors:

  • The best estimate of a measurement is determined by calculating the average of all values
  • Measures of dispersion and uncertainty of a measurement can be calculated using specific equations
• Standard error:

  • The uncertainty in the mean value of x
  • Small datasets: ∆𝑥𝑥̅ = ∆𝑥𝑥 / √𝑀𝑀
  • Large datasets: 𝑣𝑣𝑀𝑀̅ = 𝑣𝑣 / √𝑀𝑀
• Measured value:

  • Contains both the average value and the uncertainty in the mean
  • Small datasets: 𝑥𝑥𝑚𝑚 = 𝑥𝑥̅ ± ∆𝑥𝑥̅
  • Large datasets: 𝑥𝑥𝑚𝑚 = 𝑥𝑥̅ ± 𝑣𝑣𝑀𝑀̅
• How many readings to take:

  • Continue taking readings until a further reading does not alter the mean significantly
  • After about ten readings, further readings would probably have no significant effect on the mean
• Combinations of uncertainties:

  • Aim to evaluate a quantity that is a function of several measured quantities, each with its own uncertainty
  • The uncertainty of the final calculated quantity must be determined
• Rejection of outliers:

  • An outlier is a result that does not appear to belong with the other results
  • Failure to test for outliers may result in the inclusion of a result that does not belong to the set of results
• In the two standard deviation test, the largest value exceeding 2.94 must be rejected, while the smallest value below 2.85 is retained
• After rejecting an outlier, the mean and standard deviation are recalculated
• Outlier test is performed on the remaining data values after recalculating the mean and standard deviation
• For 10 data values in the Tn test at the 95% confidence level, the Tn value is 2.18
• In the Tn test, the largest and smallest allowed values are determined based on the average value and standard deviation
• Results are compared with an accepted value to assess the quality of work and procedure used
• Acceptable results should be within two standard deviations of the accepted value and have a percentage uncertainty not greater than 5%
• Graphs are a convenient way to examine experimental results and show the dependence between quantities
• When plotting a graph, the independent variable is on the horizontal axis and the dependent variable on the vertical axis
• Scales on the axes should be chosen to maximize space and have convenient units for easy reading
• In single-celled organisms, substances can easily enter the cell due to the short distance they need to cross
• In multicellular organisms, the distance for substances to enter the cell is much larger due to a higher surface area to volume ratio
• Multicellular organisms require specialised exchange surfaces for efficient gas exchange of carbon dioxide and oxygen due to their higher surface area to volume ratio
• R2 is always between 0 and 100%:

  • 0% indicates that the model explains none of the variability of the response data around its mean
  • 100% indicates that the model explains all the variability of the response data around its mean
• In general, the higher the R2, the better the model fits your data