Measurements and their Errors

• SI units


  • Mass (m): kg (kilograms)
  • Length (l): m (metres)
  • Time (t): s (seconds)
  • Amount of substance (n): mol (moles)
  • Temperature (t): K (kelvin)
  • Electric current (I): A (amperes)
• Derived SI units


  Derived from equations, e.g. F=ma gives N (newtons) as the SI unit for force
• SI units of voltage

  V = E/Q, where E is energy and Q is charge, so V has units of kg m^2 s^-3 A^-1
• SI prefixes


  • Tera (T): 10^12
  • Giga (G): 10^9
  • Mega (M): 10^6
  • Kilo (k): 10^3
  • Centi (c): 10^-2
  • Milli (m): 10^-3
  • Micro (μ): 10^-6
  • Nano (n): 10^-9
  • Pico (p): 10^-12
  • Femto (f): 10^-15
• Converting mega electron volts to joules

  1. 1 eV = 1.6x10^-19 J

  2. 76 MeV = 76 x 10^6 eV = 1.216 x 10^-11 J
• Converting kilowatt hours to joules

  1. 1 kW = 1000 J/s

  2. 1 hour = 3600 s

  3. 1 kWh = 1000 x 3600 J = 3.6 x 10^6 J = 3.6 MJ
• Random errors


  Affect precision, cause differences in measurements which causes a spread about the mean, cannot be eliminated
• Systematic errors


  Affect accuracy, cause all results to be too high or too low by the same amount each time
• Ways to reduce random errors

  • Take at least 3 repeats and calculate a mean
  • Use computers/data loggers/cameras to reduce human error
  • Use appropriate equipment with higher resolution
• Ways to reduce systematic errors

  • Calibrate apparatus by measuring a known value
  • Correct for background radiation in radiation experiments
  • Read the meniscus at eye level to reduce parallax error
  • Use controls in experiments
• Precision


  Measurements are consistent, fluctuate slightly about a mean value
• Repeatability


  Original experimenter can redo the experiment and get the same results
• Reproducibility


  Experiment is redone by a different person or with different techniques and equipment and the same results are found
• Resolution


  The smallest change in the quantity being measured that gives a recognisable change in reading
• Accuracy


  A measurement close to the true value
• Uncertainty


  The bounds in which the accurate value can be expected to lie
• Types of uncertainty


  • Absolute uncertainty: fixed quantity e.g. 7 ± 0.6 V
  • Fractional uncertainty: uncertainty as a fraction of the measurement e.g. 7 ± 3/35 V
  • Percentage uncertainty: uncertainty as a percentage of the measurement e.g. 7 ± 8.6% V
• Uncertainty in a reading

  ± half the smallest division
• Uncertainty in a measurement

  At least ± 1 smallest division
• Uncertainty in digital readings and given values

  ± the last significant digit
• Uncertainty in repeated data

  Half the range (largest - smallest value)
• Combining uncertainties


  Adding/subtracting: add absolute uncertainties

  Multiplying/dividing: add percentage uncertainties

  Raising to a power: multiply percentage uncertainty by power
• Uncertainty in graphs


  Shown as error bars, line of best fit should go through all error bars (excluding anomalous points)
• Finding uncertainty in gradient and y-intercept of a graph

  Gradient: Draw steepest and shallowest line of worst fit, calculate percentage uncertainty from best and worst gradients

  1. intercept: Calculate percentage uncertainty from best and worst y-intercepts
• Order of magnitude


  Powers of ten which describe the size of an object
• Estimating physical quantities to the nearest order of magnitude


  Calculate the value and give it as a power of ten
• Estimation is a skill physicists use to make comparisons and check if calculated values are reasonable