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Define the term systematic error and state one method by which it can be reduced.
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A systematic error causes every reading to be offset in the same direction by the same amount (or to deviate from the true value in a consistent way); reduced by calibrating the instrument / checking and correcting for zero error before use.
Distinguish between accuracy and precision of a set of measurements.
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Accuracy refers to how close the (mean) measurement is to the true value; precision refers to how close repeated readings are to each other / the small spread of the readings.
A student takes four readings of the period of a pendulum and obtains values that are very close to each other but all noticeably larger than the accepted value. Explain whether the readings are precise, accurate, or both, and suggest one possible cause.
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The readings are precise because they are close to each other / have a small spread; the readings are not accurate because the mean is far from the true value; this indicates a systematic error; possible cause: the stopwatch was started late each time / a reaction-time bias / a miscalibrated stopwatch.
The resistance $R$ of a resistor is calculated from $R=V/I$. The student measures $V=6.0\pm 0.1\text{ V}$ and $I=0.50\pm 0.02\text{ A}$. Calculate the resistance and its absolute uncertainty.
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Finding the resistance $$R=\frac{V}{I}=\frac{6.0}{0.50}=12 \Omega $$ Finding the percentage uncertainties $$\frac{\Delta V}{V}\times 100\%=\frac{0.1}{6.0}\times 100\%=1.67\%$$ $$\frac{\Delta I}{I}\times 100\%=\frac{0.02}{0.50}\times 100\%=4.00\%$$ Combining and finding the absolute uncertainty $$\frac{\Delta R}{R}\times 100\%=1.67\%+4.00\%=5.67\%$$ $$\Delta R=5.67\%\times 12=0.68 \Omega $$ The resistance is $R=(12.0\pm 0.7) \Omega $.
Show that the percentage uncertainty in the kinetic energy ${E}_{k}=\frac{1}{2}m{v}^{2}$ of a trolley is approximately $7\%$, given that $m=0.500\pm 0.005\text{ kg}$ and $v=2.00\pm 0.06{\text{ m s}}^{-1}$.
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Percentage uncertainty in $m$ $$\frac{\Delta m}{m}\times 100\%=\frac{0.005}{0.500}\times 100\%=1.0\%$$ Percentage uncertainty in $v$ (doubled for ${v}^{2}$) $$\frac{\Delta v}{v}\times 100\%=\frac{0.06}{2.00}\times 100\%=3.0\%$$ $$2\times 3.0\%=6.0\%$$ Total percentage uncertainty in ${E}_{k}$ $$\frac{\Delta {E}_{k}}{{E}_{k}}\times 100\%=1.0\%+6.0\%=7.0\%\approx 7\%$$ as required.
Discuss how a student designing an experiment to measure the density of a small metal sphere should choose between using a vernier calliper and a metre rule to measure its diameter, with reference to both random and systematic errors.
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A vernier calliper has a smaller resolution (typically 0.01 cm) than a metre rule (0.1 cm); this gives a smaller absolute uncertainty in the diameter and therefore a smaller percentage uncertainty; the diameter is squared (or cubed) in the volume formula, so its uncertainty has a magnified effect on density; the calliper therefore reduces the random error in the derived density; however, both instruments may have a zero error that must be checked and corrected before use; failing to do so introduces a systematic error that repeating readings will not remove; the calliper is the better choice provided its zero is checked.