Propagating Uncertainty — practice questions

A length is measured with a ruler marked in millimetres.

State the uncertainty in a single reading, and state the uncertainty in a length measured between two points on the ruler.

A student times five oscillations with a stopwatch that displays to $0.01$ s. Their three repeats are $6.8$ s, $7.2$ s and $6.9$ s.

Determine an appropriate uncertainty in the period of one oscillation, and explain why the stopwatch's resolution is not the appropriate uncertainty here.

In an investigation of how the period of a mass on a spring depends on the mass, a student uses a balance reading to $0.1$ g for masses between $50$ g and $500$ g, and hand-times 10 oscillations with a $\pm 0.2$ s reaction-time uncertainty for periods between $0.5$ s and $1.6$ s.

Analyse which measurement limits the precision of the final result, and justify one change that would most improve it.

A length is measured as $8.0 \pm 0.1$ cm. Calculate its percentage uncertainty.

A calculation uses $z = x^3$, where $x$ has a percentage uncertainty of $2.0\%$. Find the percentage uncertainty in $z$, and explain why raising to a power increases uncertainty so much.

Kinetic energy is calculated as $E_k = \tfrac{1}{2}mv^2$, where $m = 2.00 \pm 0.01$ kg (0.5%) and $v = 3.0 \pm 0.3$ m s⁻¹ (10%). Find the percentage uncertainty in $E_k$ and explain which measurement the student should improve first to make the biggest difference to the final uncertainty.