Explain the difference between a random error and a systematic error, and state how each can be reduced.
Explain the difference between precision and accuracy.
A digital balance with a resolution of 0.01 g reads 24.56 g. State the absolute and percentage uncertainty.
An analogue ammeter with smallest divisions of 0.1 A reads 2.4 A. State the absolute and percentage uncertainty.
Five timings are 2.1, 2.3, 2.2, 2.0 and 2.4 s. Find the mean and estimate the uncertainty using half the range.
x = 8.0 ± 0.2 cm and y = 5.0 ± 0.1 cm. Find z = x + y and its absolute uncertainty.
A current I = 2.0 ± 0.1 A passes through a resistor R = 10.0 ± 0.5 Ω. Find V = I R, its percentage uncertainty and its absolute uncertainty.
A pendulum gives g = 4 π2 l / T2. The length l has a 1 percent uncertainty and the period T has a 1 percent uncertainty. Find the percentage uncertainty in g.
A circle has radius r = 3.00 ± 0.05 cm. Find the percentage uncertainty in its area A = π r2.
A density is found from ρ = m / V with m = 120.0 ± 0.5 g and V = 50.0 ± 1.0 cm3. Calculate ρ and state it as a value plus or minus an uncertainty to an appropriate number of significant figures.
Explain why averaging many readings reduces the random error but not a systematic error.
A learner always starts a stopwatch a moment after the event begins. State the type of error this produces and its effect on the readings.
Total: 28 marks. Original work by the TheLucidSTEM team. Written in the style of the papers; no past paper question is reproduced.
Answer key · full worked solutionsclick to reveal
a random error scatters readings either side of the true value and is reduced by repeating and averaging. A systematic error shifts all readings the same way and is reduced by zeroing or calibrating the instrument and improving technique; repeating does not reduce it.
precision is how close repeated readings are to one another; accuracy is how close a reading is to the true value.
absolute uncertainty = 0.01 g (the resolution); percentage = (0.01 / 24.56) × 100 = 0.04 percent.
absolute uncertainty = half the smallest division = 0.05 A; percentage = (0.05 / 2.4) × 100 = 2 percent.
mean = (2.1 + 2.3 + 2.2 + 2.0 + 2.4) / 5 = 2.2 s; range = 2.4 − 2.0 = 0.4 s; uncertainty = half the range = 0.2 s, so 2.2 ± 0.2 s.
z = 8.0 + 5.0 = 13.0 cm; Δz = 0.2 + 0.1 = 0.3 cm, so z = 13.0 ± 0.3 cm.
percentage uncertainties: I gives 5 percent, R gives 5 percent; total = 10 percent. V = 2.0 × 10.0 = 20 V; absolute uncertainty = 10 percent of 20 = 2 V, so V = 20 ± 2 V.
g depends on T2, so its percentage uncertainty = 1 percent (from l) + 2 × 1 percent (from T) = 3 percent.
A depends on r2, so percentage uncertainty = 2 × (0.05 / 3.00) × 100 = 2 × 1.67 = 3.3 percent.
ρ = 120.0 / 50.0 = 2.40 g cm−3. Percentage uncertainties: m gives (0.5 / 120.0) × 100 = 0.4 percent; V gives (1.0 / 50.0) × 100 = 2.0 percent; total = 2.4 percent. Absolute uncertainty = 2.4 percent of 2.40 = 0.06 g cm−3, so ρ = 2.40 ± 0.06 g cm−3.
random errors are equally likely to be above or below the true value, so averaging many readings cancels much of the scatter. A systematic error shifts every reading the same way, so the shift remains in the average and is not reduced by averaging.
a systematic error. Every recorded time is shorter than the true time by roughly the same amount, so all readings are offset in the same direction.