Measurements and Uncertainties — IB Physics SL Study Guide

For: IB Physics SL candidates sitting IB Physics SL.

Covers: IB Topic 1 — fundamental SI units and prefixes, scientific notation, accuracy vs precision, absolute and percentage uncertainties, propagation in addition/multiplication/powers, graphical analysis, line of best fit and uncertainty in slope.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the IB Physics SL style for educational use. They are not reproductions of past IBO papers.

1. Why Measurements Matter

Topic 1 is the smallest unit by content (about 5% of the syllabus directly) but it underpins every experimental question on Paper 3 and the IA report. Every numerical answer should carry units, sig figs, and an uncertainty — losing marks here is the most common preventable IB Physics error.

2. SI base units and prefixes

Seven SI base units; for IB Physics, focus on:

Derived units use combinations: $\text{N}={\text{kgcdotpmcdotps}}^{-2}$, $\text{J}=\text{Ncdotpm}$, $\text{W}={\text{Jcdotps}}^{-1}$, $\text{Pa}={\text{Ncdotpm}}^{-2}$.

Common prefixes (memorise): tera ($10^{12}$), giga ($10^9$), mega ($10^6$), kilo ($10^3$), milli ($10^{-3}$), micro ($10^{-6}$), nano ($10^{-9}$), pico ($10^{-12}$).

3. Accuracy vs precision

• Accuracy = how close measurement is to true value. A poorly-calibrated voltmeter is inaccurate.
• Precision = how close repeated measurements are to each other. A jittery voltmeter is imprecise.

Random errors reduce precision (use repeated trials → average to reduce). Systematic errors reduce accuracy (calibration check is the only fix; averaging doesn't help).

4. Absolute and percentage uncertainties

For a measurement $x\pm \Delta x$:

• Absolute uncertainty: $\Delta x$ (same units as $x$).
• Percentage uncertainty: $\Delta x/x\times 100\%$.

Common rule of thumb for analogue instruments: $\Delta x=$ half the smallest scale division. Digital instruments: $\Delta x=$ smallest displayed unit.

5. Error propagation rules

For a quantity $z$ derived from measurements $x,y$:

Round the absolute uncertainty to 1 sig fig, then round the value to match the uncertainty's decimal place.

6. Graphical analysis

When a relationship is suspected as $y=mx+c$:

1. Plot $y$ vs $x$ with error bars.
2. Draw line of best fit (visually balance points above/below).
3. Determine slope from two distant points on the line: $m=(y_2-y_1)/(x_2-x_1)$.
4. Estimate uncertainty in slope: draw lines of max and min slope through the error bars; \Delta m \approx (m_\max - m_\min)/2.

For non-linear relationships, manipulate to make a linear plot — e.g. $y=a{x}^2$ → plot $y$ vs $x^2$, slope = $a$.

7. Worked Example

A pendulum's period $T$ is measured 5 times: 1.97, 2.01, 1.99, 2.03, 1.98 (in seconds). Length $L=1.00\pm 0.01$ m.

(a) Calculate the mean $T$ and its uncertainty. (b) Theory says $T=2\pi \sqrt{L/g}$. Calculate $g$ and its uncertainty.

Solution.

(a) Mean $\bar{T}=(1.97+2.01+1.99+2.03+1.98)/5=9.98/5=1.996$ s. Half-range method: \Delta T = (T_\max - T_\min)/2 = (2.03 - 1.97)/2 = 0.03 s. $T=2.00\pm 0.03$ s. (Round value to match uncertainty's decimals.)

(b) $T^2=4{\pi }^{2}L/g$, so $g=4{\pi }^{2}L/T^2$. $g=4{\pi }^2\cdot 1.00/(2.00{)}^2=39.48/4.00=9.87$ m/s².

Percentage uncertainties:

• $\Delta L/L=1\%$
• $\Delta T/T=0.03/2.00=1.5\%$, so $\Delta T^2/T^2=2\cdot 1.5\%=3\%$
• Total: $\Delta g/g=1\%+3\%=4\%$
• $\Delta g=0.04\cdot 9.87=0.4$ m/s²

Final: $g=9.9\pm 0.4$ m/s². (Round to match uncertainty.)

8. Common Pitfalls

• Sig fig drift: don't keep more sig figs than your worst measurement justifies.
• Mixing absolute and percentage uncertainty: addition uses absolute; multiplication uses percentage. Don't add absolutes for a product.
• Forgetting the factor of $n$ in $x^n$: a 5% error in $r$ becomes a 15% error in $r^3$. Easy to overlook.
• Plotting issues: error bars must be drawn; line of best fit must pass through (or close to) most error bars.

9. Practice Questions

1. A voltage is measured $V=12.0\pm 0.5$ V; current $I=2.00\pm 0.05$ A. Calculate the resistance $R=V/I$ and its absolute uncertainty.
2. The volume of a sphere is $V=\frac{4}{3}\pi r^3$. If $r=5.0\pm 0.1$ cm, find $V$ and $\Delta V$.
3. A student plots $y$ vs $x$ data and fits a line of best fit, obtaining slope $m=3.4$, with max-slope line giving 3.7 and min-slope line giving 3.1. Express the slope with uncertainty.

10. Quick Reference Cheatsheet

• 7 SI base units; common prefixes from tera ($10^{12}$) to pico ($10^{-12}$).
• Accuracy ≠ precision.
• Sum/difference: absolute uncertainties add.
• Product/quotient: percentage uncertainties add.
• Power $x^n$: percentage uncertainty $\times n$.
• Round $\Delta x$ to 1 sig fig; round $x$ to match decimals.
• Slope uncertainty: (m_\max - m_\min)/2.

11. What's Next

Measurements is the foundation for every other Topic — every quantitative answer on Paper 1, 2, 3 carries propagated uncertainty. Use Ollie for any specific propagation: "Walk me through this multi-variable uncertainty calculation" or "Should I use half-range or standard deviation here?"