Topic 1: Physical Quantities & Units
CIE 9702 Physics (2025–2027) — Learner Success Notes
Syllabus coverage (you will be tested on):
- SI base quantities & units; derived units; homogeneity (dimensional analysis).
- Scalars vs vectors (definitions and common examples).
- Prefixes, orders of magnitude, unit conversions.
- Measurement techniques: rules for analogue & digital instruments; zero error.
- Uncertainties: absolute, fractional, percentage; combining in $+,-,\times,\div,$ powers.
- Significant figures & rounding, quoting results correctly.
- Graph skills: best-fit/worst-fit, gradient & intercept with uncertainties.
- Choosing methods: repeated readings (mean & half-range), single reading (resolution).
Rapid reference
- Absolute $\Delta x$
- Fractional $\dfrac{\Delta x}{x}$
- Percentage $\dfrac{\Delta x}{x}\!\times\!100\%$
- Analogue $\Delta x=\tfrac{1}{2}$ of smallest division
- Digital $\Delta x=\text{last digit step (resolution)}$
- Add/Sub add absolute uncertainties
- Mul/Div add percentage uncertainties
- Powers if $y=x^n$, then $\dfrac{\Delta y}{y}=|n|\dfrac{\Delta x}{x}$
1) Physical quantities, SI base units & dimensions
Derived examples: speed m s$^{-1}$, force N (kg m s$^{-2}$), pressure Pa (kg m$^{-1}$ s$^{-2}$), energy J (kg m$^{2}$ s$^{-2}$), power W (kg m$^{2}$ s$^{-3}$).
Force $F=ma \Rightarrow [F]=M L T^{-2}$.
Energy $E=Fd \Rightarrow [E]=M L^{2} T^{-2}$.
Pressure $p=F/A \Rightarrow [p]=M L^{-1} T^{-2}$.
Power $P=E/t \Rightarrow [P]=M L^{2} T^{-3}$.
Common prefixes
| Prefix | Symbol | Factor | Example |
|---|---|---|---|
| micro | µ | $10^{-6}$ | µm |
| milli | m | $10^{-3}$ | mA |
| centi | c | $10^{-2}$ | cm |
| kilo | k | $10^{3}$ | kg |
| mega | M | $10^{6}$ | MW |
| giga | G | $10^{9}$ | GB |
| tera | T | $10^{12}$ | TB |
2) Measurement rules you must remember
3) Uncertainties — definitions & how to combine
Absolute uncertainty $\Delta x$ — the ± spread in the same units as $x$.
Fractional uncertainty $\dfrac{\Delta x}{x}$.
Percentage uncertainty $\dfrac{\Delta x}{x}\times 100\%$.
Combining rules:
- Add/Subtract: $y=a\pm b \Rightarrow \Delta y=\Delta a+\Delta b$ (absolute add).
- Multiply/Divide: $y=ab$ or $y=a/b \Rightarrow \dfrac{\Delta y}{y}=\dfrac{\Delta a}{a}+\dfrac{\Delta b}{b}$ (percentage add).
- Powers: $y=a^{n} \Rightarrow \dfrac{\Delta y}{y}=|n|\dfrac{\Delta a}{a}$.
- General products: $y=k a^{p} b^{q} c^{r}\Rightarrow \dfrac{\Delta y}{y}=|p|\dfrac{\Delta a}{a}+|q|\dfrac{\Delta b}{b}+|r|\dfrac{\Delta c}{c}$.
4) Exhaustive worked examples (uncertainties)
Ex.1 — Reading an analogue ruler
A ruler has 1 mm divisions. A length is read as $72.3$ mm.Resolution rule: $\Delta l=\pm0.5$ mm (half smallest division).
Quote: $l=(72.3\;\pm\;0.5)$ mm. Percentage uncertainty $=\dfrac{0.5}{72.3}\times100\%\approx0.69\%$.
Ex.2 — Reading a digital voltmeter
A DVM shows $2.34$ V with a step of $0.01$ V.Resolution rule: $\Delta V=\pm0.01$ V.
Quote: $V=(2.34\;\pm\;0.01)$ V. Percentage uncertainty $\approx0.43\%$.
Ex.3 — Repeated stopwatch readings (mean & half-range)
Times (s): 1.22, 1.25, 1.24, 1.23, 1.26.- Mean $\bar t=1.240$ s.
- Half-range $=\tfrac{1.26-1.22}{2}=0.020$ s.
Quote: $t=(1.240\;\pm\;0.020)$ s. Percentage uncertainty $=\dfrac{0.020}{1.240}\times100\%\approx1.61\%$.
Ex.4 — Area from a measured diameter
A wire diameter $D=(1.20\;\pm\;0.01)$ mm. Cross-sectional area $A=\dfrac{\pi D^{2}}{4}$.Percentage uncertainty in $A$ is $2\times\dfrac{0.01}{1.20}\times100\%=1.67\%$.
Numerical value: $A=\pi(1.20/2)^2=1.131\;\text{mm}^2=1.131\times10^{-6}\;\text{m}^2$.
Absolute uncertainty: $\Delta A=(1.67\%\;\text{of}\;A)=1.89\times10^{-2}\;\text{mm}^2$.
Ex.5 — $g$ from a pendulum: $g=\dfrac{4\pi^2 L}{T^2}$
$L=(1.000\;\pm\;0.001)$ m, $T=(2.006\;\pm\;0.005)$ s.Best value: $g=\dfrac{4\pi^2(1.000)}{(2.006)^2}=9.8107\;\text{m s}^{-2}$.
Percentage uncertainty: $\dfrac{\Delta g}{g}=\dfrac{\Delta L}{L}+2\dfrac{\Delta T}{T}=0.10\%+2\times0.249\%\approx0.599\%$.
Absolute: $\Delta g=(0.599\%\;\text{of}\;9.8107)=0.0587\;\text{m s}^{-2}$.
Quote: $g=(9.8107\;\pm\;0.0587)\;\text{m s}^{-2}$ $\approx$ $9.811\;\pm\;0.059\;\text{m s}^{-2}$ (3 s.f.).
Ex.6 — Resistivity $\rho=\dfrac{RA}{L}$ using $R$, $D$ and $L$
$R=(12.5\;\pm\;0.1)\;\Omega$, $D=(1.20\;\pm\;0.01)$ mm, $L=(1.000\;\pm\;0.001)$ m.$A=\pi(D/2)^2=1.131\times10^{-6}\;\text{m}^2$ (from Ex.4). $\,\;\dfrac{\Delta A}{A}=2\dfrac{\Delta D}{D}=1.667\%$.
Best value: $\rho=\dfrac{12.5\times1.131\times10^{-6}}{1.000}=1.414\times10^{-5}\;\Omega\,\text{m}$.
Percentage uncertainty: $\dfrac{\Delta\rho}{\rho}=\dfrac{\Delta R}{R}+\dfrac{\Delta A}{A}+\dfrac{\Delta L}{L}=0.8\%+1.667\%+0.1\%\approx2.567\%$.
Absolute: $\Delta\rho\approx2.57\%\times1.414\times10^{-5}=3.63\times10^{-7}\;\Omega\,\text{m}$.
Quote: $\rho=(1.414\;\pm\;0.036)\times10^{-5}\;\Omega\,\text{m}$.
Ex.7 — Electrical power $P=IV$
$I=(0.450\;\pm\;0.005)$ A, $V=(6.00\;\pm\;0.01)$ V.$P=2.700$ W. Percentage uncertainty $=\dfrac{0.005}{0.450}+\dfrac{0.01}{6.00}=1.278\%$.
Absolute $\Delta P=1.278\%\times2.700=0.0345$ W.
Quote: $(2.70\;\pm\;0.03)$ W (2–3 s.f. is appropriate).
Ex.8 — Density $\rho=\dfrac{m}{lwh}$ from mixed instruments
Digital mass $m=(124.56\;\pm\;0.01)$ g. Analogue dimensions: $l=(5.00\;\pm\;0.05)$ cm, $w=(2.00\;\pm\;0.05)$ cm, $h=(1.50\;\pm\;0.05)$ cm.$V=lwh=15.0$ cm$^{3}$; $\rho=8.304\;\text{g cm}^{-3}$.
Percentage $\rho$-uncertainty: $\dfrac{\Delta m}{m}+\dfrac{\Delta l}{l}+\dfrac{\Delta w}{w}+\dfrac{\Delta h}{h}=0.008\%+1.0\%+2.5\%+3.33\%\approx6.84\%$.
Absolute: $\Delta\rho=6.84\%\times8.304\approx0.568\;\text{g cm}^{-3}$.
Quote: $\rho=(8.30\;\pm\;0.57)\;\text{g cm}^{-3}$.
Ex.9 — Series vs parallel resistors
Series: $R_1=(100.0\;\pm\;0.5)\,\Omega$, $R_2=(220.0\;\pm\;0.5)\,\Omega$.
$R_{\text{S}}=320.0\,\Omega$, $\Delta R_{\text{S}}=0.5+0.5=1.0\,\Omega$ ⇒ $(320.0\;\pm\;1.0)\,\Omega$.
Parallel: $R=\left(\dfrac{1}{R_1}+\dfrac{1}{R_2}\right)^{-1}$. Use extreme values.
- $R_{\text{nom}}=68.75\,\Omega$
- $R_{\max}$ (both high): $69.035\,\Omega$; $R_{\min}$ (both low): $68.465\,\Omega$
$\Delta R=\tfrac{R_{\max}-R_{\min}}{2}=0.285\,\Omega$ ⇒ $R=(68.75\;\pm\;0.29)\,\Omega$ (about $0.42\%$).
Ex.10 — Gradient from a graph with uncertainty
Suppose a $T^{2}$ vs $L$ graph gives best-fit slope $m=4.05\;\text{s}^2\,\text{m}^{-1}$; worst acceptable slopes $m_{\min}=3.90$, $m_{\max}=4.20$.Gradient $m=(4.05\;\pm\;0.15)$; percentage $=0.15/4.05\times100\%\approx3.70\%$.
For a pendulum, $g=\dfrac{4\pi^2}{m}$ so $\dfrac{\Delta g}{g}=\dfrac{\Delta m}{m}=3.70\%$ ⇒ $g=(9.748\;\pm\;0.361)$ m s$^{-2}$.
Ex.11 — Addition/Subtraction example
A distance is the sum of three segments: $(1.200\;\pm\;0.005)$ m, $(0.750\;\pm\;0.002)$ m, $(0.305\;\pm\;0.001)$ m.Total $=2.255$ m; $\Delta=0.005+0.002+0.001=0.008$ m ⇒ $(2.255\;\pm\;0.008)$ m.
Ex.12 — Power-law (pencil rule)
If $y=kx^{n}$ (e.g., $I\propto V^{n}$), then $\dfrac{\Delta y}{y}=|n|\dfrac{\Delta x}{x}$. So, doubling the exponent doubles the percentage uncertainty carried from $x$ to $y$.Example: $A\propto D^{2}$ ⇒ a $0.8\%$ diameter uncertainty gives a $1.6\%$ area uncertainty.
5) Graph skills & reporting answers
- Draw a thin best-fit line through the scatter (balanced points above/below).
- Draw two extreme acceptable lines within all error bars to estimate gradient range.
- $\Delta m=\tfrac{m_{\max}-m_{\min}}{2}$; quote $m\pm\Delta m$.
- Match significant figures to your uncertainty (usually 1–2 s.f. in the uncertainty).
- Write value and uncertainty with the same decimal places.
- Always state units after the uncertainty: e.g., $(2.70\;\pm\;0.03)$ W.
6) Common instruments & typical uncertainties
| Instrument | Resolution | Typical $\Delta$ rule | Notes |
|---|---|---|---|
| Ruler (mm) | 1 mm | $\pm0.5$ mm | Use set square to avoid parallax. |
| Vernier caliper | 0.1 mm | $\pm0.05$ mm | Check zero error each time. |
| Micrometer | 0.01 mm | $\pm0.005$ mm | Use ratchet; measure in several orientations. |
| Stopwatch | 0.01 s | $\pm0.01$–$\pm0.02$ s | Human reaction ≈ $\pm0.1$ s dominates single readings; time many oscillations. |
| Digital voltmeter | 0.01 V (example) | $\pm$ one step | Choose range to give 2–3 s.f. |
7) Exam traps & how to avoid them
- For products/divisions, do not add absolute uncertainties — add percentages.
- Quote units with the final answer and with the uncertainty.
- Round the uncertainty first (1–2 s.f.), then round the value to the same decimal places.
- Graph ranges: error bars must reflect measurement uncertainties (not the scatter).
- Zero error: correct readings before calculations.
8) Practice set (exam-style)
- Analogue vs digital: A thermometer with 1 °C divisions reads 23.6 °C. Quote with uncertainty. A digital thermometer reads 23.6 °C with 0.1 °C steps. Quote with uncertainty.
- Percentage uncertainty: $V=(6.00\;\pm\;0.02)$ V. What is the percentage uncertainty?
- Sum of lengths: $(12.0\;\pm\;0.1)$ cm + $(8.50\;\pm\;0.05)$ cm. Quote result.
- Area of a rectangle: $l=(1.200\;\pm\;0.005)$ m, $w=(0.305\;\pm\;0.001)$ m. Find $A$ and its uncertainty.
- Wire area from diameter: $D=(0.80\;\pm\;0.01)$ mm. Find $A$ and percentage uncertainty.
- Power: $I=(0.320\;\pm\;0.003)$ A, $V=(12.0\;\pm\;0.1)$ V. Find $P$ with $\Delta P$.
- Period from 50 oscillations: Time for 50 swings $= (84.6\;\pm\;0.2)$ s. Find $T$ and its percentage uncertainty.
- Density: $m=(245.2\;\pm\;0.1)$ g; $l=(8.0\;\pm\;0.1)$ cm, $w=(3.0\;\pm\;0.1)$ cm, $h=(1.5\;\pm\;0.1)$ cm. Quote $\rho$ with uncertainty.
- Parallel resistors (extremes): $R_1=(150.0\;\pm\;0.5)\,\Omega$, $R_2=(330.0\;\pm\;0.5)\,\Omega$. Find $R_{\parallel}\pm\Delta R$.
- Graph gradient: Best-fit $m=2.60$, worst $m_{\min}=2.45$, $m_{\max}=2.76$. Quote $m$ and percentage uncertainty.
- Pendulum $g$ via gradient: Using Q10’s gradient for $T^2$ vs $L$ (slope $=4\pi^2/g$), determine $g\pm\Delta g$.
Show fully worked answers
- Analogue: $23.6\,^{\circ}\text{C}\;\pm0.5\,^{\circ}\text{C}$. Digital: $23.6\,^{\circ}\text{C}\;\pm0.1\,^{\circ}\text{C}$.
- $\dfrac{0.02}{6.00}\times100\%=0.33\%$.
- $20.50$ cm; $\Delta=0.1+0.05=0.15$ cm ⇒ $(20.50\;\pm\;0.15)$ cm.
- $A=0.3660$ m$^2$. Percent: $\dfrac{0.005}{1.200}+\dfrac{0.001}{0.305}=0.417\%+0.328\%=0.745\%$. $\Delta A=0.00273$ m$^{2}$ ⇒ $(0.3660\;\pm\;0.0027)$ m$^2$.
- $A=\pi(0.80/2)^2=0.503\,\text{mm}^2$. Percent $=2\times\dfrac{0.01}{0.80}=2.5\%$.
- $P=3.84$ W. Percent $=\dfrac{0.003}{0.320}+\dfrac{0.1}{12.0}=0.94\%+0.83\%=1.77\%$. $\Delta P=0.068$ W ⇒ $(3.84\;\pm\;0.07)$ W.
- $T=84.6/50=1.692$ s. $\Delta T/T=\Delta(\text{time})/\text{time}=0.2/84.6=0.236\%$. So $T=(1.692\;\pm\;0.004)$ s.
- $V=36.0$ cm$^3$. $\rho=6.811$ g cm$^{-3}$. Percent $=\dfrac{0.1}{245.2}+\dfrac{0.1}{8.0}+\dfrac{0.1}{3.0}+\dfrac{0.1}{1.5}=0.041\%+1.25\%+3.33\%+6.67\%\approx11.29\%$. $\Delta\rho\approx0.769$ g cm$^{-3}$.
- $R_{\text{nom}}=\big(1/150+1/330\big)^{-1}=103.1\,\Omega$. $R_{\max}$ with both high: $103.7\,\Omega$; $R_{\min}$ both low: $102.6\,\Omega$. $\Delta=0.55\,\Omega$ ⇒ $(103.1\;\pm\;0.6)\,\Omega$.
- $m=(2.60\;\pm\;0.155)$; percentage $=5.96\%$.
- $g=\dfrac{4\pi^2}{m}=\dfrac{39.478}{2.60}=15.18$; percentage same $=5.96\%$ ⇒ $\Delta g=0.905$ ⇒ $(15.2\;\pm\;0.9)$ m s$^{-2}$ (illustrative).
9) Quick self-checks
- Have I used the correct combining rule (absolute vs percentage)?
- Did I correct for zero error before computing?
- Are value and uncertainty rounded to consistent decimal places?
- Is the unit shown with the final quoted result?
- On graphs, did I use error bars and worst-fit lines to estimate $\Delta m$?