recall the following SI base quantities and their units: mass (kg), length (m), time (s), current (A), temperature (K)
Cambridge International AS & A Level Physics (9702) – SI Units, Measurement & Vectors
This note fulfils Topic 1.1‑1.4 of the Cambridge syllabus. It is written for rapid recall, exam practice and as a ready reference when solving problems in later topics (kinematics, dynamics, forces & pressure).
1. Physical Quantities and SI Units
- Physical quantity: a measurable property that always has a magnitude + unit. For example, “mass = 0.45 kg”.
- Every quantity can be estimated to an order of magnitude using everyday objects – a useful skill for the “reasonable estimates” requirement.
| Quantity | Symbol | SI Unit (name) | Unit Symbol | Typical Everyday Estimate |
|---|---|---|---|---|
| Mass | m | kilogram | kg | Textbook ≈ 0.5 kg |
| Length | ℓ | metre | m | Ruler ≈ 0.3 m |
| Time | t | second | s | Stopwatch tick ≈ 1 s |
| Electric current | I | ampere | A | AA battery ≈ 1 A (when discharged) |
| Thermodynamic temperature | T | kelvin | K | Room temperature ≈ 293 K |
2. SI Prefixes (Scaling Factors)
| Prefix | Symbol | Factor |
|---|---|---|
| tera | T | 1012 |
| giga | G | 109 |
| mega | M | 106 |
| kilo | k | 103 |
| hecto | h | 102 |
| deca | da | 101 |
| deci | d | 10-1 |
| centi | c | 10-2 |
| milli | m | 10-3 |
| micro | µ | 10-6 |
| nano | n | 10-9 |
| pico | p | 10-12 |
3. Common Derived Units
Derived units are expressed as products (or quotients) of the base units. Only the most frequently used units for the syllabus are listed.
| Quantity | Symbol | SI Unit (name) | Expression in base units |
|---|---|---|---|
| Force | F, N | newton | kg·m·s-2 |
| Energy / Work | E, J | joule | kg·m2·s-2 |
| Power | P, W | watt | kg·m2·s-3 |
| Pressure | p, Pa | pascal | kg·m-1·s-2 |
| Electric charge | Q, C | coulomb | A·s |
| Voltage | V | volt | kg·m2·s-3·A-1 |
4. Dimensional Analysis (Checking Homogeneity)
Every term in a physically correct equation must have identical dimensions. This provides a quick sanity check.
Example – free‑fall speed:
\[
v = \sqrt{2gh}
\]
- Left‑hand side: \(v\) → dimensions \(L\,T^{-1}\).
- Right‑hand side: \(2gh\) has dimensions \((L\,T^{-2})\times L = L^{2}\,T^{-2}\); the square‑root gives \(L\,T^{-1}\).
Since both sides match, the equation is dimensionally consistent.
5. Measurement Uncertainty
- Systematic error – a bias that shifts all readings in the same direction (e.g., mis‑calibrated scale).
- Random error – scatter caused by limitations of the measuring technique; reduced by repeated measurements.
- Absolute uncertainty – ± value expressed in the same units as the measurement.
- Relative (percentage) uncertainty – \(\displaystyle \frac{\Delta x}{x}\times100\%.\)
Propagation of uncertainties (for products or quotients):
\[
\frac{\Delta Q}{Q}= \sqrt{\left(\frac{\Delta A}{A}\right)^{2}+\left(\frac{\Delta B}{B}\right)^{2}}
\]
For powers, multiply the relative uncertainty by the exponent. Example for \(v = \sqrt{2gh}\):
\[
\frac{\Delta v}{v}= \tfrac12\frac{\Delta g}{g}+ \tfrac12\frac{\Delta h}{h}
\]
6. Scalars and Vectors
- Scalar – described by magnitude only (e.g., mass, temperature, energy).
- Vector – described by magnitude and direction (e.g., displacement, force, velocity).
6.1 Vector Notation
Vectors are written in bold (\(\mathbf{F}\)) or with an arrow (\(\vec{F}\)). Components are denoted \(Fx, Fy, F_z\).
6.2 Vector Addition
Tip‑to‑tail (graphical) method:
Component method (used in calculations):
\[
\vec{R}= (F{1x}+F{2x})\,\hat{i} + (F{1y}+F{2y})\,\hat{j}
\]
\[
|\vec{R}| = \sqrt{Rx^{2}+Ry^{2}},\qquad \theta = \tan^{-1}\!\left(\frac{Ry}{Rx}\right)
\]
7. Quick Recall & Practice
- Write the SI unit (symbol) for each of the five base quantities from memory.
- Express the following derived units in base‑unit form:
- Joule (J)
- Pascal (Pa)
- Coulomb (C)
- Check the dimensional consistency of the equation \(s = ut + \tfrac12 a t^{2}\).
- A mass of 0.250 kg is measured with a balance that has an absolute uncertainty of ±0.001 kg. Calculate the percentage uncertainty.
- Two forces act on a point: \(\vec{F}1 = 5\;\text{N}\) east and \(\vec{F}2 = 5\;\text{N}\) north. Determine the magnitude and direction of the resultant using both tip‑to‑tail and component methods.
- Estimate the order of magnitude (using the “reasonable estimates” skill) for:
- Mass of a school‑bag.
- Length of a typical classroom.
- Current drawn by a small LED (≈ 20 mA).
8. Links to Later Topics (Why This Matters)
- Kinematics (Topic 2) – uses the base quantity time and derived units of velocity (m s⁻¹) and acceleration (m s⁻²).
- Dynamics (Topic 3) – requires force (N) and mass (kg) to apply Newton’s laws; momentum (kg·m s⁻¹) and impulse (N·s) are derived quantities.
- Forces, Density & Pressure (Topic 4) – weight \(W = mg\) (N), density \(\rho = m/V\) (kg m⁻³), pressure \(p = \rho g h\) (Pa); all built from the base units introduced here.
- Understanding uncertainties and dimensional analysis is essential for the experimental work and data‑handling components of the syllabus.
9. Summary of Key Points
- Every physical quantity = magnitude + unit; the five SI base quantities are mass (kg), length (m), time (s), electric current (A) and temperature (K).
- SI prefixes allow convenient scaling from pico‑ (10⁻¹²) to tera‑ (10¹²).
- Derived units are constructed from base units (e.g., N = kg·m·s⁻², J = kg·m²·s⁻²).
- Dimensional analysis checks that equations are physically plausible.
- Distinguish systematic and random errors; combine uncertainties using the appropriate propagation formulas.
- Scalars have magnitude only; vectors have magnitude and direction – use tip‑to‑tail or component methods for addition.
- Reasonable estimation of magnitudes is a required skill for the exam.