express derived units as products or quotients of the SI base units and use the derived units for quantities listed in this syllabus as appropriate

Objective

Express all derived units required by the Cambridge IGCSE/A‑Level Physics syllabus (9702) as products or quotients of the seven SI base units, and apply these units correctly to the physical quantities listed in the syllabus.

1. The Seven SI Base Units

2. SI Prefixes (Commonly Used in the Syllabus)

3. Scalars, Vectors & Unit Vectors

• Scalar quantities have magnitude only (e.g. mass \$m\$, speed \$v\$, energy \$E\$). They are written in italic type.
• Vector quantities have magnitude and direction (e.g. displacement \$\mathbf{s}\$, velocity \$\mathbf{v}\$, force \$\mathbf{F}\$). They are denoted by boldface or an arrow.
• Vector addition follows the tip‑to‑tail rule. In component form:

  \$\mathbf{A}=Ax\hat{\mathbf{i}}+Ay\hat{\mathbf{j}}+A_z\hat{\mathbf{k}}\$

  where \$\hat{\mathbf{i}},\hat{\mathbf{j}},\hat{\mathbf{k}}\$ are the unit vectors along the \$x\$, \$y\$, \$z\$ axes.

• When checking units, vectors and scalars use the same dimensional expression; only the direction is omitted for scalars.

4. Dimensional Analysis & Unit Consistency

Every physical equation must be dimensionally homogeneous: the dimensions (or units) on the left‑hand side (LHS) must match those on the right‑hand side (RHS). This is a compulsory check in the exam.

• Write each symbol with its base‑unit expression, multiply or divide as the formula requires, and verify that the resulting unit matches the named derived unit.
• If the units do not match, the equation is either written incorrectly or a factor (e.g. \$2\pi\$, \$1/2\$) has been omitted.

5. Uncertainty, Errors & Significant Figures (AO2)

• Random (statistical) errors – arise from fluctuations; reduced by repeated measurements; expressed as standard deviation \$\sigma\$ or as \$\pm\$ value.
• Systematic errors – bias the result in one direction; caused by mis‑calibration, zero‑offset, etc.; identified by comparison with a known standard.
• Precision vs. Accuracy

  • Precision = closeness of repeated measurements (small random error).
  • Accuracy = closeness to the true value (small systematic error).

• Percentage (relative) uncertainty:

  \$\%\,\Delta Q = \frac{\Delta Q}{|Q|}\times100\$

• Propagation of uncertainties (for products and quotients):

  \$\frac{\Delta Q}{|Q|}\approx\frac{\Delta a}{|a|}+\frac{\Delta b}{|b|}+\dots\$

  where \$Q=a\,b\$ or \$Q=a/b\$.

• Significant figures – the number of digits that are known reliably. In calculations:

  • Multiplication / division: keep the same number of significant figures as the factor with the fewest.
  • Addition / subtraction: keep the same number of decimal places as the term with the fewest.

6. Derived Units Required by the Syllabus

7. Worked Examples – Unit Checks & Calculations

Example 1: Kinetic Energy

1. Formula: \$E_{\rm k}= \frac12 m v^{2}\$.
2. Insert base‑unit expressions:
   

   $E_{\rm k}= \frac12\;( \text{kg})\;(\text{m·s}^{-1})^{2}

   = \frac12\;\text{kg·m}^{2}\text{s}^{-2}$.

3. Recognise \$\text{kg·m}^{2}\text{s}^{-2}\$ as a joule (J).
   

   For \$m=2.0\$ kg, \$v=5.0\$ m s⁻¹: \$E_{\rm k}= \frac12(2.0)(5.0)^{2}=25\$ J.

4. Uncertainty (if \$m=2.0\pm0.1\$ kg, \$v=5.0\pm0.2\$ m s⁻¹):
   

   $\displaystyle\frac{\Delta E}{E}\approx\frac{\Delta m}{m}+2\frac{\Delta v}{v}

   =\frac{0.1}{2.0}+2\frac{0.2}{5.0}=0.05+0.08=0.13$
   

   \$\Delta E\approx0.13\times25\approx3\$ J → \$E_{\rm k}=25\pm3\$ J.

Example 2: Newton’s 2nd Law (Unit Consistency)

Equation: \$\mathbf{F}=m\mathbf{a}\$.

• \$m\$ → kg
• \$a\$ → m·s^‑2
• Product: kg·m·s^‑2 = N (newton).

Thus the units are consistent: \$[\mathbf{F}] = \text{N}\$.

Example 3: Momentum Conservation

Two carts on a frictionless track: \$m{1}=0.50\$ kg, \$u{1}=2.0\$ m s⁻¹; \$m{2}=0.80\$ kg, \$u{2}=0\$ m s⁻¹. After collision they stick together and move with speed \$v\$.

1. Conserve momentum: \$m{1}u{1}+m{2}u{2}= (m{1}+m{2})v\$.
2. Units: kg·m·s^‑1 on both sides – correct.
3. Solve: \$v=\dfrac{0.50\times2.0}{0.50+0.80}=0.71\$ m s⁻¹.

Example 4: Torque

Torque \$\tau = rF\sin\theta\$.

• \$r\$ → m, \$F\$ → N (kg·m·s^‑2), \$\sin\theta\$ is dimensionless.
• Product: m·kg·m·s^‑2 = kg·m²·s^‑2 = N m.
• If \$r=0.20\$ m and \$F=15\$ N acting perpendicular to \$r\$, \$\tau =0.20\times15=3.0\$ N m.

Example 5: Electrical Power with Uncertainty

Power \$P = VI\$ where \$V=12.0\pm0.2\$ V and \$I=1.50\pm0.05\$ A.

• Unit check: V·A = (kg·m²·s^‑3·A^‑1)·A = kg·m²·s^‑3 = W.
• Calculate \$P = 12.0\times1.50 = 18.0\$ W.
• Relative uncertainty: $\displaystyle\frac{\Delta P}{P}\approx\frac{\Delta V}{V}+\frac{\Delta I}{I}

  =\frac{0.2}{12.0}+\frac{0.05}{1.50}=0.0167+0.0333=0.050$

• Absolute uncertainty: \$\Delta P =0.05\times18.0\approx0.9\$ W → \$P=18.0\pm0.9\$ W.

8. Mapping of This Note to the Cambridge Syllabus

9. Quick Reference – Common Unit Conversions Used in the Syllabus

• Speed: \$1\;\text{m s}^{-1}=3.6\;\text{km h}^{-1}\$.
• Energy: \$1\;\text{eV}=1.602\times10^{-19}\;\text{J}\$ (useful for particle physics).
• Pressure: \$1\;\text{atm}=1.013\times10^{5}\;\text{Pa}\$.
• Power: \$1\;\text{kW}=10^{3}\;\text{W}\$, \$1\;\text{MW}=10^{6}\;\text{W}\$.
• Frequency: \$1\;\text{kHz}=10^{3}\;\text{Hz}\$, \$1\;\text{MHz}=10^{6}\;\text{Hz}\$.

10. Summary Checklist for Exam Preparation

1. Know the seven base units and be able to write any derived unit as a product/quotient of them.
2. Be fluent with the common SI prefixes and the associated powers of ten.
3. Distinguish scalars from vectors; use unit‑vector notation when required.
4. Perform a dimensional check on every equation you write.
5. Apply the correct rules for significant figures and propagate uncertainties for multiplication/division.
6. Memorise the derived‑unit table above – it covers every quantity you will meet in the syllabus.
7. Practice the worked examples; they illustrate both unit consistency and uncertainty handling.