Recall the SI base quantities and their units, express derived units as products or quotients of base units, and use base units to check the homogeneity of physical equations

What this dot point is asking

SEAB wants you to know the seven SI base quantities and their units, to express any derived unit (newton, joule, pascal and so on) as a product or quotient of base units, and to use base units to test whether a physical equation is homogeneous. This is the foundation of every numerical answer in H2 Physics, and homogeneity checks appear in Paper 1, Paper 2 and the data analysis of Paper 3.

The answer

The seven base quantities

Every physical quantity in the syllabus is built from seven base quantities, each with one SI base unit.

For H2 Physics, the first six matter constantly; the candela is rarely used in calculations.

Derived units from base units

A derived unit is any combination of base units. You build it by substituting the defining equation of the quantity.

• Speed: $v = \dfrac{\text{distance}}{\text{time}}$, so $[v] = \text{m s}^{-1}$.
• Force: $F = ma$, so $[F] = \text{kg} \times \text{m s}^{-2} = \text{kg m s}^{-2}$. This combination is named the newton, $\text{N}$.
• Energy: $E = Fd$, so $[E] = \text{kg m s}^{-2} \times \text{m} = \text{kg m}^2\text{s}^{-2}$, named the joule, $\text{J}$.
• Power: $P = \dfrac{E}{t}$, so $[P] = \text{kg m}^2\text{s}^{-3}$, named the watt, $\text{W}$.
• Pressure: $p = \dfrac{F}{A}$, so $[p] = \text{kg m}^{-1}\text{s}^{-2}$, named the pascal, $\text{Pa}$.
• Electric charge: $Q = It$, so $[Q] = \text{A s}$, named the coulomb, $\text{C}$.

Homogeneity of equations

An equation is homogeneous (dimensionally consistent) if every additive term has the same base units. To test it:

1. Write the left side in base units.
2. Write each term on the right side in base units.
3. If they all match, the equation is homogeneous.

Homogeneity is a quick error-check: if the units do not match, the equation is definitely wrong. The converse is not guaranteed, which is the key subtlety examiners probe.

What homogeneity cannot catch

A homogeneous equation may still be wrong. Three things slip through:

• A wrong pure-number coefficient (such as a missing factor of $\frac{1}{2}$ or $2\pi$), because pure numbers carry no units.
• A missing dimensionless term added to a correct one.
• An error inside a function whose argument must be dimensionless (such as $\sin$, $\ln$ or $e^x$).

So homogeneity is a necessary condition for correctness, not a sufficient one.

Examples in context

Example 1. Checking a drag equation. A student writes the terminal velocity of a sphere as $v = \dfrac{2 r^2 (\rho_s - \rho_f) g}{9 \eta}$. The right side has units $\dfrac{\text{m}^2 \cdot \text{kg m}^{-3} \cdot \text{m s}^{-2}}{\text{kg m}^{-1}\text{s}^{-1}} = \dfrac{\text{kg m}^0 \text{s}^{-2}}{\text{kg m}^{-1}\text{s}^{-1}} = \text{m s}^{-1}$, which matches a velocity. The equation passes the homogeneity test.

Example 2. Spotting an error. A candidate proposes kinetic energy as $E_k = m v$. Base units give $\text{kg} \cdot \text{m s}^{-1} = \text{kg m s}^{-1}$, which is momentum, not energy ($\text{kg m}^2\text{s}^{-2}$). The homogeneity check immediately flags the missing factor of $v$, steering the student to $E_k = \frac{1}{2}mv^2$.

Try this

Q1. State the seven SI base quantities and their units. [3 marks]

• Cue. Mass (kg), length (m), time (s), current (A), thermodynamic temperature (K), amount of substance (mol), luminous intensity (cd).

Q2. Express the joule and the watt in SI base units. [2 marks]

• Cue. $\text{J} = \text{kg m}^2\text{s}^{-2}$, $\text{W} = \text{kg m}^2\text{s}^{-3}$.

Q3. The power radiated by a black body is modelled as $P = \sigma A T^4$, where $A$ is area and $T$ is temperature. Determine the base units of the constant $\sigma$. [3 marks]

• Cue. $[\sigma] = \dfrac{[P]}{[A][T^4]} = \dfrac{\text{kg m}^2\text{s}^{-3}}{\text{m}^2 \cdot \text{K}^4} = \text{kg s}^{-3}\text{K}^{-4}$.