SI units

1. Overview

The International System of Units (SI) is the foundational language of physics, ensuring that measurements are universal, reproducible, and mathematically consistent. Every physical quantity is defined by the product of a numerical magnitude and a unit; without a unit, a number lacks physical context and meaning. The system is structured hierarchically: it begins with a small set of base units defined by fundamental physical constants, from which all other derived units are constructed through algebraic multiplication or division. This logical consistency allows physicists to verify the homogeneity of equations—a rigorous check to ensure that a mathematical model is physically possible. Mastery of SI units, including the fluent conversion of prefixes and the derivation of complex units from first principles, is a core competency required for every topic in the Cambridge 9702 syllabus, from classical mechanics to nuclear physics.

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Key Definitions

• SI Base Quantity: A fundamental physical quantity that is defined independently and cannot be expressed in terms of other quantities. The Cambridge syllabus focuses on five primary base quantities: mass, length, time, electric current, and thermodynamic temperature.
• SI Base Unit: The standard unit of measurement for a base quantity, currently defined by fixing the numerical values of seven universal constants (e.g., the Planck constant, the elementary charge).
• Derived Unit: A unit of measurement formed by the multiplication or division of SI base units. Derived units do not have independent definitions but are defined by the physical laws relating them to base quantities (e.g., the Newton is defined by $F=ma$).
• Homogeneous Equation: An equation where every additive term (quantities separated by $+$, $-$, or $=$) possesses the identical SI base units. Homogeneity is a necessary condition for any physical equation to be valid.
• Prefix: A standardized symbol and multiplier applied to a unit to represent powers of ten. Prefixes allow for the concise expression of very large or very small magnitudes in scientific notation.
• Scalar Quantity: A physical quantity described entirely by its magnitude and unit, possessing no spatial direction (e.g., mass, energy, temperature).
• Vector Quantity: A physical quantity that requires both a magnitude and a specific spatial direction for a complete description (e.g., velocity, force, acceleration).

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Content

3.1 SI Base Quantities and Units

The International System identifies seven base units, but for the 9702 syllabus, you must be able to recall and use the following five:

Base Quantity	Symbol	SI Base Unit	Unit Symbol
Mass	$m$	kilogram	$kg$
Length	$l, s, d, x$	metre	$m$
Time	$t$	second	$s$
Electric Current	$I$	ampere	$A$
Thermodynamic Temperature	$T$	kelvin	$K$

The Kilogram Exception The kilogram ($kg$) is unique as it is the only SI base unit that includes a prefix ("kilo") in its name. In all unit derivations and homogeneity checks, $kg$ must be treated as the fundamental unit. Do not convert it to grams ($g$) when asked for SI base units.

The Kelvin Scale Thermodynamic temperature must be expressed in Kelvin ($K$). Unlike the Celsius scale, the Kelvin scale is absolute, meaning $0\ K$ is absolute zero.

• Conversion: $T(K) = \theta(^\circ C) + 273.15$
• Temperature Differences: A change of $1\ K$ is equivalent to a change of $1\ ^\circ C$ ($\Delta T = \Delta \theta$).

3.2 Derived Units

Derived units are created by combining base units according to the physical definitions of the quantities. To express a derived unit in SI base units, you must follow a systematic reduction process.

The Derivation Procedure:

1. Identify the defining equation for the quantity.
2. Substitute the units for each term in the equation.
3. If a term is itself a derived unit (like Force), break it down further until only base units remain.
4. Simplify the expression using the laws of indices, ensuring all units are on a single line with negative indices for denominators (e.g., $kg \cdot m \cdot s^{-2}$).

Table of Essential Derived Units:

Quantity	Defining Equation	Derived Unit	SI Base Unit Equivalent
Frequency	$f = 1/T$	hertz ($Hz$)	$s^{-1}$
Velocity	$v = \Delta s / \Delta t$	$m \cdot s^{-1}$	$m \cdot s^{-1}$
Acceleration	$a = \Delta v / \Delta t$	$m \cdot s^{-2}$	$m \cdot s^{-2}$
Force	$F = ma$	newton ($N$)	$kg \cdot m \cdot s^{-2}$
Work / Energy	$W = Fs$	joule ($J$)	$kg \cdot m^2 \cdot s^{-2}$
Power	$P = W/t$	watt ($W$)	$kg \cdot m^2 \cdot s^{-3}$
Pressure	$p = F/A$	pascal ($Pa$)	$kg \cdot m^{-1} \cdot s^{-2}$
Charge	$Q = It$	coulomb ($C$)	$A \cdot s$
Potential Diff.	$V = W/Q$	volt ($V$)	$kg \cdot m^2 \cdot s^{-3} \cdot A^{-1}$
Resistance	$R = V/I$	ohm ($\Omega$)	$kg \cdot m^2 \cdot s^{-3} \cdot A^{-2}$
Capacitance	$C = Q/V$	farad ($F$)	$kg^{-1} \cdot m^{-2} \cdot s^4 \cdot A^2$
Magnetic Flux Density	$F = BIl$	tesla ($T$)	$kg \cdot s^{-2} \cdot A^{-1}$

3.3 Homogeneity of Equations

An equation is homogeneous if the SI base units of every term are identical. This is a fundamental requirement for any physical law.

Critical Rules for Homogeneity:

1. Terms: In the equation $A = B + C$, the units of $A$, $B$, and $C$ must all be the same. You cannot add 5 Newtons to 3 Joules.
2. Dimensionless Constants: Pure numbers (e.g., $1/2$, $2\pi$, $e$) and ratios (e.g., strain, refractive index) have no units. They are ignored during homogeneity checks.
3. Exponents: The power to which a number is raised must be dimensionless. In $e^{kt}$, the product $kt$ must have no units, meaning the unit of $k$ must be the reciprocal of the unit of $t$.
4. Trigonometric Functions: The arguments of functions like $\sin(\theta)$ or $\cos(kx)$ must be dimensionless (usually expressed in radians, which is a dimensionless ratio of lengths).

Note on Validity: Homogeneity does not guarantee an equation is correct. For example, $E_k = mv^2$ is homogeneous but physically incorrect because it lacks the dimensionless factor of $1/2$. Homogeneity only proves an equation is possible.

3.4 Prefixes and Unit Conversions

Prefixes are used to scale units by powers of ten. You must memorize the following symbols and their corresponding multipliers:

Prefix	Symbol	Multiplier	Power of 10
Tera	$T$	$1,000,000,000,000$	$10^{12}$
Giga	$G$	$1,000,000,000$	$10^{9}$
Mega	$M$	$1,000,000$	$10^{6}$
Kilo	$k$	$1,000$	$10^{3}$
Deci	$d$	$0.1$	$10^{-1}$
Centi	$c$	$0.01$	$10^{-2}$
Milli	$m$	$0.001$	$10^{-3}$
Micro	$\mu$	$0.000001$	$10^{-6}$
Nano	$n$	$0.000000001$	$10^{-9}$
Pico	$p$	$0.000000000001$	$10^{-12}$

Converting Squared and Cubed Units A common error occurs when converting units of area or volume. The prefix multiplier must be raised to the same power as the unit.

• Area: To convert $cm^2$ to $m^2$: $(10^{-2} m)^2 = 10^{-4} m^2$
• Volume: To convert $mm^3$ to $m^3$: $(10^{-3} m)^3 = 10^{-9} m^3$
• Density: To convert $g \cdot cm^{-3}$ to $kg \cdot m^{-3}$: $1 \frac{g}{cm^3} = \frac{10^{-3} kg}{(10^{-2} m)^3} = \frac{10^{-3} kg}{10^{-6} m^3} = 10^3 kg \cdot m^{-3}$

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Worked Example 1 — Deriving the Units of the Universal Gravitational Constant

Question: Newton's Law of Gravitation is given by $F = \frac{Gm_1m_2}{r^2}$, where $F$ is force, $m_1$ and $m_2$ are masses, and $r$ is the distance between their centers. Determine the SI base units of $G$.

Step 1: Rearrange for $G$. $G = \frac{Fr^2}{m_1m_2}$

Step 2: Substitute the units for each quantity.

• $F$ (Force): $kg \cdot m \cdot s^{-2}$
• $r^2$ (Distance squared): $m^2$
• $m_1, m_2$ (Masses): $kg \cdot kg = kg^2$

Step 3: Combine and simplify. $[G] = \frac{(kg \cdot m \cdot s^{-2}) \cdot m^2}{kg^2}$ $[G] = \frac{kg \cdot m^3 \cdot s^{-2}}{kg^2}$ $[G] = kg^{1-2} \cdot m^3 \cdot s^{-2}$

Final Answer: $kg^{-1} \cdot m^3 \cdot s^{-2}$

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Worked Example 2 — Checking Homogeneity of a Complex Equation

Question: The period $T$ of an oscillating liquid column is suggested to be $T = 2\pi \sqrt{\frac{h}{2g}}$, where $h$ is the height of the column and $g$ is the acceleration of free fall. Determine if this equation is homogeneous.

Step 1: Units of the Left-Hand Side (LHS). $T$ is time: $[LHS] = s$

Step 2: Units of the Right-Hand Side (RHS).

• $2\pi$ and the $2$ in the denominator are dimensionless constants.
• $h$ (height): $m$
• $g$ (acceleration): $m \cdot s^{-2}$

Step 3: Substitute into the square root. $[RHS] = \sqrt{\frac{m}{m \cdot s^{-2}}}$ $[RHS] = \sqrt{\frac{1}{s^{-2}}}$ $[RHS] = \sqrt{s^2}$ $[RHS] = s$

Step 4: Compare LHS and RHS. $LHS = s$ and $RHS = s$. Conclusion: The units are identical; the equation is homogeneous.

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Worked Example 3 — Units of a Constant in an Exponential Function

Question: The intensity $I$ of X-rays passing through a material of thickness $x$ is given by $I = I_0 e^{-\mu x}$, where $I_0$ is the initial intensity and $\mu$ is the linear absorption coefficient. Determine the SI base units of $\mu$.

Step 1: Analyze the exponential argument. In any expression $e^k$, the exponent $k$ must be dimensionless (it has no units). Therefore, the term $(-\mu x)$ must be dimensionless.

Step 2: Set up the unit equation. $[\mu] \cdot [x] = 1$ (where 1 represents no units)

Step 3: Solve for the units of $\mu$. $[\mu] = \frac{1}{[x]}$ Since $x$ is thickness (length), its unit is $m$. $[\mu] = \frac{1}{m} = m^{-1}$

Final Answer: $m^{-1}$

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Key Equations

The following equations are frequently used for unit derivations. You must know which are provided on the Data Sheet and which must be memorized.

• $F = ma$ (Force; Memorise)
• $W = Fs \cos\theta$ (Work Done; Memorise)
• $P = \frac{W}{t}$ (Power; Memorise)
• $p = \frac{F}{A}$ (Pressure; Memorise)
• $\Delta Q = I \Delta t$ (Charge; Memorise)
• $V = \frac{W}{Q}$ (Potential Difference; Memorise)
• $V = IR$ (Resistance; Memorise)
• $E_k = \frac{1}{2}mv^2$ (Kinetic Energy; Memorise)
• $\Delta E_p = mgh$ (Gravitational Potential Energy; Memorise)
• $R = \frac{\rho L}{A}$ (Resistivity; Data Sheet)
• $E = \frac{\text{stress}}{\text{strain}}$ (Young Modulus; Data Sheet)
• $F = 6\pi\eta rv$ (Stokes' Law for Viscosity; Data Sheet)

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Common Mistakes to Avoid

• ❌ Wrong: Treating the "kilo" in kilogram as a prefix to be removed during base unit derivation (e.g., converting $kg$ to $g$).
  • ✓ Right: The kilogram is the base unit. Keep it as $kg$ in all final SI base unit expressions.
• ❌ Wrong: Adding units during a homogeneity check (e.g., saying the units of $v + u$ are $2 m \cdot s^{-1}$).
  • ✓ Right: Units are not treated as algebraic variables in addition. The unit of the entire term $(v + u)$ is simply $m \cdot s^{-1}$.
• ❌ Wrong: Forgetting to apply the power to the prefix multiplier (e.g., $1\ cm^2 = 10^{-2}\ m^2$).
  • ✓ Right: $1\ cm^2 = (10^{-2}\ m)^2 = 10^{-4}\ m^2$. Always use brackets.
• ❌ Wrong: Confusing the quantity symbol with the unit symbol (e.g., using $A$ for Area in a unit derivation instead of $m^2$).
  • ✓ Right: Distinguish between the variable (e.g., $I$ for current) and the unit (e.g., $A$ for Ampere).
• ❌ Wrong: Assuming that a homogeneous equation is definitely correct.
  • ✓ Right: Homogeneity is a necessary but not sufficient condition. It cannot detect incorrect dimensionless constants like $1/2$ or $\pi$.
• ❌ Wrong: Using $^\circ C$ in equations involving thermodynamic temperature ratios.
  • ✓ Right: Always use Kelvin ($K$). While $\Delta 1\ ^\circ C = \Delta 1\ K$, the absolute values are different ($10\ ^\circ C$ is not $10$ times $1\ ^\circ C$ in terms of energy, but $10\ K$ is $10$ times $1\ K$).

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Exam Tips

1. The "Show That" Strategy: If a question asks you to "show that the units of $X$ are...", you must write down the defining equation first. Examiners award marks for the logical progression: Equation $\rightarrow$ Substitution $\rightarrow$ Simplification.
2. Use the Data Sheet: If you are asked for the units of an unfamiliar constant (like the Stefan-Boltzmann constant), find it on the Data Sheet. The units provided there (e.g., $W \cdot m^{-2} \cdot K^{-4}$) can be used as a starting point to break down into base units.
3. Negative Indices: Always express your final answer in index notation (e.g., $kg \cdot m^2 \cdot s^{-3}$) rather than using slashes ($J/s$). This is the standard expected in A-Level Physics.
4. Check the Gradient: In Paper 3 (Practical) or graph questions in Paper 2, the units of a gradient are the units of the y-axis divided by the units of the x-axis. Simplifying these to base units often reveals the physical identity of the gradient.
5. Dimensionless Ratios: Be aware of quantities that have no units, such as strain ($\Delta L / L$), refractive index ($c/v$), and relative density. In homogeneity questions, these terms are treated as the number 1.
6. Significant Figures in Conversions: When converting a value from $mm$ to $m$ in a calculation, do not change the number of significant figures. $2.0\ mm$ becomes $2.0 \times 10^{-3}\ m$, not $0.002\ m$ (which is 1 s.f.).