Physical quantities

1. Overview

In physics, a physical quantity is a property of an object or a phenomenon that can be measured or calculated. Every physical quantity is treated as a single algebraic entity consisting of two mandatory components: a numerical magnitude and a unit. This relationship is defined by the fundamental identity:

$$\text{Physical Quantity} = \text{Numerical Magnitude} \times \text{Unit}$$

The magnitude provides the "how much," while the unit provides the "of what." A value without a unit is physically ambiguous, and a unit without a magnitude is data-less. For example, stating a mass is "70" is meaningless (70 grams? 70 tonnes?), and stating a mass is "kilograms" provides no specific information.

In the Cambridge 9702 syllabus, you must treat units as algebraic symbols. This allows for complex unit manipulation, conversion using power-of-ten prefixes, and the ability to perform "sanity checks" on your calculations by estimating the order of magnitude of your results.

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Key Definitions

• Physical Quantity: A feature of an object or phenomenon that can be measured or calculated, consisting of a numerical magnitude and a unit.
• Numerical Magnitude: The number or value that indicates the size of the quantity relative to the unit.
• Unit: A standard reference used to define the magnitude of a physical quantity.
• SI Base Quantity: One of seven independent physical quantities that serve as the foundation of the International System of Units (SI). These are the "building blocks" of all other units.
• SI Base Unit: The defined standards for the base quantities (e.g., the kilogram for mass, the metre for length).
• Derived Quantity: A physical quantity that is defined in terms of the product or quotient of base quantities (e.g., force is mass multiplied by acceleration).
• Order of Magnitude: An estimate of a quantity rounded to the nearest power of ten. It is used to compare sizes and check the reasonableness of a value.
• Standard Form: A method of writing numbers as a figure between 1 and 10 multiplied by a power of 10 (e.g., $A \times 10^n$).

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Content

3.1 The Algebraic Nature of Units

Units are not merely labels; they follow the laws of algebra. When you multiply or divide physical quantities, you must apply the same operations to their units.

• Multiplication: $\text{Work} = \text{Force} \times \text{distance} = (10\text{ N}) \times (2.0\text{ m}) = 20\text{ N m}$.
• Division: $\text{Pressure} = \frac{\text{Force}}{\text{Area}} = \frac{500\text{ N}}{2.0\text{ m}^2} = 250\text{ N m}^{-2}$.

Notation Convention: In A-Level Physics, we avoid the solidus (/) for units. Instead, we use negative indices.

• Use $\text{m s}^{-1}$ instead of $\text{m/s}$.
• Use $\text{kg m}^{-3}$ instead of $\text{kg/m}^3$.
• Use $\text{J K}^{-1}\text{ mol}^{-1}$ instead of $\text{J/(K}\cdot\text{mol)}$.

3.2 The SI System (Le Système International d'Unités)

The SI system is the international standard for scientific measurement. While there are seven base units, the 9702 syllabus focuses on the first six.

Base Quantity	Symbol	SI Base Unit	Unit Abbreviation
Mass	$m$	kilogram	$\text{kg}$
Length	$l, x, s, d$	metre	$\text{m}$
Time	$t$	second	$\text{s}$
Electric Current	$I$	ampere	$\text{A}$
Thermodynamic Temperature	$T$	kelvin	$\text{K}$
Amount of Substance	$n$	mole	$\text{mol}$

Note: The seventh base unit is the candela ($\text{cd}$) for luminous intensity, which is not assessed in this syllabus.

3.3 SI Prefixes and Unit Conversion

Prefixes are used to scale units to appropriate orders of magnitude, making numbers more manageable. You must be able to convert between these fluently.

Prefix	Symbol	Multiplier	Example
Tera	$\text{T}$	$10^{12}$	$1\text{ THz} = 10^{12}\text{ Hz}$
Giga	$\text{G}$	$10^9$	$1\text{ GHz} = 10^9\text{ Hz}$
Mega	$\text{M}$	$10^6$	$1\text{ MW} = 10^6\text{ W}$
Kilo	$\text{k}$	$10^3$	$1\text{ km} = 10^3\text{ m}$
Deci	$\text{d}$	$10^{-1}$	$1\text{ dm}^3 = 10^{-3}\text{ m}^3$
Centi	$\text{c}$	$10^{-2}$	$1\text{ cm} = 10^{-2}\text{ m}$
Milli	$\text{m}$	$10^{-3}$	$1\text{ mA} = 10^{-3}\text{ A}$
Micro	$\mu$	$10^{-6}$	$1\text{ }\mu\text{m} = 10^{-6}\text{ m}$
Nano	$\text{n}$	$10^{-9}$	$1\text{ nm} = 10^{-9}\text{ m}$
Pico	$\text{p}$	$10^{-12}$	$1\text{ pF} = 10^{-12}\text{ F}$

The Squared/Cubed Unit Rule: This is a frequent source of error. When a unit with a prefix is raised to a power, the multiplier is also raised to that power.

• $1\text{ cm} = 10^{-2}\text{ m}$
• $1\text{ cm}^2 = (10^{-2}\text{ m})^2 = 10^{-4}\text{ m}^2$
• $1\text{ cm}^3 = (10^{-2}\text{ m})^3 = 10^{-6}\text{ m}^3$
• $1\text{ mm}^2 = (10^{-3}\text{ m})^2 = 10^{-6}\text{ m}^2$

3.4 Reasonable Estimates of Physical Quantities

Paper 1 (Multiple Choice) often requires you to identify a "reasonable estimate." You should memorize these benchmarks to develop physical intuition.

1. Mass ($m$)

• Electron: $\approx 10^{-30}\text{ kg}$
• Proton/Neutron: $\approx 10^{-27}\text{ kg}$
• Apple: $100\text{ g}$ ($0.1\text{ kg}$)
• Adult Human: $70\text{--}80\text{ kg}$
• Family Car: $1000\text{--}1500\text{ kg}$
• Boeing 747 (Loaded): $4 \times 10^5\text{ kg}$
• The Earth: $6 \times 10^{24}\text{ kg}$

2. Length ($L$)

• Diameter of an atom: $10^{-10}\text{ m}$ (1 Ångström)
• Diameter of a nucleus: $10^{-15}\text{ m}$ (1 femtometre)
• Thickness of a human hair: $50\text{--}100\text{ }\mu\text{m}$
• Wavelength of visible light: $400\text{ nm}$ (violet) to $700\text{ nm}$ (red)
• Height of a mountain (Everest): $8800\text{ m}$ ($\approx 10^4\text{ m}$)
• Radius of the Earth: $6.4 \times 10^6\text{ m}$ ($6400\text{ km}$)

3. Time ($t$)

• Period of visible light: $10^{-15}\text{ s}$
• Reaction time of a human: $0.2\text{ s}$
• Heartbeat: $1\text{ s}$
• One day: $8.6 \times 10^4\text{ s}$ ($\approx 10^5\text{ s}$)
• One year: $3.2 \times 10^7\text{ s}$ ($\approx \pi \times 10^7\text{ s}$)
• Age of the Universe: $4 \times 10^{17}\text{ s}$

4. Speed ($v$)

• Walking speed: $1.5\text{ m s}^{-1}$
• Sprinting speed: $10\text{ m s}^{-1}$
• Speed of sound in air: $330\text{--}340\text{ m s}^{-1}$
• Speed of a commercial jet: $250\text{ m s}^{-1}$ ($900\text{ km h}^{-1}$)
• Speed of light in vacuum ($c$): $3.00 \times 10^8\text{ m s}^{-1}$

5. Electricity and Energy

• Current in a domestic appliance: $1\text{--}13\text{ A}$
• Potential difference of a AA battery: $1.5\text{ V}$
• Mains voltage (UK/many regions): $230\text{ V}$
• Power of a domestic lightbulb: $10\text{--}60\text{ W}$
• Power of a hairdryer/kettle: $1\text{--}3\text{ kW}$
• Energy in a chocolate bar: $1\text{ MJ}$ ($10^6\text{ J}$)

6. Pressure and Density

• Atmospheric pressure: $1.0 \times 10^5\text{ Pa}$ ($101\text{ kPa}$)
• Density of air: $1.2\text{ kg m}^{-3}$
• Density of water: $1000\text{ kg m}^{-3}$
• Density of a metal (e.g., Iron): $8000\text{ kg m}^{-3}$

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Worked Example 1 — Complex Unit Conversion

Question: The thermal conductivity of a material is $150\text{ W m}^{-1}\text{ K}^{-1}$. Express this value in the SI base units of $\text{kg m s}^{-3}\text{ K}^{-1}$.

Step 1: Break down the derived unit (Watt). Power ($P$) is Work/Time. $\text{Watt (W)} = \text{J s}^{-1}$ Work ($W$) is Force $\times$ distance. $\text{Joule (J)} = \text{N m}$ Force ($F$) is mass $\times$ acceleration. $\text{Newton (N)} = \text{kg m s}^{-2}$

Step 2: Substitute base units back into the Watt. $\text{W} = (\text{kg m s}^{-2}) \times \text{m} \times \text{s}^{-1}$ $\text{W} = \text{kg m}^2\text{ s}^{-3}$

Step 3: Substitute the Watt back into the original quantity. $\text{Thermal Conductivity} = 150 \times (\text{kg m}^2\text{ s}^{-3}) \times \text{m}^{-1} \times \text{K}^{-1}$ $\text{Thermal Conductivity} = 150\text{ kg m s}^{-3}\text{ K}^{-1}$

Answer: $150\text{ kg m s}^{-3}\text{ K}^{-1}$

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Worked Example 2 — Estimation of Volume and Mass

Question: Estimate the mass of air in a typical school classroom.

Step 1: Estimate the dimensions of the room.

• Length $\approx 10\text{ m}$
• Width $\approx 8\text{ m}$
• Height $\approx 3\text{ m}$

Step 2: Calculate the estimated volume ($V$). $V = L \times W \times H = 10 \times 8 \times 3 = 240\text{ m}^3$ Order of magnitude: $\approx 10^2\text{ m}^3$

Step 3: Use the known density of air. Density of air ($\rho$) $\approx 1.2\text{ kg m}^{-3}$

Step 4: Calculate the mass ($m$). $m = \rho \times V = 1.2 \times 240 = 288\text{ kg}$

Answer: A reasonable estimate is $300\text{ kg}$ (Order of magnitude $10^2\text{ kg}$).

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Worked Example 3 — Estimating Kinetic Energy

Question: Estimate the kinetic energy of a family car travelling at highway speeds.

Step 1: Estimate the mass and speed.

• Mass of car ($m$): $\approx 1200\text{ kg}$
• Speed ($v$): $70\text{ mph} \approx 30\text{ m s}^{-1}$

Step 2: Apply the kinetic energy formula. $E_k = \frac{1}{2}mv^2$ $E_k = 0.5 \times 1200 \times (30)^2$ $E_k = 600 \times 900 = 540,000\text{ J}$

Answer: The estimate is $5.4 \times 10^5\text{ J}$ (Order of magnitude $10^5\text{--}10^6\text{ J}$).

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Key Equations

1. The Quantity Identity $Q = n[u]$

• $Q$: Physical quantity
• $n$: Numerical magnitude
• $[u]$: Unit
• Status: Fundamental concept (Memorize)

2. Density $\rho = \frac{m}{V}$

• $\rho$: Density ($\text{kg m}^{-3}$)
• $m$: Mass ($\text{kg}$)
• $V$: Volume ($\text{m}^3$)
• Status: Must memorize

3. Weight $W = mg$

• $W$: Weight ($\text{N}$)
• $m$: Mass ($\text{kg}$)
• $g$: Acceleration of free fall ($9.81\text{ m s}^{-2}$)
• Status: $g$ is on the Data Sheet; equation must be memorized

4. Pressure $p = \frac{F}{A}$

• $p$: Pressure ($\text{Pa}$ or $\text{N m}^{-2}$)
• $F$: Force ($\text{N}$)
• $A$: Area ($\text{m}^2$)
• Status: Must memorize

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Common Mistakes to Avoid

• ❌ Wrong: Treating "kg" as a unit for weight. ✓ Right: Mass is in kg; Weight is a force measured in Newtons (N). $W = mg$.
• ❌ Wrong: Forgetting to square/cube the prefix multiplier. (e.g., $1\text{ m}^2 = 100\text{ cm}^2$). ✓ Right: $1\text{ m}^2 = (100\text{ cm})^2 = 10,000\text{ cm}^2$.
• ❌ Wrong: Writing units with a plural 's' (e.g., $10\text{ ms}$ for 10 metres). ✓ Right: Units are algebraic symbols. $10\text{ m}$ is 10 metres; $10\text{ ms}$ is 10 milliseconds.
• ❌ Wrong: Using "m" for both "milli" and "metre" in a confusing way. ✓ Right: In the term $2\text{ mm}$, the first 'm' is the prefix ($10^{-3}$) and the second is the unit (metre).
• ❌ Wrong: Giving a final answer to 1 significant figure (e.g., $g = 10\text{ m s}^{-2}$). ✓ Right: Always use at least 2 or 3 significant figures. Use $g = 9.81\text{ m s}^{-2}$ unless the question data is only 1 s.f.
• ❌ Wrong: Confusing the prefix 'M' (Mega, $10^6$) with 'm' (milli, $10^{-3}$). ✓ Right: Capitalization matters. $1\text{ MW}$ is a million watts; $1\text{ mW}$ is a thousandth of a watt.

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Exam Tips

1. The "Reasonableness" Test: In Paper 2 and 4, if you calculate the mass of a car to be $1.5 \times 10^{-2}\text{ kg}$, stop immediately. You have likely made a power-of-ten error. Use the estimates in Section 3.4 to check your work.
2. Prefix Conversion First: Before plugging numbers into your calculator, convert all units to SI base units (e.g., convert $\text{mm}$ to $\text{m}$ and $\text{kN}$ to $\text{N}$). This prevents errors when squaring or cubing values in formulas like $\frac{1}{2}mv^2$ or $\pi r^2$.
3. Standard Form: Always provide very large or very small numbers in standard form (e.g., $6.67 \times 10^{-11}$ instead of $0.0000000000667$). This is standard practice in 9702 and reduces transcription errors.
4. Calculator Notation: Be careful with the "EXP" or "$\times 10^x$" button. To enter $5 \times 10^{-3}$, press 5, then EXP, then -3. Do not enter 5 \times 10 EXP -3, as this will result in $5 \times 10^{-2}$.
5. Paper 1 Strategy: For estimation questions, eliminate the "impossible" answers first. If asked for the frequency of a radio wave, and the options include $10^{-5}\text{ Hz}$ and $10^{15}\text{ Hz}$, you can eliminate them based on the known EM spectrum (radio waves are not that slow, and $10^{15}$ is UV/X-ray territory).
6. Significant Figures: Match your answer's precision to the least precise piece of data given in the question. If the input values are $2.4\text{ m}$ (2 s.f.) and $1.56\text{ s}$ (3 s.f.), your answer should be given to 2 s.f. (or 3 s.f. as a safe margin). Never give 1 s.f. or 5 s.f. unless specifically instructed.