1.1 Physical Quantities

1. Magnitude and Units

To describe the universe, we define Physical Quantities.

The Fundamental Equation

Any physical quantity $\mathrm{Q}$ must be expressed as the product of a Numerical Magnitude ($N$) and a Unit ($u$).

$$Q=N\times u$$

Example

If you measure the length of a wire: $L=5.0\times \text{metres}$

$L=5.0\mathrm{m}$

Here:

• $Q=L$ (The Physical Quantity)
• $N=5.0$ (The Magnitude)
• $u=\mathrm{m}$ (The Unit)

Tip

Mathematically, units behave like algebraic variables.

If: $Q=Nu$

Then: $N=\frac{Q}{u}$

This is why table headers in question papers are written as $Length/m$ or $V/V$. The axes are representing the numerical value only:

$\frac{\mathrm{Length}}{\mathrm{metres}}=\frac{5.0\mathrm{m}}{\mathrm{m}}=5.0$

The “Unit” Trap

Common Pitfall

Dimensionless Quantities are the exception to the rule. They have a magnitude but no unit.

Examples in Syllabus:

• Refractive Index ($n$): Ratio of speeds ($v_1/v_2$).
• Strain ($\epsilon$): Ratio of lengths ($\Delta L/L$).
• Efficiency: Ratio of energies/powers.
• Number of particles ($N$).

If you add a unit to these (e.g., “Strain = 0.02 m”), you will lose the mark.

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2. Reasonable Estimates

Estimation Strategy

When asked to estimate:

1. Give the value to 1 Significant Figure.
2. Focus on the Order of Magnitude (power of 10).
3. Use base SI units for the thought process, then convert if necessary.

Estimate Tables

Mass ($m$)

Object	Approximate Mass (kg)
Electron	$9.1\times 10^{-31}$
Proton / Neutron	$1.7\times 10^{-27}$
Speck of dust	$10^{-9}$ to $10^{-6}$
Apple / Orange	$0.1$ ($100$ g)
Adult Human	$70$
Car	$1000$ to $2000$
Earth	$6\times 10^{24}$
Sun	$2\times 10^{30}$

Length ($L$)

Object	Approximate Length (m)
Radius of a nucleus	$10^{-15}$ (1 femtometer)
Diameter of an atom	$10^{-10}$ (1 Ångström)
Wavelength of UV light	$10^{-8}$
Wavelength of Visible light	$4\times 10^{-7}$ (Violet) to $7\times 10^{-7}$ (Red)
Diameter of human hair	$10^{-4}$ ($0.1$ mm)
Height of human	$1.7$
Mt. Everest height	$10^4$
Earth Radius	$6.4\times 10^6$

Time ($t$)

Event	Approximate Time (s)
Time for light to cross nucleus	$10^{-23}$
Period of visible light	$10^{-15}$
Human reaction time	$0.2$ to $0.5$
Time for a heartbeat	$1.0$
Day	$86400$ ($\approx 10^5$)
Year	$3.16\times 10^7$ ($\approx \pi \times 10^7$)

Speeds ($v$) & Others

Quantity	Approximate Value
Speed of Continental Drift	$10^{-9}{\text{m s}}^{-1}$ (mm per year)
Walking Speed	$1-2{\text{m s}}^{-1}$
Car on Highway	$25-30{\text{m s}}^{-1}$ ($100$ km/h)
Speed of Sound (Air)	$330-340{\text{m s}}^{-1}$
Speed of Light (Vacuum)	$3.0\times 10^8{\text{m s}}^{-1}$
Atmospheric Pressure	$1.0\times 10^5\text{Pa}$
Density of Air	$1.2{\text{kg m}}^{-3}$
Density of Water	$1000{\text{kg m}}^{-3}$