Physical Quantities and Units Study Notes, Formulas & Summary

1. Physical Quantities

A physical quantity is any property of a physical system that can be measured. Every physical quantity consists of two parts: a numerical magnitude and a unit. For example, a length of 5 metres has magnitude 5 and unit metres (m). Physical quantities are categorised as either scalar or vector quantities.

Scalar quantities have only magnitude. Examples include mass, temperature, distance, speed, energy, and time.
Vector quantities have both magnitude and direction. Examples include displacement, velocity, acceleration, force, momentum, and electric field.

Vectors can be added using the parallelogram law or the triangle rule, and they can be resolved into components along perpendicular axes, which is particularly useful when analysing forces and motion in two dimensions.

2. SI Base Units

The International System of Units (SI) defines seven base units from which all other units are derived:

Quantity	Unit	Symbol
Mass	kilogram	kg
Length	metre	m
Time	second	s
Electric current	ampere	A
Temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

All other physical quantities use derived units, which are combinations of the base units. For example, the newton (N) — the unit of force — is derived as kg m s⁻². The pascal (Pa) — the unit of pressure — is N m⁻² = kg m⁻¹ s⁻².

3. Homogeneity of Equations

An equation is said to be homogeneous if the units on both sides are identical. Checking homogeneity is a powerful technique for verifying equations and detecting errors. For instance, for the kinematic equation $v = u + at$:

Left side: m s⁻¹ (velocity)
Right side: m s⁻¹ + (m s⁻²)(s) = m s⁻¹ + m s⁻¹ ✓

However, homogeneity alone cannot confirm an equation is correct — it may be missing a dimensionless constant, for example.

4. Prefixes and Orders of Magnitude

SI prefixes are used to express very large or very small quantities without writing many zeros. The most commonly used prefixes in A-Level Physics are:

Prefix	Symbol	Factor
Tera	T	10¹²
Giga	G	10⁹
Mega	M	10⁶
Kilo	k	10³
Milli	m	10⁻³
Micro	μ	10⁻⁶
Nano	n	10⁻⁹
Pico	p	10⁻¹²
Femto	f	10⁻¹⁵

Estimating the order of magnitude of a quantity — its approximate value to the nearest power of ten — is an important skill. For example, the diameter of a hydrogen atom is approximately 10⁻¹⁰ m (1 Å).

5. Measurement and Experimental Errors

In all experimental work, measurements are subject to uncertainties. Understanding and correctly handling these is essential for scientific validity.

Random errors cause measurements to scatter randomly about the true value. They can be reduced by repeating measurements and averaging. Examples include reading fluctuations or timing reaction variability.
Systematic errors cause all measurements to shift consistently in one direction. They cannot be reduced by repeating measurements. Examples include a zero error on a ruler or a calibration error in a meter.

Accuracy refers to how close a measurement is to the true value. Precision refers to how reproducible measurements are (small spread). A measurement can be precise but inaccurate if a systematic error is present.

6. Uncertainty and Significant Figures

The absolute uncertainty (Δx) is expressed in the same units as the measurement. The fractional uncertainty = Δx / x, and the percentage uncertainty = (Δx / x) × 100%.

Rules for combining uncertainties: - Adding or subtracting: Add absolute uncertainties. ΔZ = ΔA + ΔB - Multiplying or dividing: Add fractional (or percentage) uncertainties. - Raising to a power: Multiply the fractional uncertainty by the power. If Z = xⁿ, then ΔZ/Z = n(Δx/x)

Results should be quoted to an appropriate number of significant figures, consistent with the precision of the measurements. The final answer should reflect the measurement with the fewest significant figures.

7. Estimating Physical Quantities

A-Level Physics often requires estimating the order of magnitude of physical quantities. Sensible estimates require a solid understanding of the physical world:

Mass of a person: ~70 kg
Speed of sound in air: ~340 m s⁻¹
Diameter of a nucleus: ~10⁻¹⁵ m
Charge of an electron: 1.6 × 10⁻¹⁹ C
Planck's constant: 6.63 × 10⁻³⁴ J s

Being able to estimate gives you a valuable sanity check on calculated answers and is frequently tested in examinations.

graph TD
    A[Physical Quantity] --> B[Scalar: Magnitude only]
    A --> C[Vector: Magnitude + Direction]
    B --> D[Examples: mass, speed, energy]
    C --> E[Examples: force, velocity, momentum]
    C --> F[Can be resolved into components]
    F --> G[Horizontal: F cosθ]
    F --> H[Vertical: F sinθ]

Physical Quantities and Units — Study Notes