You are viewing a free preview of this lesson.
Subscribe to unlock all 3 lessons in this course and every other course on LearningBro.
Physics is a quantitative science, and every measurement we make carries some degree of uncertainty. Understanding how to measure correctly, express results in the proper units, and account for errors is fundamental to all experimental work at A-Level and beyond. This topic underpins every practical assessment and is assessed explicitly in Paper 3 (AQA) and the Practical Endorsement.
The International System of Units (SI) provides a universal framework for measurement. There are seven base quantities, each with a defined SI unit:
| Quantity | SI 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 are derived from these base units. For example, velocity has units of m s⁻¹, force has units of kg m s⁻², and energy has units of kg m² s⁻² (the joule).
SI prefixes allow us to express very large or very small quantities conveniently:
| Prefix | Symbol | Multiplier |
|---|---|---|
| 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 |
A common exam question asks you to derive the SI base units for a given quantity. The approach is to start from the equation defining the quantity and substitute base units for each term.
Worked Example — Derive the SI base units of the Young modulus.
Young modulus E = stress / strain = (F/A) / (ΔL/L)
Units of F = kg m s⁻²; units of A = m²; strain is dimensionless (m/m).
So units of E = kg m s⁻² / m² = kg m⁻¹ s⁻² (which is equivalent to Pa).
Worked Example — Derive the SI base units of electrical resistance.
R = V/I. Voltage V = W/Q = energy/charge. Energy has units kg m² s⁻²; charge Q = It has units A s.
So V has units kg m² s⁻² / (A s) = kg m² s⁻³ A⁻¹.
Therefore R = V/I has units kg m² s⁻³ A⁻¹ / A = kg m² s⁻³ A⁻² (which is the ohm, Ω).
Exam Tip: When deriving units, always start from the defining equation. Replace every quantity with its SI base units, then simplify. Never try to guess the answer — show every step.
Physicists frequently need to estimate values to check whether an answer is reasonable. An order of magnitude estimate means giving the answer to the nearest power of ten. You are expected to know sensible values for everyday quantities.
| Quantity | Approximate Value |
|---|---|
| Mass of an electron | 10⁻³⁰ kg |
| Diameter of an atom | 10⁻¹⁰ m |
| Diameter of a nucleus | 10⁻¹⁵ m |
| Height of a person | 2 m |
| Mass of a car | 10³ kg |
| Speed of sound in air | 340 m s⁻¹ |
| Speed of light | 3 × 10⁸ m s⁻¹ |
| Atmospheric pressure | 10⁵ Pa |
| Radius of the Earth | 6.4 × 10⁶ m |
| Mass of the Earth | 6 × 10²⁴ kg |
| Distance to the Sun | 1.5 × 10¹¹ m |
Worked Example — Estimate the number of atoms in a human body.
Mass of a human ≈ 70 kg. The body is mostly water, so we approximate using the mass of a water molecule (about 18 u, where 1 u ≈ 1.66 × 10⁻²⁷ kg).
Mass of one molecule ≈ 18 × 1.66 × 10⁻²⁷ ≈ 3 × 10⁻²⁶ kg.
Number of molecules ≈ 70 / (3 × 10⁻²⁶) ≈ 2 × 10²⁷.
Each water molecule has 3 atoms, so number of atoms ≈ 6 × 10²⁷, i.e. order of magnitude 10²⁸.
Exam Tip: In estimation questions, marks are awarded for a sensible method and a reasonable final answer. Do not waste time being overly precise — round aggressively and focus on powers of ten.
These two terms have distinct meanings in physics and are frequently confused.
Key Definition: Accuracy describes how close a measured value is to the true or accepted value. An accurate measurement has a small systematic error.
Key Definition: Precision describes how close repeated measurements are to each other. A precise set of measurements has small random error (i.e. low spread).
Imagine a target with a bullseye representing the true value:
| Small random error (precise) | Large random error (not precise) | |
|---|---|---|
| Small systematic error (accurate) | Best case — reliable result | Mean is correct but data is spread |
| Large systematic error (not accurate) | Consistently wrong | Unreliable in every respect |
Exam Tip: If asked how to improve accuracy, suggest calibrating the instrument or eliminating systematic errors. If asked how to improve precision, suggest using a more sensitive instrument or taking more repeat readings.
Key Definition: Resolution is the smallest change in the quantity being measured that an instrument can detect. For example, a standard ruler has a resolution of 1 mm, while a micrometer has a resolution of 0.01 mm.
Systematic errors cause all readings to be shifted in one direction — either all too high or all too low. They do not average out with repeated measurements. Examples include a zero error on a balance, parallax when reading an analogue scale, or a miscalibrated instrument.
Methods to reduce systematic errors:
Random errors cause readings to scatter about the true value. They arise from the limitations of human observation and the inherent unpredictability of the measurement process.
Methods to reduce random errors:
Every measurement should be quoted with its uncertainty. Uncertainty tells us the range within which the true value is likely to lie.
For a single reading, the uncertainty is typically ± half the smallest division (resolution) of the instrument.
For a set of repeated readings, the uncertainty is commonly taken as:
Uncertainty = (max value − min value) / 2
This is called half the range.
The absolute uncertainty has the same units as the measurement (e.g. length = 25.4 ± 0.2 cm).
The percentage uncertainty expresses the uncertainty as a fraction of the measured value:
Percentage uncertainty = (absolute uncertainty / measured value) × 100%
When quantities are added or subtracted, their absolute uncertainties are added:
Δ(A ± B) = ΔA + ΔB
When quantities are multiplied or divided, their percentage uncertainties are added:
%(A × B) = %A + %B %(A / B) = %A + %B
When a quantity is raised to a power, the percentage uncertainty is multiplied by that power:
%(Aⁿ) = n × %A
A student measures the following values for a cylindrical wire:
Calculate the resistivity ρ = RA/L = Rπr²/L and its percentage uncertainty.
Step 1: Calculate percentage uncertainties.
% uncertainty in R = (0.05/2.15) × 100 = 2.3%
% uncertainty in L = (0.001/0.750) × 100 = 0.13%
% uncertainty in r = (5 × 10⁻⁶ / 1.40 × 10⁻⁴) × 100 = 3.6%
Step 2: The formula involves r², so double the percentage uncertainty in r.
% uncertainty in r² = 2 × 3.6 = 7.1%
Step 3: Total percentage uncertainty in ρ = %R + %r² + %L = 2.3 + 7.1 + 0.13 = 9.5%
Step 4: Calculate ρ.
A = πr² = π × (1.40 × 10⁻⁴)² = 6.16 × 10⁻⁸ m²
ρ = RA/L = 2.15 × 6.16 × 10⁻⁸ / 0.750 = 1.77 × 10⁻⁷ Ω m
Step 5: Absolute uncertainty = 9.5% of 1.77 × 10⁻⁷ = ±0.17 × 10⁻⁷ Ω m
Final answer: ρ = (1.8 ± 0.2) × 10⁻⁷ Ω m
Exam Tip: The diameter of a wire is often the dominant source of uncertainty because it is small and squared. Always identify which variable contributes the most to the total uncertainty — examiners frequently ask this.
Percentage error compares your experimental result to the accepted value:
Percentage error = |experimental value − accepted value| / accepted value × 100%
If the percentage error is less than the total percentage uncertainty, the experiment is consistent with the accepted value.
When plotting data on a graph, error bars represent the uncertainty in each measurement. For a data point at (x, y), the error bar extends vertically from (x, y − Δy) to (x, y + Δy), and horizontally from (x − Δx, y) to (x + Δx, y) if both uncertainties are significant.
Described diagram — Graph with error bars: Imagine a scatter plot with data points marked as small crosses or dots. At each point, a short vertical line extends equally above and below the point, showing the uncertainty in the y-value. If x-uncertainties are significant, short horizontal lines extend equally left and right.
The line of best fit passes through or close to as many data points as possible (accounting for error bars), balancing points above and below the line.
The worst-fit line (sometimes called the worst acceptable line) is the steepest or shallowest line that still passes through all the error bars. The difference between the gradient of the best-fit line and the worst-fit line gives the uncertainty in the gradient:
Uncertainty in gradient = |gradient of best fit − gradient of worst fit|
Similarly, the uncertainty in the y-intercept can be found from the difference in intercepts.
Described diagram — Best and worst fit lines: Picture a set of data points with error bars. The best-fit line passes centrally through the data. The worst-fit line is drawn from the top of the first error bar to the bottom of the last error bar (or vice versa), giving the maximum or minimum plausible gradient.
Exam Tip: When drawing lines of best fit, use a sharp pencil and a transparent ruler. The line should pass through the error bars of all points if possible. Anomalous results should be identified and can be excluded — but always note them.
The number of significant figures in a result reflects the precision of the measurement. When performing calculations, the final answer should be given to the same number of significant figures as the least precise value used.
Exam Tip: In intermediate steps of a calculation, keep at least one extra significant figure to avoid rounding errors accumulating. Only round to the appropriate number of significant figures in the final answer.
| Concept | Key Point |
|---|---|
| SI base units | 7 base quantities; all others derived |
| Estimation | Give answers to nearest power of 10 |
| Accuracy | Closeness to true value (systematic error) |
| Precision | Closeness of repeated readings (random error) |
| Absolute uncertainty | Same units as the measurement |
| Percentage uncertainty | (Δx / x) × 100% |
| Adding/subtracting | Add absolute uncertainties |
| Multiplying/dividing | Add percentage uncertainties |
| Powers | Multiply percentage uncertainty by the power |
| Error bars | Show uncertainty on graphs |
| Best/worst fit lines | Determine uncertainty in gradient and intercept |