SI Units & Prefixes
The Seven SI Base Units
All physical quantities can be expressed in terms of seven base units. These are defined by the International System of Units (SI).
| 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 |
Derived Units
Derived units are combinations of base units. You must be able to express any derived unit in terms of base units.
| Quantity | Unit | Symbol | In base units |
|---|---|---|---|
| Force | newton | N | kg m s−2 |
| Energy | joule | J | kg m2 s−2 |
| Power | watt | W | kg m2 s−3 |
| Pressure | pascal | Pa | kg m−1 s−2 |
| Charge | coulomb | C | A s |
| Potential difference | volt | V | kg m2 s−3 A−1 |
| Resistance | ohm | Ω | kg m2 s−3 A−2 |
Worked Example
Show that the unit of energy (joule) is kg m2 s−2.
Energy = Force × distance
Force = mass × acceleration = kg × m s−2
Energy = kg m s−2 × m = kg m2 s−2
SI Prefixes
Prefixes are used to express very large or very small quantities conveniently.
| Prefix | Symbol | Multiplier |
|---|---|---|
| giga | G | 109 |
| mega | M | 106 |
| kilo | k | 103 |
| centi | c | 10−2 |
| milli | m | 10−3 |
| micro | μ | 10−6 |
| nano | n | 10−9 |
| pico | p | 10−12 |
Order of Magnitude
An order of magnitude is the power of 10 closest to a value. It is used for quick estimation and comparison of physical quantities.
Worked Example
Estimate the order of magnitude of the diameter of a human hair.
A human hair is approximately 0.1 mm = 1 × 10−4 m
Order of magnitude = 10−4 m
Key Facts
- Homogeneity of equations: both sides of a physics equation must have the same base units
- Checking units is a powerful way to verify whether an equation is correct
- Prefixes must be converted to powers of 10 before substituting into equations
Exam Tip
Always convert quantities to SI base units before substituting into equations. A common error is forgetting to convert km to m or mA to A.
Errors & Uncertainty
Random vs Systematic Errors
Random errors cause readings to scatter around the true value. They can be reduced by repeating measurements and taking an average.
Systematic errors cause all readings to shift in the same direction (always too high or too low). They cannot be reduced by repeating. Common causes include zero errors and incorrectly calibrated instruments.
| Feature | Random Error | Systematic Error |
|---|---|---|
| Effect on readings | Scatter around true value | All shifted in one direction |
| Reduced by repeating? | Yes (take mean) | No |
| Example | Timing reaction time | Zero error on a balance |
| Affects | Precision | Accuracy |
Precision vs Accuracy
Precision: How close repeated measurements are to each other. High precision means small spread (low random error).
Accuracy: How close a measurement is to the true value. High accuracy means low systematic error.
Red dot = true value. Blue dots = measurements.
Resolution and Uncertainty
The resolution of an instrument is the smallest change it can detect. The uncertainty of a single reading from an analogue scale is typically ± half the smallest division.
For digital instruments, the uncertainty is ± 1 in the last digit.
When finding a range (e.g., timing with a stopwatch), the uncertainty is half the range of the repeated readings:
Significant Figures
The number of significant figures in a result reflects the precision of the measurement. A final answer should be given to the same number of significant figures as the least precise measurement used in the calculation.
Worked Example
A force of 12.5 N acts over a distance of 3.2 m. Calculate the work done.
W = F × d = 12.5 × 3.2 = 40 J
The least precise value is 3.2 (2 s.f.), so the answer is 40 J (2 s.f.)
Key Facts
- Repeating measurements reduces random error but not systematic error
- A measurement can be precise but not accurate (and vice versa)
- Zero errors are a common source of systematic error
Exam Tip
When asked to identify the type of error in a question, look for whether the error shifts all results one way (systematic) or scatters them (random).
Using Uncertainties
Percentage Uncertainty
Percentage uncertainty expresses the absolute uncertainty as a proportion of the measured value.
Worked Example
A length is measured as 25.0 ± 0.5 cm. Find the percentage uncertainty.
Percentage uncertainty = (0.5 / 25.0) × 100% = 2.0%
Combining Uncertainties
When combining measurements in calculations, uncertainties must be combined using specific rules depending on the operation.
| Operation | Rule |
|---|---|
| Adding or subtracting: y = a ± b | Add absolute uncertainties: Δy = Δa + Δb |
| Multiplying or dividing: y = a × b or y = a / b | Add percentage uncertainties: %y = %a + %b |
| Raising to a power: y = an | Multiply percentage uncertainty by the power: %y = n × %a |
Worked Example
A cube has side length 5.0 ± 0.1 cm. Calculate the volume and its absolute uncertainty.
Volume = (5.0)3 = 125 cm3
% uncertainty in length = (0.1 / 5.0) × 100 = 2.0%
% uncertainty in volume = 3 × 2.0% = 6.0%
Absolute uncertainty = 6.0% of 125 = 7.5 cm3
Volume = 125 ± 8 cm3 (rounded to match s.f.)
Uncertainty in Graphs
When plotting data with uncertainties, use error bars to represent the uncertainty in each data point.
To find the uncertainty in the gradient, draw:
- Line of best fit: passes through or as close to all points as possible
- Worst acceptable line: steepest or shallowest line that still passes through all error bars
Key Facts
- When adding/subtracting: add absolute uncertainties
- When multiplying/dividing: add percentage uncertainties
- When raising to a power n: multiply percentage uncertainty by n
- The final uncertainty should be rounded to 1 significant figure
Exam Tip
If an exam question gives you a formula like Ek = ½mv², the percentage uncertainty in Ek = %m + 2 × %v (since v is squared). The ½ is an exact constant and does not contribute uncertainty.
Measurements & Errors Flashcards
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Measurements & Errors Quiz
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Measurements & Errors - Mock Exam Questions
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