Contents: 10 sections
Syllabus points
- Understand that a physical quantity has a numerical magnitude and a unit.
- Recall the SI base quantities and their units, and express derived units as products or quotients of base units.
- Use SI prefixes and check equations for homogeneity of units.
- Distinguish between random and systematic errors, and between precision and accuracy.
- Combine absolute and percentage uncertainties in a calculation.
- Distinguish scalars from vectors, and add and resolve vectors.
The seven base units
Everything in the course is built from six of these. Learn them as pairs of quantity and unit, because questions ask for one given the other.
| Quantity | Unit | Symbol |
|---|---|---|
| Mass | kilogram | kg |
| Length | metre | m |
| Time | second | s |
| Electric current | ampere | A |
| Thermodynamic temperature | kelvin | K |
| Amount of substance | mole | mol |
| Luminous intensity | candela | cd |
Two traps appear almost every year. The base unit of mass is the kilogram, not the gram, and it is the only base unit that already carries a prefix. And the base unit of temperature is the kelvin, written K with no degree sign; °C is a derived unit and °K does not exist at all.
Deriving a unit from an equation
Any other unit can be written in base units by taking an equation that defines it. This is a routine mark, and the method is always the same.
Example: the unit of the Young modulus.
E = stress / strain, strain has no unit, so E has the unit of stress.
Stress = force / area, so the unit is N m⁻².
N = kg m s⁻², so N m⁻² = kg m s⁻² m⁻² = kg m⁻¹ s⁻².
Work in symbols and cancel at the end. Trying to remember the answer is how people write kg m s⁻² for pressure.
Homogeneity
An equation is homogeneous when both sides have the same base units. Checking this catches a wrongly recalled formula, and questions ask you to spot the one equation that fails.
A homogeneous equation is not necessarily correct: a missing factor of ½ or 2π has no units and would not be caught. So homogeneity proves an equation can be right, not that it is.
Prefixes
| Prefix | Symbol | Factor |
|---|---|---|
| pico | p | 10⁻¹² |
| nano | n | 10⁻⁹ |
| micro | µ | 10⁻⁶ |
| milli | m | 10⁻³ |
| centi | c | 10⁻² |
| kilo | k | 10³ |
| mega | M | 10⁶ |
| giga | G | 10⁹ |
| tera | T | 10¹² |
Note the case. M is mega and m is milli, a factor of 10⁹ apart, and they are different units when written down.
Prefixes on a squared or cubed unit apply to the whole thing: 1 cm³ is (10⁻² m)³ = 10⁻⁶ m³, not 10⁻² m³. That single step is the whole of many estimation questions.
Errors: random and systematic
A random error scatters readings either side of the true value. Repeating and averaging reduces it.
A systematic error shifts every reading the same way. A zero error on a micrometer is the standard example, and repeating does not help at all, because the same offset is added every time.
Precision and accuracy
These are different things, and questions set them against each other deliberately.
- Precise readings are close to each other. Precision is about spread, and it is spoiled by random error.
- Accurate readings are close to the true value. Accuracy is spoiled by systematic error.
So a set of readings can be precise but not accurate, which is exactly what a zero error produces: five readings within 0.02 mm of each other, all 6 mm too big.
Worked example. A desk is truly 50.0 cm wide. A student records 48.01, 47.99, 48.01, 48.00, 47.99 cm.
The readings agree to within 0.02 cm, so they are precise. The mean is 48.00 cm, which is 2.00 cm from the true value, so they are not accurate. The answer is precise but not accurate, and the presence of a true value in the question is the signal that accuracy is being tested.
Combining uncertainties
Three rules, and the middle one is the one that gets forgotten.
- Adding or subtracting quantities: add the absolute uncertainties.
- Multiplying or dividing: add the percentage uncertainties.
- Raising to a power n: multiply the percentage uncertainty by n.
Worked example. A cube has side 2.0 ± 0.1 cm. What is the percentage uncertainty in its volume?
Percentage uncertainty in the side = 0.1 / 2.0 × 100 = 5%.
Volume = side³, so the percentage uncertainty is 3 × 5 = 15%.
Subtraction deserves care. If two lengths of 50.0 ± 0.5 mm and 48.0 ± 0.5 mm are subtracted, the result is 2.0 ± 1.0 mm, a percentage uncertainty of 50% from two readings that were each uncertain by 1%. Subtracting similar numbers destroys precision, and questions use this to ask which method is better.
Scalars and vectors
A scalar has magnitude only. A vector has magnitude and direction.
| Scalars | Vectors |
|---|---|
| Distance, speed, mass, time | Displacement, velocity, acceleration, force |
| Energy, work, power | Momentum, weight, electric field strength |
| Temperature, density, pressure |
Weight is a vector and mass is a scalar, and that pair is asked directly.
Adding and resolving
Two vectors add tip to tail; the resultant runs from the start of the first to the end of the second. At right angles, use Pythagoras and tan.
Resolving is the reverse. A vector F at angle θ to the horizontal has components F cos θ horizontally and F sin θ vertically. Check which angle you are given: if θ is measured from the vertical instead, the cosine and sine swap. Reading the angle off the diagram rather than assuming it is the single most common slip in this topic.
The resultant of two vectors of magnitudes a and b lies between |a − b| and a + b. That bracket alone answers several multiple-choice questions without any calculation.
Common mistakes
- Giving the gram as the base unit of mass.
- Writing the unit of temperature as °K.
- Forgetting to cube or square the prefix, so 1 cm³ becomes 10⁻² m³.
- Treating a homogeneous equation as proof that the equation is correct.
- Saying repeating readings reduces a systematic error.
- Swapping precision and accuracy, particularly when the question gives a true value.
- Adding percentage uncertainties when the quantities are being added rather than multiplied.
- Using cos where the angle is measured from the vertical.