SI units and prefixes
Section: Physical Quantities and Units | Syllabus: Cambridge AS Level Physics (9702)
Base Units
These are the fundamental units of measurement that are used to define all other units and can't be broken down into simpler units.
The International System of Units (SI) defines seven base quantities. At AS Level, you need to recall the following five:
| Base Quantity | SI Base Unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric Current | ampere | A |
| Thermodynamic Temperature | kelvin | K |
The Kelvin scale starts at absolute zero (0 K). K = °C + 273.15.
Derived Units
Derived units are units that can be expressed as combinations of the five SI base units (metre, kilogram, second, ampere, and kelvin). They are obtained by multiplying or dividing these base units.
Most physical quantities have derived units. These are units which can be expressed in terms of the five SI base units (m, kg, s, A, K).
| Quantity | Derived Unit | In SI Base Units |
|---|---|---|
| Force | Newton (N) | kg m s^-2 |
| Pressure | Pascal (Pa) | kg m^-1 s^-2 |
| Energy / Work | Joule (J) | kg m^2 s^-2 |
| Power | Watt (W) | kg m^2 s^-3 |
| Frequency | Hertz (Hz) | s^-1 |
| Charge | Coulomb (C) | A s |
| Potential Diff | Volt (V) | kg m^2 s^-3 A^-1 |
Question: Express the newton, N, in terms of base SI units.
AnswerUse the equation F = ma
In this equation F is force, m is mass and a is acceleration
The unit of mass is kg and the unit of acceleration is m s^-2
Therefore unit of F = kg m s^-2
The newton, N, expressed in base SI units is kg m s^-2
Example: Deriving the VoltVoltage = Work done / Charge
V = J / C Base units for J: kg m^2 s^-2 Base units for C: A s Base units for V: (kg m^2 s^-2) / (A s) = kg m^2 s^-3 A^-1Checking Homogeneity
An equation is homogeneous if the SI base units on both sides are the same. This is a powerful tool to check if an equation is physically possible.
Worked Example 1One of the equations of motion is v^2 = u^2 + 2as where v and u are velocities, 2 is a constant, a is acceleration and s is displacement.
Use base SI units to prove the homogeneity of this equation.
AnswerList the units for each quantity:
- v is in m s^-1 so v^2 is in m^2 s^-2
- u is in m s^-1 so u^2 is in m^2 s^-2
- 2 is a constant, so no unit
- a is in m s^-2
- s is in m
Substitute these into the equation:
m^2 s^-2 = m^2 s^-2 + (m s^-2) × (m) m^2 s^-2 = m^2 s^-2 + m^2 s^-2All terms have the same units (m^2 s^-2), which proves its homogeneity.
Worked Example 2Use base SI units to prove the homogeneity of the following equation: v = u + at where v and u are velocities, a is acceleration and t is time.
Answer- Units of v: m s^-1
- Units of u: m s^-1
- Units of at: (m s^-2) × s = m s^-1
Since all terms have the same base units (m s^-1), the equation is homogeneous.
Pro TipNumbers (like (1/2) in (1/2)mv^2 or 2 in 2as) have no units and are ignored when checking homogeneity.
When adding or subtracting terms with the same units, the result keeps those units. For example, if an equation shows m^2s^-2 = m^2s^-2 + m^2s^-2, this is still homogeneous-both sides have units of m^2s^-2.
Prefixes
Scientific notation and prefixes are used to express very large or very small quantities conveniently.
| Prefix | Symbol | Factor |
|---|---|---|
| Tera | T | 10^12 |
| Giga | G | 10^9 |
| Mega | M | 10^6 |
| Kilo | k | 10^3 |
| Deci | d | 10^-1 |
| Centi | c | 10^-2 |
| Milli | m | 10^-3 |
| Micro | μ | 10^-6 |
| Nano | n | 10^-9 |
| Pico | p | 10^-12 |
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