A physical quantity is anything that can be measured, and it always has two parts: a numerical magnitude and a unit. This topic covers the six SI base units, prefixes from nano to tera, standard form and orders of magnitude, choosing a measuring instrument by range and precision, timing pendulum oscillations, and adding scalars and vectors.
Every calculation, practical and data-response question in O-Level Physics rests on this topic. An answer with a wrong or missing unit loses the mark however good the working is, and units, measurement and vectors turn up again in every later topic, including kinematics, forces and electricity.
What these notes cover
- •Physical quantities and the six SI base units
- •Prefixes from nano to tera
- •Converting derived units: the two-step method
- •Standard form and orders of magnitude
- •Making sensible estimates
- •Choosing an instrument: range and precision
- •Zero error and parallax
- •Timing pendulum oscillations
- •Scalars and vectors
- •Adding two vectors by scale drawing
- •Common mistakes and exam technique
Key definitions
- Physical quantity
- Anything that can be measured. It is made up of a numerical magnitude (how large the quantity is) and a unit (the agreed standard it is compared against).
- Order of magnitude
- The power of ten, 10ⁿ, that appears when a quantity is written in standard form. If one quantity is 10³ times another, it is three orders of magnitude larger.
- Precision
- The smallest change in a quantity that an instrument can register: the smallest scale division on an analogue instrument, or the last digit on a digital display.
- One complete oscillation
- The motion of the pendulum bob from a chosen point until it returns to the same point, moving in the same direction.
- Scalar and vector quantities
- A scalar quantity has magnitude (size) only. A vector quantity has both magnitude and direction.
Key formulas
Standard form
N = A × 10ⁿ
1 ≤ A < 10 and n is an integer. The order of magnitude is 10ⁿ; convert to SI units before quoting it.
Prefix inside a power
1 cm³ = (10⁻² m)³ = 10⁻⁶ m³, so 1 g/cm³ = 10⁻³ kg ÷ 10⁻⁶ m³ = 10³ kg/m³
The square or cube acts on the whole unit, prefix included. To convert a density from g/cm³ to kg/m³, multiply by 1000.
Speed: km/h to m/s
speed in m/s = speed in km/h ÷ 3.6
Comes from 1 km = 10³ m and 1 h = 3600 s, e.g. 72 km/h = (72 × 10³ m) ÷ 3600 s = 20 m/s. Multiply by 3.6 to go from m/s to km/h.
Period of a pendulum
T = t ÷ N
t is the total time recorded for N complete oscillations, with N ≥ 20. Repeat the timing and average before dividing.
Reaction-time error in the period
error in T = error in t ÷ N
The reaction-time error (about 0.3 s) is roughly fixed whatever interval is timed, so timing N oscillations shrinks its effect on T by a factor of N.
Resultant of two perpendicular vectors
R = √(F₁² + F₂²)
A check for perpendicular vectors only. When the graphical method is asked for, the marks are for the accurate scale drawing.
Worked example
Aluminium has a density of 2.7 g/cm³. Convert this to kg/m³.
- 1Substitute the mass unit: 1 g = 10⁻³ kg.
- 2Substitute the volume unit, cubing the prefix too: 1 cm³ = (10⁻² m)³ = 10⁻⁶ m³.
- 3Put both into the density: 2.7 g/cm³ = (2.7 × 10⁻³ kg) ÷ (10⁻⁶ m³).
- 4Simplify the powers of ten: 2.7 × 10⁻³ × 10⁶ kg/m³ = 2.7 × 10³ kg/m³.
- 5So the density is 2700 kg/m³, which agrees with the rule that a kg/m³ value is always 1000 times the g/cm³ value.
Watch out: If your answer came out as 0.0027, you divided when you should have multiplied. Remember that 1 cm³ = 10⁻⁶ m³, not 10⁻² m³.
Common mistakes
Converting areas and volumes as if they were lengths
Wrong: 1 m² = 100 cm², because 1 m = 100 cm.
Right: Raise the conversion factor to the same power as the unit: 1 m² = (100 cm)² = 10⁴ cm² and 1 m³ = (100 cm)³ = 10⁶ cm³.
Starting the oscillation count at “one”
Wrong: Saying “one” as the stopwatch starts, so only 19 oscillations have been timed when the count reaches 20.
Right: Start the stopwatch and say “zero” as the bob passes the marker; each later pass in the same direction adds one to the count.
Giving a vector answer without a direction
Wrong: Resultant velocity = 13 m/s.
Right: State the magnitude and direction against a named reference, e.g. 13 m/s at 23° east of north.
Questions students ask
What are the six SI base units in O-Level Physics?
They are the kilogram (kg) for mass, metre (m) for length, second (s) for time, ampere (A) for electric current, kelvin (K) for thermodynamic temperature and mole (mol) for amount of substance. Every other unit, such as the newton or the joule, is a derived unit built from these. The base unit of mass is the kilogram, not the gram.
How do you choose the right measuring instrument in physics?
Ask two questions in order: does the range cover the quantity, and is the precision fine enough for the reading to have meaningful figures? An instrument that fails on range is eliminated however precise it is, so a micrometer cannot measure a pencil, and a metre rule cannot measure a 0.5 mm wire. When justifying a choice, quote both the size of the quantity and the precision of the instrument.
Why do you time 20 oscillations of a pendulum instead of one?
Human reaction time adds an error of a few tenths of a second when the stopwatch is started and stopped, and this error is roughly fixed regardless of the interval timed. Timing 20 oscillations and dividing the total time by 20 reduces the effect of this error on the period to 1/20 of its original size. In one example, the percentage uncertainty in T drops from about 20% to about 1%.
What is the difference between distance and displacement?
Distance is the total length of the path travelled by an object and is a scalar. Displacement is the straight-line distance from the starting point to the end point, together with its direction, so it is a vector. A runner who completes one lap of a 400 m track covers a distance of 400 m but has zero displacement.
In the full PDF
- •Full tables of the six SI base units and all nine prefixes from nano to tera, with conversion examples
- •The ladder of typical lengths, masses and times for order-of-magnitude estimates, with a logarithmic-scale diagram
- •The instrument precision table, plus worked choices for a wire, a pencil, a beaker, a sheet of paper, a coin and a falling ball
- •Pendulum diagram with fiducial marker, and worked examples on averaging timings and percentage uncertainty
- •Scale-drawing diagrams for tip-to-tail and parallelogram vector addition, with further worked examples
- •All 12 common mistakes with corrections, the four exam-technique non-negotiables and section summaries