10 min readConcept sheetSection A MCQ (1 mark), often a unit written in base units; Section B very short answer (2 marks): derive the unit of a quantity

Derived units, and the ones with special names (newton, joule, watt, pascal)

A derived unit is built from base units through the formula of the quantity. Some have special names: hertz, newton, pascal, joule and watt.

Derived unitA unit built by multiplying or dividing base units, using the formula of the quantity

Newton (N)Unit of force: 1 N = 1 kg m s⁻²

Joule (J) and watt (W)1 J = 1 N m = 1 kg m² s⁻²; 1 W = 1 J s⁻¹ = 1 kg m² s⁻³

Pascal (Pa)Unit of pressure: 1 Pa = 1 N m⁻² = 1 kg m⁻¹ s⁻²

How to derive any unit

1
Write the formula of the quantity, such as density = mass ÷ volume.
2
Break it down until only base quantities are left: volume = length × length × length.
3
Put in the base units: kg for mass, m for length, s for time.
4★
Simplify. Write a unit you divide by with a negative power: kg ÷ m³ = kg m⁻³.

★ key step

Common derived units

Quantity	Formula in base quantities	SI unit	Symbol
Area	length × length	square metre	m²
Volume	length × length × length	cubic metre	m³
Speed, velocity	length ÷ time	metre per second	m s⁻¹
Acceleration	velocity ÷ time	metre per second squared	m s⁻²
Density	mass ÷ volume	kilogram per cubic metre	kg m⁻³

Derived units with special names

Quantity	Formula	In base units	Special name (symbol)
Frequency	1 ÷ time	s⁻¹	hertz (Hz)
Force	mass × acceleration	kg m s⁻²	newton (N)
Pressure	force ÷ area	kg m⁻¹ s⁻²	pascal (Pa)
Work, energy	force × distance	kg m² s⁻²	joule (J)
Power	work ÷ time	kg m² s⁻³	watt (W)

Worked example: the unit of pressure

Problem

Pressure = force ÷ area. Derive the SI unit of pressure in base units.

Working

Force = mass × acceleration, so its unit is kg × m s⁻² = kg m s⁻² (the newton).
Area = length × length, so its unit is m².
Unit of pressure = kg m s⁻² ÷ m² = kg m⁻¹ s⁻² (because m¹ ÷ m² = m⁻¹).
This is the same as N m⁻², named the pascal.

Answer

1 Pa = 1 N m⁻² = 1 kg m⁻¹ s⁻²

Worked example: the watt, and a bulb's energy

Problem

Show that 1 W = 1 kg m² s⁻³. Then find the energy used by a 9 W LED bulb in 10 minutes.

Working

Work = force × distance: kg m s⁻² × m = kg m² s⁻² (the joule).
Power = work ÷ time: kg m² s⁻² ÷ s = kg m² s⁻³ (the watt).
So 1 W = 1 J s⁻¹: a 9 W bulb uses 9 J every second.
Time = 10 × 60 s = 600 s.
Energy = power × time = 9 W × 600 s = 5400 J.

Answer

1 W = 1 kg m² s⁻³; the bulb uses 5400 J (5.4 kJ) in 10 minutes

Special names you meet every day

Hertz: an FM station at 93.5 MHz sends out 93.5 million waves every second.
Watt: an LED bulb marked 9 W uses 9 J of energy each second.
Joule: many food packets show energy in kcal, and some in kJ as well.
Pascal: weather reports give air pressure in hectopascal (hPa); normal air pressure is about 1013 hPa.
Newton: a 100 g packet of biscuits weighs about 1 N on Earth.

In real life

Your electricity bill charges for energy in kilowatt hours (kWh, the 'units' on the bill). 1 kWh is the energy a 1000 W appliance uses in one hour: 1000 W × 3600 s = 3.6 × 10⁶ J.

Exam angle

1 marks: “Which physical quantity has the SI unit kg m⁻³?” Density (mass ÷ volume).

2 marks: “Express the SI unit of work in base units and give its special name.” Work = force × distance = (mass × acceleration) × length. Unit = kg × m s⁻² × m = kg m² s⁻². It is called the joule (J).

2 marks: “Is the unit of speed a base unit or a derived unit? Explain with its formula.” Derived. Speed = distance ÷ time, so its unit is m ÷ s = m s⁻¹, built from two base units, the metre and the second. A base unit, like the metre, is not built from others.

Remember it

Hertz, Newton, Pascal, Joule, Watt: 'How Nice People Just Work' = frequency, force, pressure, energy, power.