Units of Energy and Mass

From the joules in a lightning bolt to the kilojoules on a food label — energy and mass have a hierarchy of units. Convert fluently between them and you can answer any real-world measurement question.

50–55 min MS-M1 3 MC 3 SA Lesson 18 of 22 Free
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Think First

Your household electricity bill charges you by the kilowatt-hour (kWh). A standard electric kettle is rated at 2400 W. If you boil the kettle for 3 minutes, how much energy does it use — and roughly what does that cost if electricity is 30 cents per kWh? Make a rough guess before the lesson teaches you the method.

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Energy and Mass Formulas — This Lesson

$1 \text{ kJ} = 1000 \text{ J}$
J — joule, the SI unit of energy kJ — kilojoule = 1000 J; used on food labels MJ — megajoule = 1 000 000 J
$1 \text{ Cal} = 1 \text{ kcal} = 4.184 \text{ kJ}$
cal — small calorie; energy to raise 1 g of water by 1°C Cal / kcal — kilocalorie = 1000 cal; used in food/nutrition contexts $1 \text{ cal} = 4.184 \text{ J}$; so $1 \text{ kcal} = 4184 \text{ J} = 4.184 \text{ kJ}$
$E \text{ (kWh)} = P \text{ (kW)} \times t \text{ (h)}$
E — energy in kilowatt-hours (kWh) P — power in kilowatts (kW); note 1 kW = 1000 W t — time in hours; convert minutes to hours before substituting $1 \text{ kWh} = 3600 \text{ kJ} = 3\,600\,000 \text{ J}$
Mass: $1 \text{ t} = 1000 \text{ kg} = 10^6 \text{ g} = 10^9 \text{ mg}$
mg — milligram = 0.001 g; used for medication dosages g — gram kg — kilogram = 1000 g; everyday mass unit t — metric tonne = 1000 kg; used for heavy loads
UNIT CONVERSION CHAINS MASS mg ÷1000 ×1000 g ÷1000 ×1000 kg ÷1000 ×1000 t ENERGY J ÷1000 ×1000 kJ ÷4.184 ×4.184 kcal 1 kWh = 3600 kJ Solid arrow: to smaller unit (divide). Dashed arrow: to larger unit (multiply)

🧠 Know

  • The hierarchy of mass units: mg, g, kg, tonne
  • The hierarchy of energy units: J, kJ, MJ
  • The calorie conversions: 1 cal = 4.184 J; 1 kcal = 4.184 kJ
  • Power–energy–time: E (kWh) = P (kW) × t (h)
  • 1 kWh = 3600 kJ

💡 Understand

  • Why electricity is billed in kWh rather than joules
  • The difference between a calorie (small) and a Calorie/kcal (food label)
  • How to convert minutes to hours before using the energy formula

✅ Can Do

  • Convert between mass units in multi-step problems
  • Convert between joules, kilojoules, and calories
  • Calculate energy consumption using E = P × t
  • Calculate electricity costs from energy used and a given rate
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Key Terms

Joule (J) The SI unit of energy; 1 J is the energy used when a force of 1 newton acts through 1 metre
Kilowatt-hour (kWh) The practical unit of electrical energy; 1 kWh = energy used by a 1 kW device running for 1 hour = 3 600 000 J
Calorie (kcal / Cal) The Calorie on food labels is actually a kilocalorie (kcal); 1 kcal = 4.184 kJ. The small calorie (cal) = 4.184 J
Watt (W) The unit of power (rate of energy use); 1 W = 1 J per second. 1 kW = 1000 W
Metric tonne (t) A mass unit equal to 1000 kg; not to be confused with the imperial "ton"

Misconceptions to Fix

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Wrong: Capacity and volume are completely different concepts.

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Right: Capacity and volume are related; 1 litre = 1000 cm³ = 1 dm³.

Mass Unit Conversions

Key Point

Always check your units before substituting into formulas. Converting to consistent units is a common source of errors in assessment tasks.

Key Terms
FormulaA rule showing the relationship between variables using symbols.
SubstitutionReplacing variables with their known values in an equation.
Unit ConversionChanging a measurement from one unit to another.
CapacityThe amount of liquid a container can hold, measured in litres or millilitres.
PerimeterThe total distance around the outside of a shape.
AreaThe amount of space inside a two-dimensional shape.

The Mass Unit Hierarchy

Mass units follow the same pattern as length: each step up the chain multiplies by 1000. The key is always identifying which direction you are converting.

Must do: Always identify the direction of conversion first — larger unit to smaller means multiply; smaller unit to larger means divide.
UnitSymbolEquivalent in gramsCommon use
Milligrammg0.001 gMedication dosages, vitamins
Gramg1 gSmall food quantities
Kilogramkg1 000 gBody mass, everyday items
Tonnet1 000 000 gVehicles, heavy freight
Worked Example 1 Mass Unit Conversion

Problem

A medication tablet contains 250 mg of active ingredient. A patient takes 3 tablets per day for 14 days.

  • (a) How many grams of active ingredient does the patient take in total?
  • (b) Express this as kilograms.

Solution

1 Total tablets $= 3 \times 14 = 42$ tablets 3 per day for 14 days

Energy Unit Conversions

Joules, Kilojoules, and Calories

The joule is the scientific unit of energy, but real-world applications use larger units. Food labels use kilojoules (kJ) or Calories (kcal). Electricity uses kilowatt-hours (kWh).

Common error — the calorie trap: When a food label says "350 Calories", it means 350 kcal = 1463 kJ, NOT 350 J. The capital C in "Calorie" signals it is a kilocalorie. Always check which calorie is being used.
Worked Example 2 Food Energy Conversion

Problem

A muesli bar has an energy content of 756 kJ per serve.

  • (a) Convert this to joules.
  • (b) Convert this to Calories (kcal), correct to the nearest whole number.

Solution

1 (a) $756 \text{ kJ} \times 1000 = \mathbf{756\,000 \text{ J}}$ kJ → J: multiply by 1000 (going to a smaller unit)

Electrical Energy and Power

Kilowatt-Hours and Electricity Bills

Power (watts or kilowatts) tells you how fast an appliance uses energy. Energy (kWh) tells you the total amount used. Multiply power by time to get energy — but both must be in compatible units.

$$E \text{ (kWh)} = P \text{ (kW)} \times t \text{ (hours)}$$

Must do — convert units before substituting: P must be in kW and t must be in hours. A 1500 W heater is 1.5 kW; 45 minutes is 0.75 h. Substitute these converted values, not the originals.
Key insight: $1 \text{ kWh} = 3600 \text{ kJ}$. To see why: $1 \text{ kW} = 1 \text{ kJ/s}$, and there are $3600$ seconds in 1 hour. This conversion rarely appears in exam questions but explains the unit relationship.
Worked Example 3 Electricity Cost

Problem

An electric kettle has a power rating of 2400 W. It is used for 3 minutes each morning. Electricity costs 32 cents per kWh.

  • (a) How many kWh does the kettle use each morning?
  • (b) What is the daily cost of using the kettle, to the nearest tenth of a cent?
  • (c) What is the annual cost (365 days), to the nearest dollar?

Solution

1 Convert: $P = 2400 \div 1000 = 2.4 \text{ kW}$; $t = 3 \div 60 = 0.05 \text{ h}$ Watts → kilowatts (÷1000); minutes → hours (÷60)
Practice

Practice Questions

Section A — Mass Conversions

  1. Convert 3.25 kg to grams.
  2. Convert 4500 mg to grams.
  3. A truck carries 2.4 t of sand. Express this mass in kilograms.
  4. A vitamin supplement contains 0.5 mg of vitamin B12. Express this in grams.
  5. Three packages have masses of 450 g, 1.2 kg, and 800 g. Find the total mass in kilograms.

Section B — Energy Conversions

  1. Convert 5000 J to kilojoules.
  2. A chocolate bar contains 1050 kJ. Convert this to Calories (kcal), to 1 decimal place.
  3. Convert 95 Cal (kcal) to kilojoules, to 1 decimal place.
  4. A person's recommended daily intake is 8700 kJ. Express this in megajoules.

Section C — Electricity

  1. A 1200 W toaster is used for 4 minutes. Calculate the energy used in kWh.
  2. A pool pump is rated at 0.75 kW and runs for 8 hours per day. How many kWh does it use per day?
  3. Using Q11, if electricity costs 28 cents per kWh, find the weekly cost of running the pump.
  4. A household uses 22 kWh of electricity in a day. If electricity costs 35 cents per kWh, find the daily cost in dollars.

Q1

$3.25 \times 1000 = \mathbf{3250 \text{ g}}$

Q2

$4500 \div 1000 = \mathbf{4.5 \text{ g}}$

Q3

$2.4 \times 1000 = \mathbf{2400 \text{ kg}}$

Q4

$0.5 \div 1000 = \mathbf{0.0005 \text{ g}}$

Q5

$450 \text{ g} = 0.45 \text{ kg}$; total $= 0.45 + 1.2 + 0.8 = \mathbf{2.45 \text{ kg}}$

Q6

$5000 \div 1000 = \mathbf{5 \text{ kJ}}$

Q7

$1050 \div 4.184 \approx \mathbf{250.9 \text{ Cal}}$

Q8

$95 \times 4.184 = \mathbf{397.5 \text{ kJ}}$

Q9

$8700 \div 1000 = \mathbf{8.7 \text{ MJ}}$

Q10

$P = 1.2 \text{ kW}$; $t = 4/60 \approx 0.0\overline{6} \text{ h}$; $E = 1.2 \times (4/60) = \mathbf{0.08 \text{ kWh}}$

Q11

$E = 0.75 \times 8 = \mathbf{6 \text{ kWh per day}}$

Q12

Weekly: $6 \times 7 = 42 \text{ kWh}$; cost $= 42 \times 28 = 1176 \text{ c} = \mathbf{\$11.76}$

Q13

$22 \times 0.35 = \mathbf{\$7.70}$

Revisit Your Initial Thinking

Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?

Multiple Choice

1 A food label shows an energy content of 520 Cal. Expressed in kilojoules (to the nearest kJ), this is:

A   124 kJ
B   2176 kJ
C   520 kJ
D   124 000 kJ

? Regarding this topic, 1 A food label shows an energy content of 520 Cal. Expressed in kilojoules (to the nearest kJ), this is:

A     124 kJ
B     2176 kJ
C     520 kJ
D     124 000 kJ
B - Correct!
B — 520 Cal = 520 kcal; $520 \times 4.184 = 2175.68 \approx 2176$ kJ. Option A reverses the conversion (÷4.184 instead of ×4.184). Option C confuses Cal with kJ.

2 A 600 W refrigerator runs continuously. How much electrical energy does it use in 24 hours?

A   14 400 kWh
B   14.4 kWh
C   1.44 kWh
D   144 kWh

? Regarding this topic, 2 A 600 W refrigerator runs continuously. How much electrical energy does it use in 24 hours?

A     14 400 kWh
B     14.4 kWh
C     1.44 kWh
D     144 kWh
B - Correct!
B — $P = 600 \div 1000 = 0.6 \text{ kW}$; $E = 0.6 \times 24 = 14.4 \text{ kWh}$. Option A forgets to convert W to kW.

3 A delivery truck carries 3.6 t of goods. The truck itself has a mass of 8500 kg. What is the total mass in tonnes?

A   8503.6 t
B   12.1 t
C   121 t
D   12 100 t

? 3 A delivery truck carries 3.6 t of goods. The truck itself has a mass of 8500 kg. Identify the total mass in tonnes?

A     8503.6 t
B     12.1 t
C     121 t
D     12 100 t
B - Correct!
B — Convert 8500 kg to tonnes: $8500 \div 1000 = 8.5 \text{ t}$; total $= 3.6 + 8.5 = 12.1 \text{ t}$. Option A adds 8500 kg directly to 3.6 t without conversion.

Short Answer

01

SA 4 3 marks A portable speaker is charged using a 5 W USB cable for 2.5 hours.

(a) Calculate the energy used to charge the speaker in kWh.  (1 mark)

(b) Convert this energy to kilojoules.  (1 mark)

(c) If electricity costs 30 cents per kWh, find the cost to charge the speaker.  (1 mark)

Work in your book
Saved

(a)

$P = 5 \div 1000 = 0.005 \text{ kW}$; $E = 0.005 \times 2.5 = \mathbf{0.0125 \text{ kWh}}$

(b)

$0.0125 \times 3600 = \mathbf{45 \text{ kJ}}$

(c)

$0.0125 \times 30 = \mathbf{0.375 \text{ cents}}$ (less than half a cent)

02

SA 5 4 marks A household appliance runs for 3 hours at a power of 800 W. Electricity is charged at 34 cents per kWh.

(a) Find the energy used in kWh.  (1 mark)

(b) Find the cost of running the appliance.  (1 mark)

(c) If the appliance is used every day, find the annual cost to the nearest dollar.  (2 marks)

Work in your book
Saved

(a)

$P = 0.8 \text{ kW}$; $E = 0.8 \times 3 = \mathbf{2.4 \text{ kWh}}$

(b)

$2.4 \times 34 = 81.6 \text{ c} = \mathbf{\$0.816} \approx \mathbf{81.6 \text{ cents}}$

(c)

Annual: $0.816 \times 365 = 297.84 \approx \mathbf{\$298}$

03

SA 6 4 marks An athlete burns 2800 Cal (kcal) per day through exercise and metabolism.

(a) Convert 2800 Cal to kilojoules.  (1 mark)

(b) Convert your answer to megajoules, correct to 2 decimal places.  (1 mark)

(c) If the athlete's daily food provides 12 500 kJ, by how many kilojoules does their intake fall short of their energy requirement?  (2 marks)

Work in your book
Saved

(a)

$2800 \times 4.184 = \mathbf{11\,715.2 \text{ kJ}}$

(b)

$11\,715.2 \div 1000 = \mathbf{11.72 \text{ MJ}}$

(c)

Shortfall $= 11\,715.2 - 12\,500 = -784.8$; the intake actually exceeds requirements by $\mathbf{784.8 \text{ kJ}}$. (Or: if comparing requirement to intake: intake 12 500 kJ exceeds the 11 715.2 kJ needed by 784.8 kJ.)

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