📈 Lifted up: GPE gained
When a crane lifts a steel beam, the beam's gravitational store fills up. The energy gained is m × g × h, where h is how far the beam goes up.
Work and Power » Gravitational Potential Energy
What you'll learn this session
Study time: 30 minutes
Edexcel spec: 4.13
- What gravitational potential energy (GPE) is, and when it is gained or lost
- How to use GPE = m × g × h
- How to change grams to kilograms and centimetres to metres first
- How to rearrange the equation to find a mass or a height
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Unlock This CourseWhat is gravitational potential energy?
Remember, an object that has been lifted up has energy in its gravitational store. The amount of energy in that store is called its gravitational potential energy, or GPE for short.
Lift an object higher and its GPE goes up. Let it come back down and its GPE goes down again. How much it changes depends on three things: the mass of the object, the gravitational field strength, and how far up or down it moves.
You need to know this relationship by heart. It is not given on the formulae page of the exam paper.
gravitational potential energy = mass × gravitational field strength × height
GPE = m × g × h
| Quantity | Symbol | Unit |
|---|---|---|
| Gravitational potential energy | GPE | joule (J) |
| Mass | m | kilogram (kg) |
| Gravitational field strength | g | newton per kilogram (N/kg) |
| Height | h | metre (m) |
On Earth, use g = 10 N/kg unless the question gives you a different value.
Key terms:
- Gravitational potential energy (GPE): the energy an object has in its gravitational store because of its height. It is measured in joules.
- Height (h): in this equation, the vertical distance the object moves up or down.
Gaining and losing GPE
The equation tells you the change in GPE when an object changes height. It does not matter where you start measuring from. Only the change in height counts.
📉 Falling down: GPE lost
When a coconut drops from a palm tree, its gravitational store empties. The energy lost is m × g × h, where h is how far the coconut comes down.
The equation also shows two simple patterns. GPE is directly proportional to mass, so lifting twice the mass through the same height gives twice the GPE. GPE is also directly proportional to height, so lifting the same mass twice as high gives twice the GPE.
Using GPE = m × g × h
Set out every calculation in three lines: the equation, the numbers, then the answer with its unit.
Worked example 1: energy gained
A crane lifts a steel beam of mass 400 kg through a height of 15 m. Calculate the gravitational potential energy gained by the beam.
GPE = m × g × h
GPE = 400 × 10 × 15
GPE = 60 000 J
Worked example 2: energy lost
A coconut of mass 1.5 kg falls 8 m from a palm tree to the sand. Calculate the gravitational potential energy it loses.
GPE = m × g × h
GPE = 1.5 × 10 × 8
GPE = 120 J
The coconut's gravitational store has 120 J less energy at the bottom than at the top. What happens to that energy is covered in Energy Changes: GPE, KE and Work.
Check the units first
The equation only gives joules if the mass is in kilograms and the height is in metres. Exam questions often give grams or centimetres to catch you out, so convert before you put any numbers in.
| Given | Change it to | How |
|---|---|---|
| 600 g | 0.6 kg | divide by 1000 |
| 150 cm | 1.5 m | divide by 100 |
Worked example 3: grams to kilograms
A squirrel of mass 600 g climbs 8 m up a tree. Calculate the gravitational potential energy it gains.
600 g = 600 ÷ 1000 = 0.6 kg
GPE = m × g × h
GPE = 0.6 × 10 × 8
GPE = 48 J
Worked example 4: grams and centimetres
A tin of beans of mass 400 g is lifted 150 cm from the floor to a kitchen worktop. Calculate the gravitational potential energy it gains.
400 g = 0.4 kg and 150 cm = 1.5 m
GPE = m × g × h
GPE = 0.4 × 10 × 1.5
GPE = 6 J
Which height do I use?
Gravity pulls straight down, so h is always the vertical change in height. If an object moves up a slope, a ramp or a winding path, ignore the distance along the path. Use only how much higher it ends up.
Worked example 5: a sloping path
A hiker of mass 60 kg walks 2000 m along a path up a hill. At the top she is 300 m higher than where she started. Calculate the gravitational potential energy she gains.
Use the vertical height: 300 m, not 2000 m.
GPE = m × g × h
GPE = 60 × 10 × 300
GPE = 180 000 J
There is a neat link here to two earlier lessons. Remember, weight = mass × g. So m × g in the GPE equation is just the object's weight, and GPE = weight × height. That is the same as the work done lifting the object at a steady speed. The work you do lifting something is the energy its gravitational store gains.
Rearranging the equation
| To find | Use |
|---|---|
| Gravitational potential energy | GPE = m × g × h |
| Mass | m = GPE ÷ (g × h) |
| Height | h = GPE ÷ (m × g) |
The rule is simple: divide the GPE by the other two quantities multiplied together. Work out the bracket first.
Worked example 6: finding the mass
A climber gains 9600 J of gravitational potential energy climbing 12 m up a climbing wall. Calculate the mass of the climber.
m = GPE ÷ (g × h)
m = 9600 ÷ (10 × 12)
m = 9600 ÷ 120
m = 80 kg
Worked example 7: finding the height
A pole vaulter of mass 70 kg gains 3500 J of gravitational potential energy between the ground and the top of her jump. Calculate the height she rises through.
h = GPE ÷ (m × g)
h = 3500 ÷ (70 × 10)
h = 3500 ÷ 700
h = 5 m
Always check your answer makes sense. A climber with a mass of 80 kg is sensible. If you had got 0.08 kg or 8000 kg, you would know to look for a slip.
Common mistakes
- Leaving out g. Mass × height is not an energy. You must multiply by 10 N/kg as well.
- Leaving the mass in grams. A 400 g tin is 0.4 kg. Using 400 gives an answer 1000 times too big.
- Leaving the height in centimetres. 150 cm is 1.5 m. Using 150 gives an answer 100 times too big.
- Using the distance along a slope. h is the vertical change in height only.
- Multiplying a weight by g. If the question gives the weight in newtons, g is already included. Just use GPE = weight × height.
- Rearranging in the wrong order. To find m or h, the GPE always goes on top of the division.
Exam-style question
A cable car carries people up a mountain. The cable car has a mass of 1200 kg. It travels 1500 m along the cable and finishes 450 m higher than it started.
(a) State the equation linking gravitational potential energy, mass, gravitational field strength and height. [1 mark]
(b) Calculate the gravitational potential energy gained by the cable car. [2 marks]
(c) Explain why the distance of 1500 m is not used in part (b). [1 mark]
(d) A passenger in the cable car gains 292 500 J of gravitational potential energy during the same journey. Calculate the mass of the passenger. [3 marks]
(e) On the way back down, the cable car stops at a station. By then it has lost 1 800 000 J of gravitational potential energy. Calculate the vertical height it has come down. [3 marks]
Model answer
(a) gravitational potential energy = mass × gravitational field strength × height, or GPE = m × g × h (1)
(b) GPE = 1200 × 10 × 450 (1) = 5 400 000 J (1)
(c) The height in the equation is the vertical change in height, and 1500 m is the distance along the cable (1).
(d) m = GPE ÷ (g × h) (1) = 292 500 ÷ (10 × 450) (1) = 65 kg (1)
(e) h = GPE ÷ (m × g) (1) = 1 800 000 ÷ (1200 × 10) (1) = 150 m (1)
Exam tip
When a question gives two distances, as this one does, underline the vertical one before you start. In parts (d) and (e), write the rearranged equation first, because it earns a mark even if you slip on the arithmetic.