Higher tier: Gravitational potential energy
Any object lifted above the ground has gravitational potential energy (\(E_{p}\) or GPE).
The amount of gravitational potential energy an object has on Earth depends on its:
- mass;
- height above the ground.
In the diagram:
- all the books on a shelf have GPE;
- books A and B have more GPE than book C because they are higher;
- book B has more GPE than book A because it has a greater mass.
Calculating gravitational potential energy
The gravitational potential energy of an object raised above the Earth鈥檚 surface can be calculated using the equation:
gravitational potential energy = mass x gravitational field strength x vertical height raised
gravitational potential energy = mgh.
or
\(E_{p}\) = mgh
where:
\(E_{p}\) is the gravitational potential energy in joules, J
m is the mass in kilograms, kg
g is the gravitational field strength in newtons per kilogram, N/kg
On Earth, g = 10 N/kg.
h is the change in height in metres, m.
Example
A book with a mass of 0.25 kg is lifted 2 m onto a bookshelf.
If g is 10 N/kg, how much gravitational potential energy does it gain?
Answer
\(E_{p}\) = mgh
m = 0.25 kg
g = 10 N/kg
h = 2 m
\(E_{p}\) = 0.25 x 10 x 2
\(E_{p}\) = 5 J
The gravitational potential energy gained by the book is 5 J.
Question
A book of mass 600 g has 12 J of gravitational potential energy. How high is it above the Earth鈥檚 surface? (g = 10 N/kg)?
The book has mass 600 g.
This must be converted into kg to use in the equation for gravitational potential energy.
To change from g to kg you divide by 1000.
600 g = \(\frac{600}{1000}\) = 0.6 kg
\(E_{p}\) = mgh
\(E_{p}\) = 12 J
m = 0.6 kg
g = 10 N/kg
12 = 0.6 x 10 x h
h = \(\frac {12}{{0.6}\times{10}}\)
h = 2 m
The book is 2 m above the surface of the Earth.
Conservation of energy and gravitational potential energy
The book in the above question has 12 J of The energy stored by an object lifted up against the force of gravity. Also known as GPE. when it is 2 m above the ground.
If the book falls to the floor, it loses its GPE.
So, where has it gone?
From the principle of conservation of energy, energy can never be destroyed; it can only be transferred from one form to another.
When the book starts to fall its gravitational potential energy is converted to The energy an object possesses due to its motion. energy.
As the book gets faster it gains more kinetic energy and loses more potential energy.
Just before it hits the floor all the gravitational potential energy has been converted to kinetic energy.
When the book hits the floor it stops 鈥 the kinetic energy of the book is converted into heat and sound energy.
As the book falls:
Gravitational potential energy \(\rightarrow\) kinetic energy.
When the book hits the ground:
Kinetic energy \(\rightarrow\) heat energy + sound energy.
Rollercoasters use these energy transfers too.
The rollercoaster car gains GPE as it travels to the top.
Once over the top, the car gains speed as GPE is transferred to kinetic energy.
As it travels to the top of another loop, kinetic energy is transferred to GPE.
Not all the energy is transferred to or from GPE 鈥 some is transferred to the surroundings as heat and sound.
Other theme park rides use the transfer of gravitational potential energy to kinetic energy and kinetic energy to gravitational potential energy.
As the pirate ship falls, GPE is transferred into kinetic energy.
At the bottom of the swing, it's travelling at its highest speed.
As it swings back up the other side it slows down as its kinetic energy is transferred back into GPE.