Potential energy

Energy that waits

A stone lies on a roof, doing nothing. If it falls on a clay pot, the pot breaks. The stone can do work because of where it is. This stored energy is potential energy.

• Gravitational potential energy: because of height.
• Elastic potential energy: because something is stretched, squeezed or bent, like a rubber band, a bow or a spring.

Gravitational potential energy

To lift a mass $m$ to a height $h$, we do work $mgh$ against gravity. That work is stored in the object:

with $m$ in kilograms, $g = 9.8\ \text{m/s}^2$, $h$ in metres and $PE$ in joules.

The graph is a straight line through zero: twice the height, twice the energy.

You should find that the energy grows in step with the height. On the Moon the same lift stores only about a sixth as much, because $g$ is smaller.

From where do we measure the height?

We measure $h$ from a reference level that we choose, such as the floor or the ground. The potential energy is zero there. Another level gives other values, but the change in potential energy between two heights stays the same, and the change is what counts.

The path does not matter either: lifted straight up, or pushed up a smooth ramp, the object at the same height has the same $mgh$.

Elastic potential energy

When you stretch a rubber band, you do work on it, and the band stores it. Let go, and the band does work: it throws a paper pellet. A wound spring and a drawn bow do the same. We need no formula for this yet.

Watch out. $h$ is a vertical height. For a ramp, use how high the top is, not how long the ramp is.

A worked example

A $15\ \text{kg}$ bucket of water is carried up to a roof $6\ \text{m}$ above the ground. A balcony is $2.5\ \text{m}$ above the ground.

1. From the ground. $PE = 15\ \text{kg} \times 9.8\ \text{m/s}^2 \times 6\ \text{m} = 882\ \text{J}$.
2. From the balcony. The roof is $3.5\ \text{m}$ above it: $PE = 15\ \text{kg} \times 9.8\ \text{m/s}^2 \times 3.5\ \text{m} = 514.5\ \text{J}$.
3. The change. From the ground, the balcony is at $15\ \text{kg} \times 9.8\ \text{m/s}^2 \times 2.5\ \text{m} = 367.5\ \text{J}$ and the roof at $882\ \text{J}$. The change, $882\ \text{J} - 367.5\ \text{J} = 514.5\ \text{J}$, is the same as from the balcony.

Practise with new numbers

Try it in Python

The program finds the potential energy of the same object on four worlds.

Remember these

• Potential energy is stored energy of position or shape.
• $PE = mgh$ for a height $h$ above the level we choose.
• Only the vertical height counts, not the path.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the potential energy of an $8\ \text{kg}$ bucket of water $3\ \text{m}$ above the ground.

2. A ball $4\ \text{m}$ up has $49\ \text{J}$ of potential energy. What is its mass?

3. Name two things that store elastic potential energy.

4. Two students measure the potential energy of the same book, one from the floor and one from a table top. Who is right?