Gravitational Potential Energy

Heave a box up onto a high shelf and let go — it crashes back to the floor. Pull a rollercoaster car slowly up to the top of the first big hill and release it — it comes screaming down. Hold a pile of water back behind a tall dam — open a valve and it thunders out through the turbines. In every case, lifting something up "loaded" it with energy, and letting it come back down handed that energy back.

That stored-up, waiting-to-be-released energy is called gravitational potential energy — E_p for short. It is the energy an object has because of how high it has been raised in a gravitational field. Raise something higher and you store more of it; let it fall and the store empties out again.

Why lifting stores energy

Gravity is always pulling an object down with a force equal to its weight, W = m\,g. To raise the object you must push up against that pull the whole way — you do work against gravity. That work does not vanish: it is stored in the object's raised position as gravitational potential energy, ready to be released the instant you let go.

Work done is force times distance. The force you fight is the weight m\,g, and the distance you lift it is the height h, so the energy stored is force \times distance = m\,g \times h. That single line is the GPE formula.

The gravitational potential energy gained by an object of mass m raised through a height h is:

E_p = m\,g\,h.

E_p — gravitational potential energy, in joules (J)
m — mass, in kilograms (kg)
g — gravitational field strength, about 9.8\ \text{N/kg} on Earth (often rounded to 10\ \text{N/kg} for quick sums)
h — the height it is raised, in metres (m)

Read the formula and you can already predict everything: double the mass and you double the stored energy; double the height and you double it too; carry the same lift out on the Moon, where g is six times smaller, and you store six times less.

Worked example: energy to lift a box

A removal worker lifts a 4\ \text{kg} box from the floor onto a shelf 2\ \text{m} high. Taking g = 9.8\ \text{N/kg}, how much gravitational potential energy does the box gain?

Step 1 — write the formula.

E_p = m\,g\,h.

Step 2 — put the numbers in.

E_p = 4 \times 9.8 \times 2.

Step 3 — work it out.

E_p = 78.4\ \text{J}.

So lifting the box takes 78.4 J of work, and all of it is now stored as GPE. Knock the box off the shelf and gravity will pay every joule straight back.

Working backwards: find the height, or the mass

E_p = m\,g\,h links four quantities, so knowing any three lets you find the fourth. Rearrange it like any equation:

h = \frac{E_p}{m\,g}, \qquad\qquad m = \frac{E_p}{g\,h}.

Find the height. A 3\ \text{kg} book gains 180\ \text{J} of GPE when it is lifted onto a shelf. Taking g = 10\ \text{N/kg}, how high is the shelf?

h = \frac{E_p}{m\,g} = \frac{180}{3 \times 10} = \frac{180}{30} = 6\ \text{m}.

Find the mass. Raising an object 5\ \text{m} stores 250\ \text{J} of GPE (g = 10\ \text{N/kg}). What is its mass?

m = \frac{E_p}{g\,h} = \frac{250}{10 \times 5} = \frac{250}{50} = 5\ \text{kg}.

Lift it, then drop it: GPE becomes kinetic

Stored energy is only interesting because it can be released. Let a raised object fall and its gravitational potential energy is transferred, joule for joule, into kinetic energy — the energy of movement, E_k = \tfrac{1}{2}mv^2. Ignoring air resistance, all the GPE it had at the top becomes kinetic energy at the bottom:

m\,g\,h = \tfrac{1}{2}mv^2.

Notice the mass m cancels from both sides, leaving v = \sqrt{2gh} — which is exactly why a heavy ball and a light ball dropped from the same height hit the ground at the same speed.

Worked example. A 2\ \text{kg} ball is raised 5\ \text{m} and dropped (g = 10\ \text{N/kg}). How fast is it going just before it lands?

Step 1 — find the GPE at the top.

E_p = m\,g\,h = 2 \times 10 \times 5 = 100\ \text{J}.

Step 2 — all of it becomes kinetic energy at the bottom.

\tfrac{1}{2}mv^2 = 100 \;\Rightarrow\; \tfrac{1}{2}\times 2 \times v^2 = 100 \;\Rightarrow\; v^2 = 100.

Step 3 — take the square root.

v = \sqrt{100} = 10\ \text{m/s}.

The same answer drops straight out of v = \sqrt{2gh} = \sqrt{2 \times 10 \times 5} = \sqrt{100} = 10\ \text{m/s} — the store you filled by lifting has been paid back entirely as speed.

Fill the store yourself

In the box below, the square is an object being raised above the ground and the tall bar on the right is its gravitational potential energy store. Slide the height up and the store fills; slide it back down and it empties — GPE depends on how high you have lifted it. Now change the mass: a heavier object stores more energy at the very same height. The readout tracks E_p = m\,g\,h live, with g = 9.8\ \text{N/kg}.

Giant stores: reservoirs and rollercoasters

The same idea runs some enormous machines. A hydroelectric dam holds a whole lake of water high up behind it. Every kilogram of that water is a little packet of GPE, m\,g\,h, waiting at the top. Open the gates and the water rushes down, its GPE turning into kinetic energy that spins turbines and generates electricity for a city. A pumped-storage station even runs it in reverse: when electricity is cheap it pumps water back up to the high reservoir — deliberately storing GPE to release again at teatime when everyone switches the kettle on.

A rollercoaster plays the same trick for fun. A motor drags the car slowly up the tall first hill, patiently loading it with GPE. From then on no engine is needed: at the top the car is a full store of energy, and the whole ride is just that GPE sloshing back and forth — into kinetic energy plunging down each drop, and back into GPE climbing the next hill, a little lost to friction each time until the ride finally coasts to a stop.

These four catch nearly everyone out — check yourself against all of them:

It is the change in height that matters. GPE is always measured relative to some starting level — usually the ground or floor. Carrying a book across a flat room changes its GPE by nothing at all, because h (the height gained) is zero. Only lifting it higher (or letting it drop lower) changes the store.
GPE is not the same as weight. Weight W = m\,g is a force, measured in newtons; GPE E_p = m\,g\,h is energy, measured in joules. An object resting on the floor still has a weight, but (measuring from the floor) it has no GPE until you raise it.
You need mass, g and height — all three. Leave any one out and the answer is meaningless. A huge height with a tiny mass, or a big mass left on the ground (h = 0), both store little or nothing.
Keep the units straight. Mass in kilograms, height in metres, g in N/kg — then the answer comes out in joules. Height in centimetres or mass in grams will wreck the sum.

An apple has a mass of about 0.1\ \text{kg}. Lift it 1\ \text{m} onto a table and you store E_p = 0.1 \times 9.8 \times 1 \approx 1\ \text{J} of GPE — which is roughly where the joule gets its "everyday feel": one joule is about the energy of raising a small apple by a metre. Tiny.

Now scale up. The water behind a large dam might be millions of tonnes, held a hundred metres high — m\,g\,h then runs into the trillions of joules, enough to light a city. Same formula, same physics as the apple; the only difference is a colossal mass raised a long way up. That is the quiet power of storing energy simply by lifting things.