Gravitational Potential Energy

Far apart counts as zero, so every bound pair has negative energy.

11 minUpdated 21 October 2026gravitational potential energyzero at infinitysystem of particles

Lesson at a glanceShow the key points

Change in PE when separation goes from r₁ to r₂: U(r₂) − U(r₁) = Gm₁m₂(1/r₁ − 1/r₂) (11.2).
With U = 0 at infinite separation: U(r) = −Gm₁m₂/r (11.3).
For many particles, add the PE of every pair: N particles have N(N − 1)/2 pairs.
Bringing particles together from far away, gravity does positive work and U becomes more negative.
Near the surface, U changes by approximately mgh for a small height h.

Near the ground, lifting something by h costs mgh. But g weakens with distance, so for satellites and rockets we need the exact potential energy of two bodies at any separation.

Moving one particle away from another

Recall from the chapter on energy that for a conservative force, $U_f - U_i = -\int_i^f \vec F\cdot d\vec r$ . Hold particle m₁ fixed and move m₂ outward so the separation grows from r to r + dr. Gravity pulls m₂ back toward m₁ with $Gm_1m_2/r^2$ , so it does work $dW = -\dfrac{Gm_1m_2}{r^2}dr$ , and the potential energy rises by $dU = -dW = \dfrac{Gm_1m_2}{r^2}dr$ . From r₁ to r₂:

U(r_2) - U(r_1) = Gm_1m_2\int_{r_1}^{r_2}\frac{dr}{r^2} = Gm_1m_2\left(\frac1{r_1} - \frac1{r_2}\right) \qquad (11.2)

Where to put zero

Only differences matter, so we may choose where U = 0. The natural choice is infinite separation, where the particles no longer interact. Put r₁ → ∞ and r₂ = r:

Gravitational potential energy of a pair(11.3)

U(r) = -\frac{Gm_1m_2}{r}

zero at infinite separation; negative at every finite distance

U depends only on the separation, not on which particle moved, so the formula holds generally. For three or more particles, add the potential energy of every pair: N particles form N(N − 1)/2 pairs.

Interactive

Every pair counts

Identical masses m sit at the corners of a regular polygon of side a. The system's potential energy is the sum of −Gm²/r over every pair (each line). N particles make N(N − 1)/2 pairs.

N(N-1)/2

3.00

U

−3.000 Gm²/a

Number of particles

N

Worked example

Assembling a square

Four 200 g balls are brought from very far apart to the corners of a square of side 10 cm. Find the gravitational potential energy of the arrangement and the work done by gravity.

Where mgh comes from

For a body of mass m lifted from the surface (r = R) to a height h, the exact formula gives

\Delta U = GMm\left(\frac1R - \frac1{R+h}\right) = \frac{GMm\,h}{R(R+h)} \approx \frac{GMm}{R^2}h = mgh \quad (h \ll R)

Interactive

How deep is the gravitational well?

U(r) = −GMm/r, plotted in units of GMm/R with r in planet radii. Drag the two points: the rise in potential energy (gold) is always less than the mgh estimate (blue), which assumes g stays at its surface value. For small heights the two agree.

exact ΔU0.333GMm/R

mgh estimate0.500GMm/R

mgh overestimates by50.0 %

Start

r_1

1.00R

End

r_2

1.50R

Worked example

Lifting a satellite

How much energy is needed just to lift a 1000 kg satellite from the Earth’s surface to a height equal to the Earth’s radius (6400 km)? Compare with mgh.

What to remember

1 $U(r_2) - U(r_1) = Gm_1m_2(1/r_1 - 1/r_2)$ .
2With U = 0 at infinity: $U = -Gm_1m_2/r$ , always negative.
3Systems: add U over all N(N − 1)/2 pairs.
4mgh is the small-height approximation of the exact formula.

Check yourself

Two particles move from separation r to 2r. Their potential energy…

How many pairs are there among 5 particles?

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

ReadGravitational potential energy and total energyOpenStax University Physics §13.3 ReadGravitational potential energyHyperPhysics ReadPotential energyWikipedia

PreviousWeighing the Earth: Measuring G Next Gravitational Potential

Gravitational Potential Energy

Far apart counts as zero, so every bound pair has negative energy.

11 minUpdated 21 October 2026gravitational potential energyzero at infinitysystem of particles

Lesson at a glanceShow the key points

Change in PE when separation goes from r₁ to r₂: U(r₂) − U(r₁) = Gm₁m₂(1/r₁ − 1/r₂) (11.2).
With U = 0 at infinite separation: U(r) = −Gm₁m₂/r (11.3).
For many particles, add the PE of every pair: N particles have N(N − 1)/2 pairs.
Bringing particles together from far away, gravity does positive work and U becomes more negative.
Near the surface, U changes by approximately mgh for a small height h.

Near the ground, lifting something by h costs mgh. But g weakens with distance, so for satellites and rockets we need the exact potential energy of two bodies at any separation.

Moving one particle away from another

U(r_2) - U(r_1) = Gm_1m_2\int_{r_1}^{r_2}\frac{dr}{r^2} = Gm_1m_2\left(\frac1{r_1} - \frac1{r_2}\right) \qquad (11.2)

Where to put zero

Only differences matter, so we may choose where U = 0. The natural choice is infinite separation, where the particles no longer interact. Put r₁ → ∞ and r₂ = r:

Gravitational potential energy of a pair(11.3)

U(r) = -\frac{Gm_1m_2}{r}

zero at infinite separation; negative at every finite distance

Interactive

Every pair counts

Identical masses m sit at the corners of a regular polygon of side a. The system's potential energy is the sum of −Gm²/r over every pair (each line). N particles make N(N − 1)/2 pairs.

N(N-1)/2

3.00

U

−3.000 Gm²/a

Number of particles

N

Worked example

Assembling a square

Four 200 g balls are brought from very far apart to the corners of a square of side 10 cm. Find the gravitational potential energy of the arrangement and the work done by gravity.

Where mgh comes from

For a body of mass m lifted from the surface (r = R) to a height h, the exact formula gives

\Delta U = GMm\left(\frac1R - \frac1{R+h}\right) = \frac{GMm\,h}{R(R+h)} \approx \frac{GMm}{R^2}h = mgh \quad (h \ll R)

Interactive

How deep is the gravitational well?

exact ΔU0.333GMm/R

mgh estimate0.500GMm/R

mgh overestimates by50.0 %

Start

r_1

1.00R

End

r_2

1.50R

Worked example

Lifting a satellite

How much energy is needed just to lift a 1000 kg satellite from the Earth’s surface to a height equal to the Earth’s radius (6400 km)? Compare with mgh.

What to remember

1 $U(r_2) - U(r_1) = Gm_1m_2(1/r_1 - 1/r_2)$ .
2With U = 0 at infinity: $U = -Gm_1m_2/r$ , always negative.
3Systems: add U over all N(N − 1)/2 pairs.
4mgh is the small-height approximation of the exact formula.

Check yourself

Two particles move from separation r to 2r. Their potential energy…

How many pairs are there among 5 particles?

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

ReadGravitational potential energy and total energyOpenStax University Physics §13.3 ReadGravitational potential energyHyperPhysics ReadPotential energyWikipedia

PreviousWeighing the Earth: Measuring G Next Gravitational Potential

Moving one particle away from another#

Where to put zero#

Where mgh comes from#

Check yourself

Keep exploring

Moving one particle away from another#

Where to put zero#

Where mgh comes from#

Check yourself

Keep exploring

Moving one particle away from another

Where to put zero

Where mgh comes from

Moving one particle away from another

Where to put zero

Where mgh comes from