Gravitational Potential

Gravitational potential describes the gravitational environment at a particular point in space in terms of energy per unit mass.

It is defined as:

the gravitational potential energy per unit mass of a small test mass placed at that point.

Gravitational potential is represented by:

V

and:

V = Eₚ/m

where:

• V = gravitational potential in J/kg
• Eₚ = gravitational potential energy in J
• m = test mass in kg

Unlike gravitational field strength, gravitational potential is a scalar quantity. It has magnitude but no direction.

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Potential Versus Potential Energy

Gravitational potential and gravitational potential energy are closely related, but they are not the same quantity.

Gravitational potential, V, describes the field at a particular location.

Gravitational potential energy, Eₚ, describes the energy associated with a particular mass placed at that location.

They are related by:

Eₚ = mV

or:

V = Eₚ/m

This distinction is similar to the difference between gravitational field strength and gravitational force.

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Gravitational Potential Is a Property of Position

Suppose a point in space has gravitational potential:

V = −2.0 × 10⁷ J/kg

This value is the same regardless of which small test mass is placed there.

A 1 kg mass would have:

Eₚ = mV

= 1 × (−2.0 × 10⁷)

= −2.0 × 10⁷ J

A 5 kg mass would have:

Eₚ = 5 × (−2.0 × 10⁷)

= −1.0 × 10⁸ J

The potential is unchanged.

The potential energy changes because the test mass is different.

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Calculating Gravitational Potential

Outside a spherical mass:

V = −GM/r

where:

• V = gravitational potential in J/kg
• G = gravitational constant = 6.67 × 10⁻¹¹ N·m²/kg²
• M = mass producing the gravitational field in kg
• r = distance from the centre of the mass in m

Notice that the mass of the test object does not appear in this equation.

This is because gravitational potential describes the field itself, not a particular test mass.

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Deriving the Potential Equation

Gravitational potential energy in a radial gravitational field is:

Eₚ = −GMm/r

Gravitational potential is:

V = Eₚ/m

Substitute:

V = (−GMm/r)/m

The test mass cancels:

V = −GM/r

This demonstrates why gravitational potential does not depend on the mass used to measure it.

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Worked Example: Potential Near a Planet

A planet has mass:

M = 6.0 × 10²⁴ kg

Find the gravitational potential at:

r = 8.0 × 10⁶ m

Use:

V = −GM/r

V = −(6.67 × 10⁻¹¹)(6.0 × 10²⁴)/(8.0 × 10⁶)

V ≈ −5.0 × 10⁷ J/kg

This means that every kilogram placed at that location has approximately:

−5.0 × 10⁷ J

of gravitational potential energy relative to the conventional zero at infinity.

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Gravitational Potential at Earth's Surface

Using approximate values:

Earth's mass:

M = 5.97 × 10²⁴ kg

Earth's radius:

R = 6.37 × 10⁶ m

At Earth's surface:

V = −GM/R

V = −(6.67 × 10⁻¹¹)(5.97 × 10²⁴)/(6.37 × 10⁶)

V ≈ −6.25 × 10⁷ J/kg

Therefore, Earth's surface gravitational potential is approximately:

−62.5 MJ/kg

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What Does −62.5 MJ/kg Mean?

It means that a 1 kg mass at Earth's surface has gravitational potential energy of approximately:

−62.5 MJ

relative to the conventional zero at infinite separation.

To move that kilogram infinitely far from Earth, at least approximately:

62.5 MJ

of energy must be supplied in the idealized Earth-only model.

This gives gravitational potential a useful physical interpretation.

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Why Is Gravitational Potential Negative?

For isolated gravitational systems, we conventionally choose:

V = 0 at infinity

At any finite distance from an attractive mass:

V < 0

because energy must be supplied to move a test mass from that location all the way to infinity.

The negative sign does not mean gravitational potential is somehow physically impossible or that energy has been destroyed.

It tells us that the point lies below the chosen zero-potential level.

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Thinking of a Gravitational Potential Well

A useful conceptual model is a potential well.

Imagine infinity as:

V = 0

Moving toward a planet takes us deeper into the gravitational potential well.

Far from the planet:

V is close to zero.

Closer to the planet:

V becomes more negative.

A more massive or more compact object can produce a deeper gravitational potential well.

This idea is especially useful in astronomy.

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Potential Changes with Distance

For a spherical mass:

V = −GM/r

As r increases:

• V becomes less negative
• V approaches zero

As r decreases:

• V becomes more negative

Therefore:

moving away from a planet increases gravitational potential

while:

moving toward a planet decreases gravitational potential.

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Comparing Two Distances

Suppose the potential at distance r is:

V = −40 MJ/kg

At distance 2r:

V₂ = −GM/(2r)

Therefore:

V₂ = −20 MJ/kg

At 4r:

V₄ = −10 MJ/kg

Notice that potential follows:

V ∝ −1/r

This is different from gravitational field strength:

g ∝ 1/r²

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Potential and Field Strength Change Differently

This distinction is important.

For a spherical mass:

g = GM/r²

but:

V = −GM/r

Therefore, if distance doubles:

field strength becomes:

g/4

while the magnitude of potential becomes:

|V|/2

Field strength decreases faster with distance than the magnitude of potential.

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Potential Is Scalar

Gravitational field strength is a vector.

It has:

• magnitude
• direction

Gravitational potential is a scalar.

It has:

• magnitude/value
• no direction

Therefore, gravitational potentials can be added algebraically without resolving them into directions.

This becomes particularly useful when several masses are present.

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Potential Due to Several Masses

Suppose several masses produce gravitational potentials at point P.

The total potential is:

V_total = V₁ + V₂ + V₃ + ...

For point masses:

V_total = −G(M₁/r₁ + M₂/r₂ + M₃/r₃ + ...)

Because potential is scalar, we simply add the numerical values.

No vector components are required.

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Worked Example: Two Masses

At point P:

Mass A produces:

V₁ = −3.0 × 10⁶ J/kg

Mass B produces:

V₂ = −2.0 × 10⁶ J/kg

Then:

V_total = V₁ + V₂

= −3.0 × 10⁶ − 2.0 × 10⁶

= −5.0 × 10⁶ J/kg

Notice that there is no concept of "left potential" or "right potential."

Potential has no direction.

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Zero Field Does Not Mean Zero Potential

This is a very important distinction.

Consider two identical masses with a point exactly halfway between them.

At the midpoint:

• the gravitational field from one mass points left
• the gravitational field from the other points right
• the field strengths are equal

Therefore:

g_net = 0

But each mass produces a negative gravitational potential.

The potentials add:

V_total = V₁ + V₂

Therefore:

V_total < 0

So:

zero gravitational field does not necessarily mean zero gravitational potential.

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Worked Example: Zero Field but Non-Zero Potential

Two identical masses each produce a potential of:

−4 MJ/kg

at their midpoint.

Because potential is scalar:

V_total = −4 + (−4)

= −8 MJ/kg

However, their gravitational fields point in opposite directions and cancel.

Therefore:

g_net = 0

but:

V_total = −8 MJ/kg

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Potential Difference

The difference in gravitational potential between two locations is:

ΔV = V₂ − V₁

Potential difference tells us how much gravitational potential energy changes per kilogram when moving between the locations.

Its unit is:

J/kg

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Potential Difference and Energy

Because:

Eₚ = mV

the change in gravitational potential energy is:

ΔEₚ = mΔV

Therefore:

ΔEₚ = m(V₂ − V₁)

This equation is useful when moving objects through gravitational fields.

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Worked Example: Potential Difference

A spacecraft moves from a location where:

V₁ = −50 MJ/kg

to:

V₂ = −20 MJ/kg

Then:

ΔV = V₂ − V₁

= −20 − (−50)

= +30 MJ/kg

The gravitational potential has increased by:

30 MJ/kg

If the spacecraft has mass:

1000 kg

then:

ΔEₚ = mΔV

= 1000 × 30 × 10⁶

= 3.0 × 10¹⁰ J

Its gravitational potential energy has increased by:

3.0 × 10¹⁰ J

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Gravitational Potential and Work Done

Potential difference is directly related to the work required to move a mass through a gravitational field.

If an external force slowly moves a mass without changing its kinetic energy:

work done by the external force = increase in gravitational potential energy

Therefore:

W_external = mΔV

or:

ΔV = W_external/m

This gives another definition of gravitational potential difference:

the work done per unit mass in moving a small test mass between two points without changing its kinetic energy.

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Work Done by Gravity

If gravity itself moves an object:

W_gravity = −ΔEₚ

Since:

ΔEₚ = mΔV

then:

W_gravity = −mΔV

This negative relationship is important.

When gravitational potential decreases, gravity does positive work.

When gravitational potential increases, gravity does negative work.

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Example: Falling Toward a Planet

Suppose a 2 kg object moves from:

V₁ = −3 MJ/kg

to:

V₂ = −7 MJ/kg

Potential difference:

ΔV = −7 − (−3)

= −4 MJ/kg

Therefore:

ΔEₚ = mΔV

= 2 × (−4 MJ/kg)

= −8 MJ

The system loses 8 MJ of gravitational potential energy.

Gravity does:

+8 MJ

of work.

If no other energy transfers occur, this energy appears as increased kinetic energy.

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Moving Away from a Planet

Now suppose a spacecraft moves outward.

Its potential changes from:

−8 MJ/kg

to:

−3 MJ/kg

Then:

ΔV = −3 − (−8)

= +5 MJ/kg

Potential increases.

Therefore, gravitational potential energy also increases.

An external energy source, such as a spacecraft engine, may provide the required energy.

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Relationship to Conservation of Energy

If gravity is the only force doing work:

Eₖ + Eₚ = constant

Since:

Eₚ = mV

we can write:

Eₖ + mV = constant

As an object falls:

• V decreases
• Eₚ decreases
• Eₖ increases

As an object moves outward:

• V increases
• Eₚ increases
• Eₖ decreases

This provides an energy-based way of analyzing motion in gravitational fields.

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Worked Example: Speed from Potential Difference

A spacecraft begins from rest at a location where:

V₁ = −8.0 × 10⁷ J/kg

It falls to a location where:

V₂ = −9.8 × 10⁷ J/kg

Ignore other forces.

Potential change per kilogram:

ΔV = −9.8 × 10⁷ − (−8.0 × 10⁷)

= −1.8 × 10⁷ J/kg

The decrease in gravitational potential becomes kinetic energy per kilogram:

Eₖ/m = 1.8 × 10⁷ J/kg

Since:

Eₖ/m = 1/2 v²

then:

1/2 v² = 1.8 × 10⁷

v² = 3.6 × 10⁷

v = 6000 m/s

Notice that the spacecraft's mass cancels.

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Potential Near Earth's Surface

Near Earth's surface, g is approximately constant.

For a small vertical change:

ΔV ≈ gΔh

This is the potential-per-unit-mass version of:

ΔEₚ = mgΔh

Since:

ΔEₚ = mΔV

and:

ΔEₚ = mgΔh

we get:

ΔV = gΔh

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Worked Example Near Earth's Surface

A student climbs:

12 m

vertically.

Using:

g = 9.8 N/kg

the increase in gravitational potential is:

ΔV = gΔh

= 9.8 × 12

= 117.6 J/kg

Every kilogram of the student gains:

117.6 J

of gravitational potential energy.

If the student has mass 60 kg:

ΔEₚ = 60 × 117.6

= 7056 J

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Equipotential Surfaces

An equipotential surface is a surface on which gravitational potential is the same everywhere.

Around an isolated spherical mass, equipotential surfaces are spherical.

In a two-dimensional diagram, they appear as concentric circles.

Moving along an equipotential surface does not change gravitational potential.

Therefore:

ΔV = 0

and:

ΔEₚ = 0

for movement along that surface.

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Field Lines and Equipotentials

Gravitational field lines cross equipotential surfaces at:

90°

The gravitational field points in the direction in which potential decreases most rapidly.

Around a planet:

• field lines point inward
• gravitational potential becomes more negative inward
• equipotential surfaces surround the planet

This relationship provides a useful connection between field diagrams and energy diagrams.

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Closely Spaced Equipotentials

If equipotential surfaces representing equal potential differences are very close together, the gravitational field is strong.

If they are farther apart, the gravitational field is weaker.

Conceptually:

large change in V over small distance → strong field

small change in V over large distance → weak field

This links gravitational potential directly to gravitational field strength.

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Potential and Field Strength

In one radial dimension, gravitational field strength is related to the rate at which potential changes with distance.

Conceptually:

g = rate of decrease of V with distance in the field direction

For a radial coordinate measured outward from a spherical mass:

g_r = −dV/dr

The minus sign indicates that the gravitational field points toward decreasing potential.

At a more introductory level, the key idea is:

the steeper the potential changes with position, the stronger the gravitational field.

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Gravitational Potential Graph

For a spherical mass:

V = −GM/r

A graph of V against r has several important features:

• V is negative
• V becomes very negative close to the mass
• V increases as r increases
• V approaches zero as r approaches infinity
• the curve becomes flatter at large distances

A steep potential graph corresponds to a strong gravitational field.

A flatter potential graph corresponds to a weaker field.

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Comparing Potential and GPE Graphs

For a fixed test mass:

Eₚ = mV

Therefore, gravitational potential and gravitational potential energy have the same general shape when plotted against distance.

However:

• V is measured in J/kg
• Eₚ is measured in J
• V does not depend on the test mass
• Eₚ does depend on the test mass

For a larger test mass, the GPE graph is scaled by m.

The potential graph remains unchanged.

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Potential Around Different Planets

At the surface of a spherical planet:

V = −GM/R

Therefore, surface gravitational potential depends on:

• planetary mass
• planetary radius

A planet with large mass and small radius can have a very negative surface potential.

This means a large amount of energy per kilogram is required to escape its gravitational influence.

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Comparing Two Planets

Suppose Planet B has:

4 times the mass of Planet A

and:

2 times the radius.

For Planet A:

VA = −GMA/RA

For Planet B:

VB = −G(4MA)/(2RA)

VB = 2VA

Because both values are negative, Planet B's surface potential is:

twice as negative

as Planet A's.

Notice that their surface field strengths would actually be equal:

gB/gA = 4/2²

= 1

So two planets can have the same surface gravitational field strength but different gravitational potentials.

This is an important distinction.

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Same g Does Not Mean Same V

This result may initially seem surprising.

Field strength depends on:

g ∝ M/R²

Potential depends on:

V ∝ −M/R

Therefore, two planets can have:

the same surface g

but:

different surface V

Field strength describes the local force per kilogram.

Potential describes the energy per kilogram relative to infinity.

They answer different physical questions.

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Gravitational Potential and Escape

Suppose an object is at a location with gravitational potential:

V

Its gravitational potential energy per kilogram is V.

To reach infinity with zero final speed:

final total energy per kilogram = 0

Therefore, the required kinetic energy per kilogram must compensate for the negative potential.

This connects gravitational potential directly to escape speed.

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Escape Speed from Potential

Per kilogram:

initial kinetic energy + initial potential = 0

So:

1/2 v_escape² + V = 0

Therefore:

v_escape = √(−2V)

Since V is negative, −2V is positive.

For Earth's surface:

V ≈ −6.25 × 10⁷ J/kg

v_escape = √(1.25 × 10⁸)

≈ 1.12 × 10⁴ m/s

≈ 11.2 km/s

This agrees with Earth's familiar idealized escape speed.

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Potential in Multi-Body Systems

Because potential is scalar, multi-body gravitational systems are often easier to analyze using potential than using field vectors.

For several masses:

V = −G(M₁/r₁ + M₂/r₂ + M₃/r₃ + ...)

Every positive source mass contributes a negative potential when zero is chosen at infinity.

The contributions simply add.

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Example: Potential Between Two Masses

Suppose point P is:

• 4 m from Mass A
• 6 m from Mass B

where:

MA = 3.0 × 10¹¹ kg

MB = 5.0 × 10¹¹ kg

Then:

V = −G(MA/rA + MB/rB)

V = −6.67 × 10⁻¹¹[(3.0 × 10¹¹/4) + (5.0 × 10¹¹/6)]

V ≈ −10.6 J/kg

No directions need to be assigned because potential is scalar.

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Potential at a Neutral Field Point

A neutral field point is where:

g_net = 0

But the gravitational potential there will generally still be negative.

This is because:

• fields are vectors and may cancel
• potentials are scalars and add

Therefore:

g = 0 does not imply V = 0

This is one of the most important conceptual differences between gravitational field strength and gravitational potential.

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Potential and Work: A Useful Interpretation

Suppose:

V = −20 MJ/kg

at one point and:

V = −5 MJ/kg

at another.

Moving 1 kg slowly from the first point to the second requires an increase in gravitational potential energy of:

15 MJ

Therefore, an external agent must do:

15 MJ of work per kilogram

in the idealized case.

This is why gravitational potential can be interpreted as an energy-per-unit-mass landscape.

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Common Mistakes

Confusing potential with potential energy

Potential is measured in J/kg.

Potential energy is measured in J.

Thinking potential depends on the test mass

Gravitational potential is a property of position in the field.

Forgetting the negative sign

For an isolated spherical mass with zero potential at infinity:

V = −GM/r

Thinking more negative means greater potential

−20 MJ/kg is a lower potential than −5 MJ/kg.

Thinking potential becomes more negative when moving away

Moving away makes potential less negative, so potential increases.

Confusing the distance relationships

Potential:

V ∝ −1/r

Field strength:

g ∝ 1/r²

Thinking zero field means zero potential

Fields can cancel vectorially while potentials remain negative.

Adding gravitational potentials as vectors

Potential is scalar. Simply add the numerical contributions.

Using altitude instead of distance from the centre

In:

V = −GM/r

r is measured from the centre of the source mass.

Thinking negative potential means negative mass

The negative sign results from the chosen zero at infinity and the attractive nature of gravity.

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Key Terms

Gravitational potential (V) — gravitational potential energy per unit mass at a point in a gravitational field.

Gravitational potential energy (Eₚ) — energy associated with the configuration of interacting masses.

Potential difference (ΔV) — difference in gravitational potential between two locations.

Joule per kilogram (J/kg) — the SI unit of gravitational potential.

Scalar — a quantity with magnitude but no direction.

Test mass — a small mass used conceptually to investigate a gravitational field.

Reference level — the chosen position at which potential is defined to have a particular value.

Infinity reference — the conventional choice V = 0 at infinite distance from an isolated gravitational source.

Equipotential surface — a surface on which gravitational potential has the same value everywhere.

Potential well — a representation of a region of negative gravitational potential around an attractive mass.

Work done — energy transferred when a force acts through a displacement.

Neutral field point — a point where the net gravitational field is zero.

Escape speed — the minimum idealized initial speed required to reach infinity with zero remaining speed.

Potential gradient — the rate at which gravitational potential changes with position, related to gravitational field strength.

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Key Takeaways

• Gravitational potential is gravitational potential energy per unit mass.
• It is represented by V and measured in J/kg.
• Gravitational potential is a property of a location in a gravitational field.
• Gravitational potential energy depends on the mass placed at that location.
• The quantities are related by Eₚ = mV.
• Outside a spherical mass, V = −GM/r.
• Gravitational potential is a scalar, while gravitational field strength is a vector.
• Potential contributions from several masses can therefore be added directly.
• Zero potential is conventionally chosen at infinity for isolated gravitational systems.
• Potential is negative at finite distances because energy must be supplied to move a mass completely away from an attractive gravitational source.
• Moving toward a planet decreases gravitational potential.
• Moving away from a planet increases gravitational potential.
• Potential varies as −1/r, while field strength varies as 1/r² in magnitude.
• Potential difference represents a change in gravitational energy per unit mass.
• ΔEₚ = mΔV connects potential difference with energy.
• Work done by an external force in slowly moving a mass is W_external = mΔV.
• Work done by gravity is W_gravity = −mΔV.
• Equipotential surfaces have constant gravitational potential.
• Gravitational field lines are perpendicular to equipotential surfaces.
• A point can have zero net gravitational field while still having a non-zero gravitational potential.
• Gravitational potential provides a powerful energy-based way of analyzing planets, satellites, spacecraft, and multi-body gravitational systems.

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Check Your Understanding

1. Define gravitational potential and state its SI unit.
2. Explain the difference between gravitational potential and gravitational potential energy.
3. A 4 kg object has gravitational potential energy of −2.4 × 10⁸ J. Calculate the gravitational potential at its location.
4. A point has gravitational potential −3.0 × 10⁷ J/kg. Calculate the gravitational potential energy of a 6 kg mass placed there.
5. Calculate the gravitational potential at a distance of 8.0 × 10⁶ m from the centre of a planet of mass 5.0 × 10²⁴ kg. Use G = 6.67 × 10⁻¹¹ N·m²/kg².
6. Explain why gravitational potential is negative when zero potential is defined at infinity.
7. The gravitational potential at distance r from a planet is −24 MJ/kg. What is the potential at 2r? What about 4r?
8. Explain why gravitational field strength decreases with 1/r² while gravitational potential varies with −1/r.
9. A spacecraft moves from a point where V = −60 MJ/kg to a point where V = −20 MJ/kg. Calculate the change in gravitational potential.
10. If the spacecraft in Question 9 has a mass of 1500 kg, calculate its change in gravitational potential energy.
11. Explain the relationship between gravitational potential difference and the work required to move a mass slowly through a gravitational field.
12. Two identical masses produce equal and opposite gravitational fields at their midpoint. Explain why the net gravitational field is zero there but the gravitational potential is not zero.
13. Explain what is meant by an equipotential surface and describe its relationship to gravitational field lines.
14. Planet B has four times the mass and twice the radius of Planet A. Compare their surface gravitational field strengths and their surface gravitational potentials.
15. A spacecraft falls from a region of higher gravitational potential to a region of lower gravitational potential. Explain how its gravitational potential energy, kinetic energy, and speed change if no other forces transfer energy.