Equipotential Surfaces

An equipotential surface is a surface on which the gravitational potential has the same value at every point.

If two points, A and B, lie on the same equipotential surface:

VA = VB

Therefore:

ΔV = 0

Since the change in gravitational potential energy is:

ΔEₚ = mΔV

moving a mass from A to B along the equipotential gives:

ΔEₚ = 0

This means that gravity does no net work on an object moving along an equipotential surface.

 

 

 

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Understanding Equipotential Surfaces

The word equipotential can be broken into:

• equi = equal
• potential = gravitational potential

Every point on one equipotential surface therefore has the same gravitational potential.

For example, imagine a spherical equipotential around a planet where:

V = −40 MJ/kg

Any point on that surface has:

V = −40 MJ/kg

Another equipotential farther away might have:

V = −30 MJ/kg

and another:

V = −20 MJ/kg

Each surface represents a different value of gravitational potential.

Surfaces Versus Lines

Gravitational equipotentials are actually three-dimensional surfaces.

Around an isolated spherical mass, they are spherical shells surrounding the mass.

However, textbooks usually show a two-dimensional cross-section.

A spherical equipotential then appears as a circle.

 

 

 

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So when you see concentric circles around a mass in a two-dimensional diagram, imagine them as cross-sections through spherical surfaces.

Equipotentials Around a Spherical Mass

For an isolated spherical mass:

V = −GM/r

Potential depends only on:

r, the distance from the centre.

Every point the same distance r from the centre therefore has the same potential.

This produces spherical equipotential surfaces.

For example:

At radius r:

V = −GM/r

Every point on that sphere has exactly the same gravitational potential.

Why Equipotentials Are Spherical

Imagine a planet that can be approximated as a sphere.

At a particular distance r from its centre:

• M is constant
• G is constant
• r is constant

Therefore:

V = −GM/r

has the same value everywhere on that spherical surface.

Changing direction around the planet does not change V.

Changing distance from the centre does.

Equipotential Surfaces and Field Lines

Gravitational field lines and equipotential surfaces have a very important geometric relationship:

Gravitational field lines always cross equipotential surfaces at 90°.

In other words, they are perpendicular.

Around a spherical mass:

• gravitational field lines point radially inward
• equipotential surfaces form concentric spheres
• the field lines cross those surfaces at right angles

 

 

 

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Why Are They Perpendicular?

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

If you move along an equipotential:

potential does not change.

Therefore, the gravitational field cannot have a component along the equipotential surface.

If it did, gravity would do work and the potential would change.

The field must therefore point perpendicular to the surface.

No Work Along an Equipotential

Suppose a mass m moves from point A to point B along the same equipotential.

Then:

VA = VB

Therefore:

ΔV = VB − VA

= 0

Since:

ΔEₚ = mΔV

then:

ΔEₚ = m(0)

= 0

The work done by gravity is:

W_gravity = −ΔEₚ

Therefore:

W_gravity = 0

Another Explanation Using Force and Displacement

Work can also be described by:

W = Fd cos θ

When an object moves along an equipotential surface:

• gravitational force points perpendicular to the surface
• displacement is along the surface
• the angle between force and displacement is 90°

Since:

cos 90° = 0

then:

W = 0

So gravity does no work during motion exactly along an equipotential.

Example: Moving Around a Planet

Imagine a spacecraft moving sideways along a spherical surface at a constant distance from the centre of a planet.

If:

r remains constant

then:

V = −GM/r

also remains constant.

Therefore:

ΔV = 0

and:

ΔEₚ = 0

Gravity does no work for an infinitesimal displacement exactly tangent to that equipotential.

This idea is closely related to circular orbital motion.

Circular Orbits and Equipotentials

In a perfectly circular orbit:

• the satellite remains at constant r
• gravitational potential remains constant
• gravitational potential energy remains constant
• gravitational force points toward the centre
• instantaneous velocity is tangent to the orbit

Therefore, gravitational force is perpendicular to the instantaneous displacement.

Gravity changes the direction of the satellite's velocity but not its speed in the ideal circular-orbit case.

 

 

 

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Moving Between Equipotentials

Now suppose an object moves from one equipotential surface to another.

Then:

V changes

so:

ΔV ≠ 0

Therefore:

ΔEₚ = mΔV

is also non-zero.

Gravity can now do work on the object.

Moving Toward a Mass

Suppose a spacecraft moves toward a planet.

As r decreases:

V = −GM/r

becomes more negative.

Therefore:

gravitational potential decreases.

Its gravitational potential energy also decreases:

ΔEₚ < 0

Gravity does positive work:

W_gravity > 0

If other energy transfers are negligible, kinetic energy increases.

The spacecraft speeds up.

Moving Away from a Mass

If a spacecraft moves away from a planet:

• r increases
• V becomes less negative
• gravitational potential increases
• gravitational potential energy increases

Gravity does negative work.

If the spacecraft's engines are off and other influences are negligible, its kinetic energy decreases.

The spacecraft slows down.

Worked Example: Moving Between Equipotentials

A 5 kg object moves from:

V₁ = −8.0 × 10⁶ J/kg

to:

V₂ = −5.0 × 10⁶ J/kg

Potential difference:

ΔV = V₂ − V₁

= −5.0 × 10⁶ − (−8.0 × 10⁶)

= +3.0 × 10⁶ J/kg

Change in GPE:

ΔEₚ = mΔV

= 5 × 3.0 × 10⁶

= 1.5 × 10⁷ J

The gravitational potential energy increases by:

15 MJ

The object has moved to a higher gravitational potential.

Worked Example: Motion Along an Equipotential

A 10 kg spacecraft moves from A to B.

At A:

VA = −4.5 × 10⁷ J/kg

At B:

VB = −4.5 × 10⁷ J/kg

Then:

ΔV = 0

Therefore:

ΔEₚ = 10 × 0

= 0 J

The gravitational field does:

0 J

of work associated with this displacement.

Equipotential Diagrams

An equipotential diagram usually contains several lines, each representing a particular potential.

For example:

−60 MJ/kg

−50 MJ/kg

−40 MJ/kg

−30 MJ/kg

The diagram allows us to determine:

• where potential is high or low
• the direction of the gravitational field
• where the field is strongest
• how much energy changes when a mass moves between points

 

 

 

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Reading Potential Values

Remember that gravitational potential around an isolated mass is normally negative.

Suppose:

Point A:

V = −80 MJ/kg

Point B:

V = −40 MJ/kg

Point B has the higher gravitational potential.

This is because:

−40 > −80

It is less negative.

An object moving from A to B therefore moves to a higher gravitational potential.

Determining Field Direction from Equipotentials

The gravitational field points:

from higher potential toward lower potential

and crosses equipotential surfaces at right angles.

For example:

Outer equipotential:

−20 MJ/kg

Inner equipotential:

−40 MJ/kg

Still farther inward:

−60 MJ/kg

The gravitational field points inward:

−20 → −40 → −60 MJ/kg

because potential decreases in that direction.

A Useful Rule

When reading an equipotential diagram:

Field direction = perpendicular to equipotentials, toward decreasing V

This rule allows the direction of a gravitational field to be determined even if no field arrows are shown.

Equipotential Spacing and Field Strength

The spacing between equipotential surfaces tells us something about gravitational field strength.

If equal changes in potential occur over a short distance, the gravitational field is strong.

If the same changes occur over a large distance, the field is weaker.

Conceptually:

closely spaced equipotentials → strong field

widely spaced equipotentials → weak field

 

 

 

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Why Spacing Indicates Field Strength

Gravitational field strength is related to how rapidly potential changes with distance.

For motion perpendicular to closely spaced equipotentials:

a large potential change occurs over a small distance.

This means the potential gradient is large.

A large potential gradient corresponds to a strong gravitational field.

Potential Gradient

For a one-dimensional change:

g ≈ −ΔV/Δr

when the interval is sufficiently small and the displacement is along the radial direction.

More generally:

gravitational field = negative gradient of gravitational potential

The negative sign tells us that the field points toward decreasing potential.

Worked Example: Estimating Field Strength

Two nearby equipotential surfaces have values:

−40 J/kg

and:

−50 J/kg

They are separated by:

2.0 m

The magnitude of the potential change is:

|ΔV| = 10 J/kg

Therefore:

g ≈ |ΔV|/Δr

= 10/2.0

= 5 N/kg

So the gravitational field strength in that region is approximately:

5 N/kg

Why the Units Work

Potential gradient has units:

(J/kg)/m

Since:

1 J = 1 N·m

then:

(J/kg)/m = (N·m/kg)/m

= N/kg

which is exactly the unit of gravitational field strength.

Uniform Gravitational Fields

In a uniform gravitational field:

• field strength is constant
• field direction is constant
• field lines are parallel
• equally spaced equipotential surfaces are parallel

Near Earth's surface, the gravitational field can often be approximated as uniform.

 

If equal potential differences are shown, the equipotential lines will be equally spaced.

Equipotentials Near Earth's Surface

Near Earth's surface:

ΔV ≈ gΔh

If g is approximately constant, equal changes in height produce equal changes in potential.

For example, using:

g = 9.8 N/kg

a vertical rise of:

10 m

produces approximately:

ΔV = 9.8 × 10

= 98 J/kg

So every kilogram gains approximately 98 J of gravitational potential energy.

Worked Example Near Earth

Two horizontal equipotential surfaces are separated vertically by:

5.0 m

Take:

g = 9.8 N/kg

Then:

ΔV = gΔh

= 9.8 × 5.0

= 49 J/kg

For a 3 kg object:

ΔEₚ = mΔV

= 3 × 49

= 147 J

Radial Gravitational Fields

Around an isolated spherical mass:

• field lines are radial
• field lines point inward
• equipotential surfaces are concentric spheres

Because:

V = −GM/r

potential changes non-linearly with distance.

The equipotentials representing equal potential differences are therefore not equally spaced.

Equipotential Spacing Around a Planet

Near the planet:

• the field is strong
• potential changes rapidly
• equal-potential intervals occur over shorter distances

Farther away:

• the field is weaker
• potential changes more slowly
• equal-potential intervals occur over larger distances

This produces increasingly spaced equipotential surfaces as distance increases.

Field Lines and Equipotential Lines Never Run Parallel

A gravitational field line cannot run along an equipotential.

If it did, gravity would point along a direction where potential was not changing.

Instead:

field lines intersect equipotential surfaces perpendicularly.

This gives a useful check when interpreting or drawing diagrams.

Equipotentials Cannot Cross

Two equipotential surfaces representing different potentials cannot cross.

Suppose:

one surface represents:

V = −20 J/kg

and another represents:

V = −30 J/kg

If they crossed, the intersection would have to possess both potentials simultaneously.

That is impossible.

Therefore:

equipotential surfaces never intersect one another.

Multiple Masses

Equipotential surfaces become more complicated when several masses are present.

Each mass contributes to the gravitational potential.

Because potential is scalar:

V_total = V₁ + V₂ + V₃ + ...

The resulting equipotential surfaces represent the combined gravitational potential.

 

Instead of simple circles, the contours may become distorted and curved.

Two Equal Masses

Consider two identical masses.

Far from both masses, their combined equipotentials may approximately surround the pair.

Closer to the masses, the surfaces become distorted by the contribution from each individual mass.

At the midpoint:

g_net = 0

because the two gravitational fields cancel.

However:

V_total ≠ 0

because the two negative gravitational potentials add.

This is an important distinction.

Zero Field and Equipotential Diagrams

At a point where:

g = 0

the potential does not necessarily equal zero.

Instead, the potential gradient is zero at that point.

In other words, the potential is locally not changing in the direction of the net field because there is no net field at that exact point.

This is why field strength and potential must not be treated as the same quantity.

Equipotential Maps Are Like Contour Maps

Equipotential diagrams are similar to geographical contour maps.

On a topographic map:

• each contour represents equal elevation
• close contours indicate steep terrain
• widely spaced contours indicate gentle terrain

On a gravitational equipotential map:

• each contour represents equal gravitational potential
• close contours indicate a strong field
• widely spaced contours indicate a weaker field

This analogy can make equipotential diagrams much easier to interpret.

 

A Potential Landscape

Another useful way to imagine gravitational potential is as a landscape.

Moving along a contour line:

potential stays constant.

Moving across contour lines:

potential changes.

Moving toward lower potential is analogous to moving downhill.

The gravitational field points in the direction of the steepest decrease in potential.

This analogy is not a literal picture of curved space, but it is useful for interpreting potential maps.

Determining Work from an Equipotential Diagram

Suppose a 4 kg object moves from:

VA = −100 J/kg

to:

VB = −70 J/kg

Potential difference:

ΔV = VB − VA

= −70 − (−100)

= +30 J/kg

Change in gravitational potential energy:

ΔEₚ = mΔV

= 4 × 30

= 120 J

Therefore, gravity does:

W_gravity = −120 J

of work during the movement.

An external agent moving the object slowly would need to do approximately:

+120 J

of work.

Same Start and Finish, Different Paths

Suppose an object moves from A to B through a gravitational field.

Path 1 is short.

Path 2 curves around before reaching B.

If both paths begin at the same potential and finish at the same potential:

ΔEₚ is identical for both paths.

Therefore, the work done by gravity is identical.

Gravity is a conservative force.

The energy change depends on the endpoints, not the path.

Moving Along and Across Equipotentials

Imagine a spacecraft following three possible motions.

Motion A: along an equipotential

ΔV = 0

No change in GPE.

Motion B: toward lower potential

ΔV < 0

GPE decreases.

Gravity can increase the spacecraft's kinetic energy.

Motion C: toward higher potential

ΔV > 0

GPE increases.

Energy must come from kinetic energy or another source.

Using Equipotential Diagrams to Predict Motion

If an object is released from rest in a gravitational field, it initially accelerates:

• perpendicular to the local equipotential
• toward lower gravitational potential

As it moves:

• gravitational potential decreases
• gravitational potential energy decreases
• kinetic energy increases

Thus an equipotential diagram can provide information about both force direction and energy changes.

Field Strength from a Potential Map

Suppose an equipotential map has contours differing by:

20 J/kg

Region A:

contours are 2 m apart.

Region B:

contours are 10 m apart.

Approximate field strength in A:

gA ≈ 20/2

= 10 N/kg

Approximate field strength in B:

gB ≈ 20/10

= 2 N/kg

Therefore:

Region A has the stronger gravitational field.

Interpreting an Equipotential Diagram Step by Step

When given an equipotential diagram, first examine the numbers.

Determine which regions have higher and lower potential.

Then examine the spacing.

Closer spacing indicates a larger potential gradient and therefore a stronger field.

Next determine the field direction.

The gravitational field points:

perpendicular to the equipotentials toward decreasing potential.

Finally, consider any motion between points.

Use:

ΔV = V₂ − V₁

and, if necessary:

ΔEₚ = mΔV

Drawing Equipotential Surfaces

When drawing equipotentials, remember:

• each line represents one value of V
• equipotentials never cross
• field lines cross equipotentials at 90°
• close spacing represents stronger fields when the potential intervals are equal
• around a spherical isolated mass, equipotentials are concentric
• in a uniform field, equipotentials are parallel
• around multiple masses, equipotentials become distorted

Equipotentials in Astronomy

Equipotential surfaces are useful for analyzing complex gravitational systems such as:

• planets and moons
• binary stars
• star clusters
• spacecraft trajectories
• rotating gravitational systems

They provide an energy-based picture of gravitational interactions.

Instead of examining forces at every point, scientists can study the shape of the gravitational potential landscape.

Earth and Moon

The Earth-Moon system creates a combined gravitational potential.

Near Earth, Earth's contribution dominates.

Near the Moon, the Moon's contribution becomes increasingly important.

Between them, the potential landscape reflects contributions from both bodies.

 

 

 

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In more advanced orbital mechanics, effective potential in a rotating reference frame becomes important when studying locations such as Lagrange points. That model includes effects beyond the simple gravitational equipotentials discussed here.

Equipotential Surfaces and Energy Efficiency

Suppose a spacecraft moves sideways along an equipotential.

Gravity does no work during that displacement.

However, this does not mean a real spacecraft necessarily requires no fuel.

Fuel may still be required to:

• change direction
• overcome drag
• maintain a non-natural trajectory
• correct its orbit

The statement specifically means:

gravity does no work for displacement along an equipotential.

Common Mistakes

Thinking an equipotential is a field line

Field lines show the direction of gravitational force.

Equipotentials show locations of equal gravitational potential.

Drawing field lines parallel to equipotentials

They must intersect at 90°.

Thinking no work means no force

Gravity can still exert a force while doing zero work if force is perpendicular to displacement.

Thinking potential is zero along every equipotential

An equipotential has constant potential, not necessarily zero potential.

Assuming equally drawn spacing automatically means equal potential intervals

Always check the numerical labels on the diagram.

Thinking close equipotentials mean weak gravity

For equal potential intervals, close spacing indicates a stronger field.

Thinking potential increases toward a planet

With zero at infinity, potential becomes more negative and therefore decreases toward the planet.

Confusing gravitational potential with GPE

Potential is measured in J/kg.

GPE is measured in J.

Thinking zero net field means zero potential

Fields are vectors and can cancel. Potentials are scalars and generally add.

Thinking equipotential surfaces can intersect

A point cannot simultaneously have two different gravitational potentials.

Key Terms

Equipotential surface — a surface on which gravitational potential has the same value at every point.

Gravitational potential (V) — gravitational potential energy per unit mass, measured in J/kg.

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

Gravitational potential energy (Eₚ) — energy associated with the relative positions of masses in a gravitational field.

Gravitational field strength (g) — gravitational force per unit mass at a location, measured in N/kg.

Field line — a line showing the direction of the gravitational field.

Potential gradient — the rate at which gravitational potential changes with position.

Conservative force — a force for which work done between two points depends only on the endpoints and not on the path.

Radial field — a field directed toward or away from a central point; gravitational radial fields point inward.

Uniform field — a field with constant magnitude and direction.

Contour — a line connecting points having the same value of a quantity.

Work done by gravity — energy transferred by gravitational force as an object moves through a field.

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

Key Takeaways

• An equipotential surface contains points with the same gravitational potential.
• Around an isolated spherical mass, equipotentials are concentric spherical surfaces.
• In two-dimensional diagrams, these surfaces appear as concentric circles.
• Moving along an equipotential gives ΔV = 0.
• Therefore, gravitational potential energy does not change when moving along an equipotential.
• Gravity does zero work for displacement exactly along an equipotential.
• This occurs because gravitational force is perpendicular to displacement.
• Gravitational field lines cross equipotential surfaces at 90°.
• The gravitational field points toward decreasing gravitational potential.
• Moving toward a mass normally means moving to more negative gravitational potential.
• Moving away means moving to less negative, higher gravitational potential.
• Closely spaced equipotentials indicate a strong gravitational field when the potential intervals are equal.
• Widely spaced equipotentials indicate a weaker field.
• Field strength is related to the gradient of gravitational potential.
• Equipotential surfaces never cross.
• In a uniform gravitational field, equipotentials are parallel and equally spaced for equal potential differences.
• Around multiple masses, equipotential surfaces become distorted.
• Zero gravitational field does not necessarily mean zero gravitational potential.
• Equipotential diagrams can be used to determine field direction, compare field strength, calculate energy changes, and predict motion.

Check Your Understanding

1. Define an equipotential surface.
2. Explain why gravitational potential does not change when an object moves along an equipotential.
3. Use ΔEₚ = mΔV to explain why gravity does no work when a mass moves along an equipotential.
4. Explain why gravitational field lines cross equipotential surfaces at 90°.
5. Describe the shape of the equipotential surfaces surrounding an isolated spherical planet.
6. An object moves from a point where V = −30 J/kg to another point on the same equipotential. Calculate ΔV and the change in GPE of a 5 kg object.
7. A 4 kg object moves from −80 J/kg to −50 J/kg. Calculate its change in gravitational potential energy.
8. Two equipotential surfaces differ by 24 J/kg and are separated by 3.0 m. Estimate the gravitational field strength between them.
9. Another region has the same 24 J/kg potential difference, but the surfaces are 12 m apart. Compare the field strength with that in Question 8.
10. An equipotential diagram is labelled −20, −40, −60 and −80 J/kg moving inward toward a planet. State the direction of the gravitational field and explain your reasoning.
11. Explain why equipotential surfaces cannot cross one another.
12. Describe the equipotential surfaces expected in a uniform gravitational field and compare them with those around a spherical planet.
13. Two identical masses produce zero net gravitational field at their midpoint. Explain why the gravitational potential at that point is still negative.
14. A spacecraft moves from one equipotential to another with a lower potential while its engines are off. Describe the changes in its gravitational potential energy and kinetic energy.
15. A diagram shows equipotential contours that become progressively closer together toward a massive object. Explain what this tells you about the magnitude and direction of the gravitational field.

Energy Transformations During Falling

Consider an object falling toward Earth.

As it falls:

• height decreases
• gravitational potential energy decreases
• speed increases
• kinetic energy increases

Ignoring air resistance:

decrease in Eₚ = increase in Eₖ

The energy is not created as the object speeds up. It is transferred from gravitational potential energy into kinetic energy.

Worked Example: Falling Ball

A 3.0 kg ball is dropped from a height of 10 m.

Use:

g = 9.8 N/kg

Initial GPE relative to the ground:

Eₚ = mgh

= 3.0 × 9.8 × 10

= 294 J

Initially:

Eₖ = 0

So:

E_total = 294 J

Immediately before reaching the ground, ignoring air resistance:

Eₚ ≈ 0

Therefore:

Eₖ ≈ 294 J

The total mechanical energy has remained constant.

Finding Speed from Energy

For the falling ball:

mgh = 1/2 mv²

Mass cancels:

gh = 1/2 v²

Therefore:

v = √(2gh)

For h = 10 m:

v = √(2 × 9.8 × 10)

≈ 14.0 m/s

Energy conservation allows us to determine the final speed without directly calculating acceleration and time.

Energy During Upward Motion

If an object is thrown upward:

• kinetic energy decreases
• gravitational potential energy increases
• speed decreases

At the maximum height:

• vertical velocity is momentarily zero
• vertical kinetic energy is zero
• gravitational potential energy has increased

During the downward journey, the transformation reverses.

 

 

 

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Worked Example: Maximum Height

A ball is launched vertically upward at:

20 m/s

Ignore air resistance.

Initial kinetic energy per kilogram:

Eₖ/m = 1/2 v²

= 1/2 × 20²

= 200 J/kg

At maximum height, this has become additional gravitational potential energy per kilogram:

gΔh = 200

9.8Δh = 200

Δh ≈ 20.4 m

The ball rises approximately 20.4 m above its launch position.

Energy Conservation Does Not Mean Each Energy Is Constant

A common misunderstanding is that conservation of energy means kinetic and potential energy remain unchanged.

They do not.

Instead:

their total remains constant when no energy leaves or enters the mechanical system.

For example:

Energy changes form while the total remains constant.

Real Systems

In real situations, some mechanical energy may be transferred to other energy stores.

For example, air resistance can transfer mechanical energy into:

• thermal energy of the air
• thermal energy of the moving object
• sound

Then:

Eₖ + Eₚ

may decrease.

However, the total energy of the complete system remains conserved.

Energy in Radial Gravitational Fields

For satellites and spacecraft, gravitational field strength changes significantly with distance.

We therefore use:

Eₚ = −GMm/r

rather than the near-surface approximation mgh.

Total mechanical energy becomes:

E_total = 1/2 mv² − GMm/r

This equation can describe many gravitational motions.

Worked Example: Total Mechanical Energy

A spacecraft has:

m = 1000 kg
v = 5000 m/s

At its position:

Eₚ = −2.0 × 10¹⁰ J

Calculate its kinetic energy:

Eₖ = 1/2 mv²

= 1/2 × 1000 × 5000²

= 1.25 × 10¹⁰ J

Therefore:

E_total = Eₖ + Eₚ

= 1.25 × 10¹⁰ − 2.0 × 10¹⁰

= −7.5 × 10⁹ J

The negative total energy indicates that the spacecraft is gravitationally bound in this simplified system.

What Total Energy Tells Us

When the zero of gravitational potential energy is chosen at infinity, total mechanical energy provides useful information about gravitational motion.

E_total < 0

The object is gravitationally bound.

It does not have enough mechanical energy to escape to infinity.

E_total = 0

The object is at the minimum-energy escape condition.

It could theoretically reach infinity with its speed approaching zero.

E_total > 0

The object has enough mechanical energy to escape with kinetic energy remaining at very large distance.

This is a powerful way to classify gravitational trajectories.

Circular Orbits

A satellite in a circular orbit moves at constant speed while gravity continually changes the direction of its velocity.

Gravity provides the centripetal acceleration.

For a circular orbit:

GMm/r² = mv²/r

Cancel m:

GM/r² = v²/r

Therefore:

v² = GM/r

and:

v = √(GM/r)

Kinetic Energy in a Circular Orbit

Since:

Eₖ = 1/2 mv²

and:

v² = GM/r

then:

Eₖ = 1/2 m(GM/r)

Therefore:

Eₖ = GMm/(2r)

The kinetic energy of a satellite in a circular orbit is positive.

Potential Energy in a Circular Orbit

The gravitational potential energy is:

Eₚ = −GMm/r

Compare this with kinetic energy:

Eₖ = GMm/(2r)

Therefore:

Eₚ = −2Eₖ

or:

Eₖ = −Eₚ/2

This is an important relationship for circular gravitational orbits.

 

 

 

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Total Energy of a Circular Orbit

Total mechanical energy is:

E_total = Eₖ + Eₚ

Substitute:

E_total = GMm/(2r) − GMm/r

Therefore:

E_total = −GMm/(2r)

This means:

E_total = Eₚ/2

and:

E_total = −Eₖ

So for a circular orbit:

Eₚ = −2Eₖ

E_total = −Eₖ

E_total = Eₚ/2

These relationships are useful shortcuts.

Example of Orbital Energies

Suppose a satellite in a circular orbit has:

Eₖ = 4.0 × 10⁹ J

Then:

Eₚ = −2Eₖ

= −8.0 × 10⁹ J

Total energy:

E_total = Eₖ + Eₚ

= 4.0 × 10⁹ − 8.0 × 10⁹

= −4.0 × 10⁹ J

The satellite is gravitationally bound.

Why Circular-Orbit Energy Is Negative

A satellite has positive kinetic energy but larger-magnitude negative potential energy.

For example:

Eₖ = +5 units

Eₚ = −10 units

Therefore:

E_total = −5 units

Energy must be supplied to increase the total energy to zero before the satellite can completely escape the planet's gravitational influence.

Higher Circular Orbits

Consider moving a satellite from a lower circular orbit to a higher circular orbit.

At larger r:

Eₚ = −GMm/r

becomes less negative.

So:

potential energy increases.

But:

Eₖ = GMm/(2r)

decreases.

Therefore, a satellite in a higher circular orbit moves more slowly.

The Surprising Result

Moving a satellite into a higher circular orbit requires energy even though the satellite ends up moving more slowly.

Why?

Because the increase in gravitational potential energy is larger than the decrease in kinetic energy.

Therefore:

total mechanical energy increases.

The final total energy is less negative.

Comparing Two Circular Orbits

Suppose Satellite A orbits at radius r.

Satellite B orbits at radius 2r.

For Satellite A:

Eₖ,A = GMm/(2r)

Eₚ,A = −GMm/r

E_total,A = −GMm/(2r)

At 2r:

Eₖ,B = GMm/(4r)

Eₚ,B = −GMm/(2r)

E_total,B = −GMm/(4r)

Therefore, compared with the lower orbit:

• KE is halved
• magnitude of GPE is halved
• magnitude of total energy is halved
• total energy has increased because it is less negative

Energy Changes During Orbital Transfers

A spacecraft cannot normally move from one circular orbit to another simply by "lifting" vertically.

Engines are used to change its velocity and therefore its orbital energy.

During an orbital transfer:

• engines can add kinetic energy
• the spacecraft moves into a different orbit
• gravitational potential energy changes
• kinetic energy may change as the spacecraft moves
• another engine burn may be needed to establish the final orbit

 

 

 

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The detailed trajectory can be complex, but conservation of energy remains central to understanding the motion between engine burns.

Elliptical Orbits

Not all orbits are circular.

In an elliptical orbit, the distance from the central mass changes continuously.

Therefore:

• gravitational potential energy changes
• kinetic energy changes
• speed changes
• total mechanical energy remains constant if no external work is done

Periapsis and Apoapsis

The point closest to the central body is called periapsis.

The farthest point is called apoapsis.

At periapsis:

• r is smallest
• GPE is most negative
• speed is greatest
• KE is greatest

At apoapsis:

• r is largest
• GPE is less negative
• speed is smallest
• KE is smallest

 

Throughout the ideal orbit:

E_total remains constant.

Energy Transformations in an Elliptical Orbit

As a satellite falls from apoapsis toward periapsis:

Eₚ decreases → Eₖ increases

The satellite speeds up.

As it moves from periapsis toward apoapsis:

Eₖ decreases → Eₚ increases

The satellite slows down.

This is an orbital version of the same energy transformation seen when an object falls near Earth's surface.

Worked Example: Energy Change in an Elliptical Orbit

Suppose a spacecraft's GPE changes from:

−4.0 × 10¹⁰ J

at apoapsis to:

−7.0 × 10¹⁰ J

at periapsis.

Change in GPE:

ΔEₚ = −7.0 × 10¹⁰ − (−4.0 × 10¹⁰)

= −3.0 × 10¹⁰ J

If mechanical energy is conserved:

ΔEₖ = +3.0 × 10¹⁰ J

The spacecraft gains 3.0 × 10¹⁰ J of kinetic energy as it approaches periapsis.

Energy Diagrams

Energy diagrams provide a visual way to represent gravitational interactions.

A simple diagram may show:

• kinetic energy
• gravitational potential energy
• total mechanical energy

For an ideal system:

E_total remains constant

while Eₖ and Eₚ change.

 

 

 

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Interpreting Negative Potential Energy

Because:

Eₚ = 0 at infinity

gravitational potential energy is negative at finite distances.

Suppose:

Eₚ = −100 J

and later:

Eₚ = −40 J

The potential energy has increased by 60 J.

Although both values are negative:

−40 J > −100 J

Moving upward on an energy diagram corresponds to increasing energy.

Potential Energy Curve

For two gravitationally interacting masses:

Eₚ = −GMm/r

The Eₚ-r graph:

• is strongly negative at small r
• rises as r increases
• approaches zero at infinity
• never reaches positive values in the simple Newtonian model with zero at infinity

The curve becomes flatter as r increases because the gravitational force becomes weaker.

Force and the Energy Curve

The gravitational force is related to the gradient of the potential-energy graph:

F_r = −dEₚ/dr

Therefore:

• steep Eₚ curve → strong force
• shallow Eₚ curve → weak force

The negative sign indicates that gravity acts toward decreasing potential energy.

Total Energy on an Energy Diagram

Suppose a horizontal line represents:

E_total = −20 J

At a particular position:

Eₚ = −50 J

Then:

Eₖ = E_total − Eₚ

= −20 − (−50)

= 30 J

Since kinetic energy must satisfy:

Eₖ ≥ 0

an object can only occupy positions where:

E_total ≥ Eₚ

This makes energy diagrams useful for identifying allowed and forbidden regions of motion.

Turning Points

At a turning point:

v = 0

Therefore:

Eₖ = 0

So:

E_total = Eₚ

On an energy diagram, turning points occur where the total-energy line intersects the potential-energy curve.

At that position, the object momentarily stops moving radially before reversing direction.

Bound Motion on an Energy Diagram

If:

E_total < 0

the object is generally gravitationally bound in a simple isolated two-body system.

It cannot reach infinity because at infinity:

Eₚ → 0

But if total energy is negative, reaching infinity would require negative kinetic energy:

Eₖ = E_total − 0

which is impossible.

Therefore, the object must remain bound.

Escape on an Energy Diagram

If:

E_total = 0

the object has exactly enough energy to escape and reach infinity with:

v → 0

If:

E_total > 0

the object can reach infinity with kinetic energy remaining.

This provides an energy-based explanation of escape velocity.

Escape Velocity

At the minimum escape condition:

E_total = 0

At the surface of a planet:

1/2 mv² − GMm/R = 0

Therefore:

v_escape = √(2GM/R)

The mass of the escaping object cancels.

Escape speed therefore depends on:

• mass of the planet
• radius of the planet

but not on the mass of the escaping object.

Worked Escape Example

Suppose a planet has:

M = 6.0 × 10²⁴ kg

R = 6.4 × 10⁶ m

Using:

G = 6.67 × 10⁻¹¹ N·m²/kg²

v_escape = √(2GM/R)

≈ √[2(6.67 × 10⁻¹¹)(6.0 × 10²⁴)/(6.4 × 10⁶)]

≈ 1.12 × 10⁴ m/s

or approximately:

11.2 km/s

This is close to Earth's escape speed.

Energy Per Unit Mass

For spacecraft and astronomical systems, it is often useful to divide all energies by the object's mass.

Specific kinetic energy:

Eₖ/m = 1/2 v²

Gravitational potential:

Eₚ/m = V = −GM/r

Specific total mechanical energy:

E_total/m = 1/2 v² − GM/r

The unit is:

J/kg

Using energy per kilogram often simplifies gravitational calculations because the spacecraft's mass cancels.

Worked Example Using Energy per Kilogram

At one location, a spacecraft has:

V = −5.0 × 10⁷ J/kg

and:

v = 6000 m/s

Kinetic energy per kilogram:

Eₖ/m = 1/2 × 6000²

= 1.8 × 10⁷ J/kg

Total energy per kilogram:

E_total/m = 1.8 × 10⁷ − 5.0 × 10⁷

= −3.2 × 10⁷ J/kg

Because the result is negative, the spacecraft is gravitationally bound in this simplified system.

Energy and Planetary Orbits

The same principles apply to planets orbiting stars.

As a planet moves through an elliptical orbit:

• GPE decreases as it approaches the star
• KE increases
• speed increases

As it moves away:

• GPE increases
• KE decreases
• speed decreases

This explains why planets move faster when closer to the Sun and slower when farther away.

Energy in Binary Systems

Two stars orbiting one another also exchange kinetic and gravitational potential energy as their separation changes.

The gravitational potential energy of the pair can be described approximately by:

Eₚ = −GM₁M₂/r

where r is the separation between their centres.

The energy belongs to the gravitational system, not exclusively to either star.

Gravitational Collapse

Energy transformations in gravitational systems are also important during star formation.

A large cloud of gas can contract under gravity.

As the cloud collapses:

• gravitational potential energy decreases
• particles accelerate
• kinetic energy increases
• collisions convert some kinetic energy into thermal energy
• the gas becomes hotter

 

 

 

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Gravitational potential energy is therefore an important source of heating during the early stages of star formation.

Accretion and Gravitational Energy

Matter falling toward:

• stars
• neutron stars
• black holes

can lose enormous amounts of gravitational potential energy.

Some of this energy becomes kinetic energy and thermal energy.

Hot accreting matter can then emit large amounts of electromagnetic radiation.

Gravity is not creating energy; the radiation ultimately comes from changes in the energy of the gravitational system.

Conservation of Energy with Engines

If a spacecraft fires its engines, mechanical energy is not necessarily constant.

Chemical, nuclear, or electrical energy may be transferred into the spacecraft's mechanical energy.

For example, an engine burn may increase:

Eₖ

which can then allow the spacecraft to reach a region with greater:

Eₚ

Therefore, when engines operate:

Eₖ + Eₚ

for the spacecraft-planet mechanical system does not necessarily remain constant.

Energy conservation still holds when all relevant energy stores are included.

Conservation of Energy with Atmospheric Drag

Suppose a spacecraft enters an atmosphere.

Drag transfers mechanical energy into:

• thermal energy of the atmosphere
• thermal energy of the spacecraft
• sound
• deformation or other processes

Therefore:

Eₖ + Eₚ decreases

even though total energy is still conserved.

This is why spacecraft require heat shields during atmospheric entry.

Worked Example with Energy Loss

A falling object loses:

5000 J

of gravitational potential energy.

Its kinetic energy increases by:

4200 J

The difference is:

5000 − 4200

= 800 J

Approximately 800 J has been transferred into other forms such as thermal energy and sound.

Total energy remains conserved.

Choosing the Correct Energy Model

For motion close to Earth's surface:

ΔEₚ = mgΔh

is usually sufficient.

For large changes in distance:

Eₚ = −GMm/r

should be used.

For speed:

Eₖ = 1/2 mv²

For an isolated mechanical system:

E_total = Eₖ + Eₚ

For circular orbits:

Eₖ = GMm/(2r)

Eₚ = −GMm/r

E_total = −GMm/(2r)

Choosing the appropriate model is an important part of solving gravitational problems.

A General Problem-Solving Strategy

When solving an energy problem, first identify the initial state and final state.

Write the relevant energies at each position:

Initial:

Eₖ₁ + Eₚ₁

Final:

Eₖ₂ + Eₚ₂

If mechanical energy is conserved:

Eₖ₁ + Eₚ₁ = Eₖ₂ + Eₚ₂

Then substitute the appropriate expressions.

For near-surface problems, use height differences.

For large-scale gravitational problems, use distance from the centre of the source mass.

Finally, check whether the signs and physical interpretation make sense.

Common Mistakes

Thinking negative potential energy means negative total energy in every situation

A system can have negative Eₚ but enough positive kinetic energy to make E_total positive.

Thinking conservation means KE and GPE stay constant

They can change while their total remains constant.

Using mgh for satellite-scale distances

The field varies significantly, so use Eₚ = −GMm/r.

Forgetting that gravitational potential energy is negative in the radial-field convention

Zero is chosen at infinity.

Thinking −20 J is less energy than −100 J by magnitude alone

Numerically:

−20 J > −100 J

so −20 J represents greater energy.

Thinking a higher circular orbit has greater orbital speed

Higher circular orbits have lower orbital speeds.

Thinking a higher orbit requires no energy because the final speed is lower

The increase in GPE is larger than the decrease in KE, so total energy increases.

Assuming astronauts in orbit have no gravitational potential energy

Orbiting objects possess both GPE and KE.

Forgetting the factor of 1/2 in circular-orbit kinetic energy

For a circular orbit:

Eₖ = GMm/(2r)

not GMm/r.

Confusing total orbital energy with potential energy

For a circular orbit:

E_total = −GMm/(2r)

while:

Eₚ = −GMm/r

Key Terms

Kinetic energy (Eₖ) — energy associated with motion.

Gravitational potential energy (Eₚ) — energy associated with the configuration of masses in a gravitational field.

Mechanical energy — the sum of kinetic and gravitational potential energy.

Conservation of mechanical energy — constancy of Eₖ + Eₚ when no non-conservative interaction transfers energy into or out of the mechanical system.

Circular orbit — orbit in which an object remains at constant distance from the central mass.

Elliptical orbit — orbit in which distance from the central mass varies.

Periapsis — closest point in an orbit to the central body.

Apoapsis — farthest point in an orbit from the central body.

Bound system — gravitational system with negative total mechanical energy under the usual zero-at-infinity convention.

Escape velocity — minimum idealized initial speed needed to escape to infinity with no remaining speed.

Potential-energy curve — graph showing how potential energy varies with position.

Turning point — position where kinetic energy becomes zero and motion reverses in the relevant direction.

Specific energy — energy per unit mass, usually measured in J/kg.

Orbital energy — mechanical energy associated with an orbiting object's motion and gravitational interaction.

Gravitational collapse — contraction of matter under gravitational attraction, often accompanied by conversion of GPE into kinetic and thermal energy.

Key Takeaways

• Gravitational systems involve both kinetic energy and gravitational potential energy.
• Kinetic energy is calculated using Eₖ = 1/2 mv².
• Near Earth's surface, GPE changes can be approximated using ΔEₚ = mgΔh.
• In radial gravitational fields, Eₚ = −GMm/r.
• Total mechanical energy is E_total = Eₖ + Eₚ.
• When gravity is the only force doing work, total mechanical energy remains constant.
• Falling objects lose GPE and gain KE.
• Objects moving away from a gravitational source generally gain GPE and lose KE if no external energy is supplied.
• In circular orbit, Eₚ = −2Eₖ.
• The total energy of a circular orbit is E_total = −GMm/(2r).
• A higher circular orbit has greater GPE but lower KE.
• Moving a satellite to a higher circular orbit requires an increase in total mechanical energy.
• In elliptical orbits, KE and GPE continually transform into one another while total mechanical energy remains constant.
• Objects move fastest near periapsis and slowest near apoapsis.
• Negative total energy represents a gravitationally bound system under the usual convention.
• Zero total energy corresponds to the minimum idealized escape condition.
• Positive total energy allows an object to escape with kinetic energy remaining.
• Energy diagrams can identify kinetic energy, turning points, bound motion, and escape conditions.
• Energy methods provide a powerful way to analyze satellites, spacecraft, planets, stars, and other gravitational systems.

Check Your Understanding

1. Define kinetic energy, gravitational potential energy, and total mechanical energy.
2. A 4 kg object is dropped from a height of 15 m. Calculate its initial GPE using g = 9.8 N/kg.
3. Ignoring air resistance, calculate the kinetic energy of the object in Question 2 immediately before it reaches the ground.
4. Use conservation of energy to calculate its speed immediately before reaching the ground.
5. Explain how kinetic and gravitational potential energy change as an object is thrown vertically upward.
6. A spacecraft has Eₖ = 3.0 × 10⁹ J and Eₚ = −8.0 × 10⁹ J. Calculate its total mechanical energy and state whether it is gravitationally bound in the simple isolated model.
7. A satellite in a circular orbit has kinetic energy 5.0 × 10⁸ J. Determine its gravitational potential energy and total mechanical energy.
8. Explain why a satellite in a higher circular orbit moves more slowly but has greater total mechanical energy.
9. A spacecraft's GPE changes from −3.0 × 10¹⁰ J to −7.0 × 10¹⁰ J as it moves through an elliptical orbit. If total mechanical energy is conserved, calculate the change in kinetic energy.
10. Describe the changes in KE, GPE, and speed as a satellite moves from apoapsis toward periapsis.
11. Explain why negative total mechanical energy indicates a bound gravitational system.
12. An energy diagram shows E_total = −30 J and Eₚ = −70 J at a particular position. Calculate the object's kinetic energy.
13. Explain how a turning point can be identified on a potential-energy diagram.
14. Describe what happens to mechanical energy when a spacecraft experiences significant atmospheric drag. Explain why this does not violate conservation of energy.
15. Explain how the conversion of gravitational potential energy can contribute to the heating of a collapsing gas cloud during star formation.