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.

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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.

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

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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

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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.

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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.

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.

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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.

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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.

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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.

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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.

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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.