Radial and Uniform Gravitational Fields

Gravitational fields describe how a mass would experience gravitational force at different locations in space. Two important models are the radial gravitational field and the uniform gravitational field.

A radial field is the more complete model for the gravitational field around an isolated spherical object such as a planet or star.

A uniform field is a useful approximation when the field strength and direction change so little across a region that we can treat them as constant.

Understanding the difference between these models allows us to interpret field diagrams and decide which model is appropriate for a particular situation.

---

Radial Gravitational Fields

A radial gravitational field is a field whose direction is along lines directed toward a central mass.

Around an isolated spherical planet, gravitational field lines:

• point toward the centre of the planet
• are radial
• become more widely spaced with increasing distance
• form a symmetric pattern around the planet

The arrows point inward because gravity between ordinary masses is attractive.

At every point in the field, the arrow shows the direction in which a small test mass would initially accelerate.

---

Why a Planet Produces a Radial Field

For a spherically symmetric planet, gravitational attraction is directed toward its centre.

Imagine placing small test masses at several positions around Earth.

A test mass above the North Pole would accelerate toward Earth's centre.

A test mass above the equator would also accelerate toward Earth's centre.

The same is true at every location around Earth.

Therefore, the gravitational field vectors point inward from all directions.

This produces the characteristic radial pattern.

---

Direction in a Radial Field

Gravitational field strength is a vector quantity.

This means it has:

• magnitude
• direction

At any point around a spherical planet, the direction of the gravitational field is:

toward the centre of the planet.

The field direction at a point is tangent to the field line passing through that point.

For a radial field, this tangent points directly toward the central mass.

---

Radial Field Lines Spread Out

Near the planet, gravitational field lines are represented as relatively close together.

Farther from the planet, they are represented as more widely spaced.

This indicates that:

the gravitational field becomes weaker with increasing distance.

The change is not linear.

Instead, outside a spherical mass the field follows an inverse-square relationship.

---

Field Strength in a Radial Field

Outside a spherical mass:

g = GM/r²

where:

• g = gravitational field strength in N/kg
• G = gravitational constant
• M = mass producing the field in kg
• r = distance from the centre of the mass in m

This equation shows that:

g ∝ 1/r²

if M remains constant.

---

The Inverse-Square Relationship

Suppose the gravitational field strength at distance r is:

g

At distance 2r:

field strength = g/4

At distance 3r:

field strength = g/9

At distance 4r:

field strength = g/16

This occurs because distance is squared in:

g = GM/r²

The field becomes weaker rapidly as distance increases.

---

Worked Example: Doubling the Distance

At distance r from a planet's centre:

g = 20 N/kg

A spacecraft moves to distance:

2r

Because:

g ∝ 1/r²

the new field strength is:

g₂ = 20/2²

= 20/4

= 5 N/kg

Doubling the distance does not halve the field strength.

It reduces it to one-quarter.

---

Worked Example: Tripling the Distance

Suppose:

g = 45 N/kg

at distance r.

At distance 3r:

g₂ = 45/3²

= 45/9

= 5 N/kg

So increasing the distance by a factor of 3 reduces the field strength by a factor of 9.

---

Comparing Field Strength at Two Distances

Sometimes we do not need G or the mass of the planet.

Since:

g ∝ 1/r²

we can write:

g₁r₁² = g₂r₂²

or:

g₂/g₁ = r₁²/r₂²

Example

A gravitational field has strength:

g₁ = 12 N/kg

at:

r₁ = 4.0 × 10⁶ m

Find the field strength at:

r₂ = 8.0 × 10⁶ m

Since the distance has doubled:

g₂ = 12 × (4.0 × 10⁶ / 8.0 × 10⁶)²

= 12 × (1/2)²

= 12 × 1/4

= 3 N/kg

---

Distance Is Measured from the Centre

In a radial field:

r means distance from the centre of the spherical mass.

This is important.

Suppose a spacecraft is 2,000 km above a planet whose radius is 6,000 km.

The spacecraft's distance from the planet's centre is:

r = 6,000 + 2,000

= 8,000 km

not 2,000 km.

This centre-to-centre distance must be used in gravitational field calculations.

---

Radial Fields Extend Through Space

A planet's gravitational field does not suddenly end.

In the Newtonian model, as distance increases:

g becomes smaller and smaller

but does not reach exactly zero at any finite distance from an isolated mass.

This is why Earth can exert gravitational effects on objects far beyond its atmosphere.

The Moon, satellites, and spacecraft can all be affected by Earth's gravitational field.

---

Uniform Gravitational Fields

A uniform gravitational field is one in which gravitational field strength is approximately constant in both:

• magnitude
• direction

A uniform field is represented by:

parallel, equally spaced field lines.

The parallel lines show that the direction does not change.

The equal spacing indicates that the field strength is constant.

---

Earth's Surface as an Approximately Uniform Field

Earth's actual gravitational field is radial.

However, over a small region close to Earth's surface, the radial field lines are almost parallel.

Why?

Earth's radius is enormous compared with everyday distances.

Earth's mean radius is about:

6.37 × 10⁶ m

or:

6,370 km

A classroom only a few metres across is tiny compared with this distance.

Across such a small region:

• the direction toward Earth's centre changes by a negligible amount
• the distance from Earth's centre changes very little
• gravitational field strength changes very little

Therefore, we can approximate the field as uniform.

---

Local Versus Global Models

This gives us two useful ways to represent Earth's gravitational field.

Global View

When considering Earth as a whole:

use a radial field.

Field lines converge toward Earth's centre.

Local View

When considering a small region near Earth's surface:

use an approximately uniform field.

Field lines can be treated as parallel and equally spaced.

Both models describe the same gravitational field at different scales.

---

Why "Uniform" Is an Approximation

Near Earth's surface:

g ≈ 9.8 N/kg

But g is not exactly identical everywhere.

It changes slightly with:

• altitude
• latitude
• Earth's rotation
• Earth's shape
• local geology

For many introductory physics problems, these differences are small enough to ignore.

We therefore use:

g ≈ 9.8 N/kg

throughout the region being studied.

---

Uniform Field-Line Diagram

A uniform gravitational field might be drawn as:

↓ ↓ ↓ ↓ ↓

↓ ↓ ↓ ↓ ↓

↓ ↓ ↓ ↓ ↓

The arrows are:

parallel

because the direction is constant.

They are:

equally spaced

because the magnitude of the field is constant.

They point:

downward

when representing gravity near Earth's surface.

---

Radial Field-Line Diagram

A simplified radial field around a planet has arrows directed inward from every direction.

The field lines:

• converge toward the centre
• are not parallel
• become more widely spaced farther away
• indicate changing direction
• indicate decreasing field strength with distance

This is fundamentally different from the uniform-field model.

---

Comparing Radial and Uniform Fields

This comparison is especially useful when interpreting field diagrams.

---

Reading Field-Line Direction

The arrows on a gravitational field diagram show the direction in which a small positive test mass would accelerate.

For gravity, this means the arrows point:

toward the mass producing the field.

Around a planet:

arrows point inward.

Near Earth's surface:

arrows point downward toward Earth.

Field lines should not point outward from an isolated ordinary mass.

---

Reading Field-Line Spacing

In conventional field diagrams, field-line spacing represents relative field strength.

Closer field lines → stronger field

More widely spaced field lines → weaker field

In a radial field, lines spread farther apart as distance increases.

Therefore:

field strength decreases with distance.

In a uniform field, the lines remain equally spaced.

Therefore:

field strength remains approximately constant.

---

Field Lines Are Not Physical Objects

Field lines are a visual model.

There are not actual strings or lines extending through space.

A diagram could be drawn using:

8 field lines

or:

16 field lines

or:

32 field lines.

The physical field has not changed.

What matters is the pattern:

• direction
• symmetry
• relative spacing

Field lines allow us to visualize a continuous vector field.

---

Field Vectors

Another way to represent gravitational fields is with arrows at selected points.

Each arrow represents the gravitational field vector at that position.

The arrow's:

direction shows the direction of g.

Its:

length can be used to represent the magnitude of g.

Near a planet:

• arrows point toward the centre
• arrows become shorter farther away

In a uniform field:

• arrows have the same direction
• arrows have approximately the same length

---

Why Radial Field Lines Spread Out

Imagine a spherical surface surrounding a planet.

As the radius of this imaginary sphere increases, its surface area increases according to:

A = 4πr²

The same gravitational influence is spread across increasingly large spherical surfaces.

Because area increases with r², field strength decreases with:

1/r²

This provides a useful geometric way to understand the inverse-square relationship.

---

Surface Area and the Inverse-Square Law

Suppose the distance doubles.

Original spherical area:

A₁ = 4πr²

At distance 2r:

A₂ = 4π(2r)²

= 16πr²

Therefore:

A₂ = 4A₁

The corresponding field strength becomes:

1/4 as large.

If distance triples:

area increases by:

3² = 9

so field strength becomes:

1/9 as large.

---

Field Strength and Altitude

As an object moves upward from Earth's surface, its distance from Earth's centre increases.

Therefore:

g decreases.

However, small changes in altitude produce only small changes in g because Earth's radius is so large.

This is why a person standing:

• on the ground
• on the second floor
• on the roof of a building

experiences almost the same gravitational field strength.

For everyday situations, the uniform-field approximation is excellent.

---

Example: A Small Change in Height

Suppose an object moves upward by 10 m.

Earth's radius is approximately:

6,370,000 m

The distance changes from approximately:

6,370,000 m

to:

6,370,010 m

This is a tiny fractional change.

Therefore, the corresponding change in gravitational field strength is also tiny.

We can reasonably treat:

g ≈ constant

over this height.

---

When the Uniform Approximation Breaks Down

The uniform approximation becomes less appropriate when distances become large compared with the planet's radius.

Examples include:

• high-altitude satellites
• spacecraft travelling far from Earth
• motion between planets
• comparing surface gravity with gravity far above a planet

In these situations, use the radial-field model and:

g = GM/r²

---

Example: Choosing the Correct Model

A Ball Falling 2 m in a Classroom

Use:

uniform field approximation

because the change in distance from Earth's centre is negligible.

A Satellite Orbiting Earth

Use:

radial field

because field direction changes around the orbit and field strength depends on distance from Earth's centre.

A Skydiver Falling Several Kilometres

For many introductory calculations:

uniform field approximation

may still be acceptable.

For high-precision calculations:

changes in g could be considered.

A Spacecraft Travelling from Earth Toward the Moon

Use:

radial gravitational fields from both bodies.

---

Comparing Motion in the Two Models

In a uniform field, an object moving vertically experiences approximately constant gravitational acceleration:

a ≈ g

Near Earth's surface:

a ≈ 9.8 m/s² downward

In a radial field, acceleration changes as the object's distance from the central mass changes:

a = g = GM/r²

Therefore, a spacecraft moving away from a planet experiences progressively smaller gravitational acceleration.

---

Uniform Fields and Potential Energy

Near Earth's surface, because g is approximately constant, changes in gravitational potential energy can often be calculated using:

ΔEₚ = mgΔh

This works well for relatively small changes in height.

For very large changes in distance from a planet, g changes significantly, so the simple near-surface equation becomes less appropriate.

This is another example of how the uniform-field approximation simplifies calculations.

---

Radial Fields and Orbits

Satellites move through Earth's radial gravitational field.

At every point in a circular orbit, the gravitational force points toward Earth's centre.

The satellite's velocity is tangent to the orbit, while gravitational acceleration points inward.

Gravity continually changes the direction of the satellite's velocity.

This inward gravitational acceleration allows the satellite to follow a curved path around Earth.

---

Field Strength at the Surface of Different Planets

At the surface of a spherical planet:

g = GM/R²

where R is the planet's radius.

This means surface gravity depends on both:

• planetary mass
• planetary radius

A planet with greater mass does not necessarily have greater surface gravity if it also has a much larger radius.

---

Worked Comparison of Two Planets

Suppose Planet B has:

4 times the mass of Planet A

but:

2 times the radius.

For Planet A:

gA = GMA/RA²

For Planet B:

gB = G(4MA)/(2RA)²

gB = 4GMA/4RA²

Therefore:

gB = gA

Despite having four times the mass, Planet B has the same surface gravitational field strength because its radius is twice as large.

---

Changing Mass but Keeping Radius Constant

Suppose two planets have the same radius.

Planet B has twice the mass of Planet A.

Since:

g = GM/R²

and R is unchanged:

gB = 2gA

At the same distance, doubling the source mass doubles the gravitational field strength.

This is different from changing distance, where the inverse-square relationship applies.

---

Multiple Radial Fields

When several astronomical objects are present, each produces its own radial gravitational field.

For example:

• Earth produces a field.
• The Moon produces a field.
• The Sun produces a field.

At any point in space, the total gravitational field is the vector sum of all relevant fields.

This is called superposition.

g_net = g₁ + g₂ + g₃ + ...

Direction must be considered because gravitational field strength is a vector.

---

Neutral Points

Between two masses, their gravitational fields may oppose each other.

At some point, their magnitudes may become equal:

g₁ = g₂

If they point in opposite directions:

g_net = 0

This is a neutral point in the gravitational field.

For two equal isolated masses, the neutral point is at the midpoint.

For unequal masses, the neutral point between them is closer to the smaller mass.

---

Identifying Fields from Diagrams

When shown an unfamiliar field diagram, use the following clues.

Radial Field

Look for:

• lines converging toward a central object
• changing direction
• changing line spacing
• spherical symmetry

Uniform Field

Look for:

• straight parallel lines
• equal spacing
• identical direction throughout the region

You should be able to identify the type of field without knowing which planet or object created it.

---

A Useful Decision Rule

Ask:

Is the region being considered small compared with the radius of the planet?

If yes:

uniform-field approximation may be appropriate.

If no:

use the radial-field model.

Then ask:

Does g change significantly over the distances involved?

If yes, the radial model is required.

---

Common Mistakes

Thinking Earth's field is actually uniform

Earth's gravitational field is radial. Uniform gravity is an approximation used over relatively small regions near the surface.

Drawing radial field arrows outward

Gravitational field arrows point toward the mass.

Drawing uniform field lines that converge

Uniform field lines should be parallel.

Drawing uniform field lines with changing spacing

Changing spacing suggests changing field strength.

Thinking gravitational field strength decreases linearly with distance

Outside a spherical mass:

g ∝ 1/r²

not 1/r.

Thinking twice the distance means half the gravity

Twice the distance means:

1/4 the field strength.

Using altitude instead of distance from the centre

For radial-field calculations:

r = planetary radius + altitude

when the object is above the surface.

Assuming field lines physically exist

Field lines are a visual model used to represent a continuous gravitational field.

Thinking the field suddenly ends

A gravitational field becomes progressively weaker but does not abruptly stop.

Using the radial model unnecessarily

For small distances near Earth's surface, treating g as constant is often much simpler and sufficiently accurate.

---

Key Terms

Gravitational field — a region in which a mass experiences gravitational force.

Gravitational field strength (g) — gravitational force per unit mass at a location.

Radial field — a field whose direction is along radial lines toward a central mass.

Uniform field — a field whose magnitude and direction are approximately constant throughout the region considered.

Field line — an imaginary line representing the direction of a field.

Field vector — an arrow representing the magnitude and direction of gravitational field strength at a particular point.

Inverse-square law — a relationship in which a quantity is proportional to 1/r².

Distance from centre (r) — the radial distance between a point and the centre of the spherical mass producing the field.

Altitude — height above the surface of a planet or other astronomical body.

Uniform-field approximation — treating a slightly changing gravitational field as constant over a sufficiently small region.

Radial symmetry — a pattern that is equivalent in every direction from a central point for an isolated spherical source.

Superposition — combining individual gravitational field vectors to determine the net field.

Neutral point — a point where opposing gravitational fields cancel to give zero net field.

Gravitational constant (G) — the constant in Newton's law of gravitation, approximately 6.67 × 10⁻¹¹ N·m²/kg².

---

Key Takeaways

• The gravitational field around an isolated spherical planet is a radial field.
• Radial gravitational field lines point inward toward the centre because gravity is attractive.
• Field direction changes from place to place in a radial field.
• Gravitational field strength decreases as distance from the central mass increases.
• Outside a spherical mass, g = GM/r².
• Therefore, gravitational field strength follows an inverse-square relationship with distance.
• Doubling distance from the centre reduces field strength to one-quarter.
• Tripling distance reduces field strength to one-ninth.
• The distance r is measured from the centre of the spherical mass.
• A uniform field has approximately constant magnitude and direction.
• Uniform fields are represented by parallel, equally spaced field lines.
• Earth's gravitational field can be approximated as uniform over small regions near its surface.
• Earth's complete gravitational field is radial, not uniform.
• Radial and uniform models describe the same gravitational interaction at different scales.
• Field-line spacing can represent relative field strength.
• The appropriate model depends on the size of the region being considered compared with the size of the planet.

---

Check Your Understanding

1. Describe the shape and direction of the gravitational field around an isolated spherical planet.
2. Explain why gravitational field lines around a planet point inward rather than outward.
3. Describe how the spacing of field lines changes as distance from a planet increases and explain what this indicates.
4. Define a uniform gravitational field and describe how it is represented in a field-line diagram.
5. Explain why Earth's gravitational field can be approximated as uniform inside a classroom even though Earth's actual field is radial.
6. A planet produces a gravitational field strength of 24 N/kg at distance r. Calculate the field strength at 2r.
7. The gravitational field strength at distance r is 90 N/kg. Calculate the field strength at 3r.
8. A spacecraft moves from distance r to distance 5r from a planet's centre. By what factor does the gravitational field strength change?
9. A spacecraft is 1,500 km above the surface of a planet with radius 5,000 km. Determine the value of r that should be used in a radial-field calculation.
10. Explain why doubling the distance from a planet does not simply halve the gravitational field strength.
11. Compare a radial gravitational field and a uniform gravitational field in terms of field-line direction, field-line spacing, and field strength.
12. A student draws Earth's gravitational field using parallel lines from Earth all the way to the Moon. Explain why this model is inappropriate.
13. Planet B has four times the mass and twice the radius of Planet A. Compare their surface gravitational field strengths.
14. Explain why a ball falling a few metres can usually be analysed using a uniform gravitational field, while an orbiting satellite should be analysed using a radial field.
15. Two identical planets are separated in space. Describe the gravitational fields at the midpoint between them and explain why the net gravitational field strength can be zero even though both planets produce a gravitational field there.

Field Superposition

A gravitational field can be produced by a single mass, but real systems usually contain many masses. Earth and the Moon both create gravitational fields. The Sun, Earth, Moon, planets, satellites, and every other mass contribute to the gravitational field around them.

The principle of superposition allows us to determine the combined gravitational field produced by multiple masses.

The central idea is:

The net gravitational field at a point is the vector sum of the gravitational fields produced by all individual masses.

For two masses:

g_net = g₁ + g₂

For several masses:

g_net = g₁ + g₂ + g₃ + ...

Because gravitational field strength is a vector, both magnitude and direction must be considered.

---

What Is Superposition?

Suppose two masses, A and B, are located near a point P.

Mass A creates a gravitational field at P:

gA

Mass B also creates a gravitational field at P:

gB

Neither field disappears because the other mass is present.

Instead, both fields exist simultaneously.

The total field is found by combining them:

g_net = gA + gB

This is called superposition.

The same principle works for three, four, or any number of masses.

---

Each Mass Produces Its Own Field

The gravitational field produced by a spherical mass is:

g = GM/r²

where:

• g = gravitational field strength in N/kg
• G = gravitational constant
• M = mass producing the field
• r = distance from the centre of that mass

When several masses are present, calculate the field produced by each mass separately.

Then combine the resulting vectors.

This is usually easier than trying to think about the entire system at once.

---

Direction Matters

Gravitational fields always point toward the mass producing them.

Imagine a point P between two masses:

Mass A ← P → Mass B

At P:

• Mass A produces a field toward A
• Mass B produces a field toward B

The fields therefore point in opposite directions.

In this situation, their magnitudes are subtracted.

If:

gA = 8 N/kg left

and:

gB = 3 N/kg right

then:

g_net = 5 N/kg left

The stronger field determines the direction of the net field.

---

Fields in the Same Direction

Now imagine point P lies to the left of both masses:

P Mass A Mass B

Both masses attract a test mass toward the right.

Suppose:

gA = 6 N/kg right

and:

gB = 2 N/kg right

Then:

g_net = 6 + 2

g_net = 8 N/kg right

When gravitational field vectors point in the same direction, their magnitudes add.

---

A Simple Rule

For fields along one straight line:

Same direction → add

Opposite directions → subtract

But always determine the direction of each field before doing the arithmetic.

Do not automatically add or subtract simply because there are two masses.

---

Worked Example: Opposing Fields

At a point between two planets:

Planet A produces:

gA = 12 N/kg left

Planet B produces:

gB = 7 N/kg right

Choose right as positive.

Then:

gA = −12 N/kg

gB = +7 N/kg

Therefore:

g_net = −12 + 7

= −5 N/kg

The negative sign indicates left.

So:

g_net = 5 N/kg toward Planet A

---

Worked Example: Fields in the Same Direction

A spacecraft is located to the right of two moons.

Moon A produces a field of:

2.5 N/kg left

Moon B produces:

1.2 N/kg left

Both fields point left.

Therefore:

g_net = 2.5 + 1.2

= 3.7 N/kg left

---

Why Gravitational Fields Combine

Imagine placing a small test mass at a point influenced by several objects.

Each surrounding mass attracts the test mass.

The test mass therefore experiences several gravitational forces simultaneously.

Because:

g = F/m

and forces combine vectorially, gravitational fields must also combine vectorially.

Superposition is therefore closely related to the idea of resultant force.

---

Superposition and Resultant Force

Suppose a 2 kg object is placed where:

g_net = 4 N/kg right

Its gravitational force is:

F = mg

F = 2 × 4

= 8 N right

The net gravitational field determines the net gravitational force acting on a mass placed at that location.

---

Equal Masses

Consider two identical spherical masses:

M₁ = M₂

At their exact midpoint, the distance to each mass is also equal.

Since:

g = GM/r²

both masses produce gravitational fields of equal magnitude.

However, their directions are opposite.

Therefore:

g₁ = −g₂

and:

g_net = 0

The midpoint is therefore a gravitational neutral point.

---

What Is a Neutral Point?

A neutral point is a location where the net gravitational field is zero.

At this point:

g_net = 0

This does not mean that no gravitational fields exist there.

Instead:

• one mass produces a field
• another mass produces an opposing field
• the fields have equal magnitude
• the vectors cancel

This distinction is very important.

---

Zero Net Field Does Not Mean Zero Gravity Sources

Suppose:

gA = 5 N/kg left

and:

gB = 5 N/kg right

Then:

g_net = −5 + 5

= 0 N/kg

Both gravitational fields still exist.

Their effects simply cancel at that particular point.

Moving even a small distance away from the balance point generally produces a non-zero net field.

---

Unequal Masses

Now suppose:

M₁ > M₂

At the midpoint between them, the larger mass produces the stronger gravitational field because the distances are equal.

Therefore, the neutral point cannot be at the midpoint.

To balance the fields, we must move closer to the smaller mass.

Why?

Because:

g = GM/r²

Moving closer to the smaller mass increases its gravitational field strength.

Moving farther from the larger mass decreases the larger mass's field strength.

Eventually, the two fields may become equal.

Therefore:

For two unequal isolated masses, the neutral point between them lies closer to the smaller mass.

---

Finding a Neutral Point

At a neutral point between two masses:

g₁ = g₂

Using:

g = GM/r²

we can write:

GM₁/r₁² = GM₂/r₂²

G appears on both sides, so it cancels:

M₁/r₁² = M₂/r₂²

This equation can be used to determine the position of the neutral point.

---

Worked Example: Equal Masses

Two identical masses are separated by:

100 m

Because the masses are equal, the neutral point is halfway between them.

Therefore:

r₁ = 50 m

r₂ = 50 m

The fields are equal in magnitude and opposite in direction.

So:

g_net = 0

---

Worked Example: Unequal Masses

Mass A is:

4M

Mass B is:

M

They are separated by:

30 m

Let x be the distance from the smaller mass B to the neutral point.

Then the distance from the larger mass A is:

30 − x

At the neutral point:

4M/(30 − x)² = M/x²

Cancel M:

4/(30 − x)² = 1/x²

Take the positive square root:

2/(30 − x) = 1/x

Cross multiply:

2x = 30 − x

3x = 30

x = 10 m

Therefore, the neutral point is:

10 m from the smaller mass

and:

20 m from the larger mass

As expected, it is closer to the smaller mass.

---

A Useful Shortcut for Balance Problems

Starting from:

M₁/r₁² = M₂/r₂²

we can rearrange:

r₁/r₂ = √(M₁/M₂)

This tells us how the distances from the balance point compare.

For example, if:

M₁ = 9M₂

then:

r₁/r₂ = √9

= 3

The neutral point must be three times farther from the larger mass than from the smaller mass.

---

Example Using the Distance Ratio

Two masses are:

9M and M

and are separated by:

40 m

At the neutral point:

r_large/r_small = √(9M/M)

= 3

So:

r_large = 3r_small

The total distance is:

r_large + r_small = 40

3r_small + r_small = 40

4r_small = 40

r_small = 10 m

Therefore:

r_large = 30 m

The neutral point is 10 m from the smaller mass.

---

Field Diagrams for Two Masses

Field-line diagrams become more complex when more than one mass is present.

Every gravitational field line ultimately points toward a mass.

Around two masses:

• field lines curve because both masses influence the field
• the pattern depends on the masses
• more field lines may be associated with the larger mass in a qualitative diagram
• there may be a neutral point between the masses

A field diagram represents the net gravitational field, not two completely independent diagrams drawn on top of each other.

---

Field Direction Between Two Masses

Consider:

A -------- P -------- B

At P:

Field due to A points:

toward A

Field due to B points:

toward B

Therefore, the fields oppose one another.

The net field points toward whichever mass produces the stronger field at P.

Remember that field strength depends on both:

mass

and:

distance

A smaller but much closer mass can produce a stronger field than a larger, more distant mass.

---

Example: Mass Versus Distance

Suppose at point P:

Mass A:

M = 8 × 10¹⁰ kg

Distance to P:

r = 4 m

Mass B:

M = 2 × 10¹⁰ kg

Distance to P:

r = 2 m

Ignoring the common factor G:

For A:

M/r² = 8 × 10¹⁰ / 16

= 0.5 × 10¹⁰

For B:

M/r² = 2 × 10¹⁰ / 4

= 0.5 × 10¹⁰

Therefore:

gA = gB

If P lies between the masses, the fields point in opposite directions.

So:

g_net = 0

The masses do not have to be equal for their fields to balance.

---

Points Outside Two Masses

Consider:

P ---- A -------- B

Point P is outside the two masses.

Both A and B attract a test mass at P toward the right.

Therefore, their fields point in the same direction.

So:

g_net = gA + gB

This is why it is essential to sketch the system before calculating.

Between two masses, fields usually oppose.

Outside the pair, they usually point in the same direction.

---

A Four-Step Method for One-Dimensional Superposition

For masses arranged along a straight line:

Identify the Point

Mark the location where you want to calculate the field.

Determine Each Direction

For every source mass, draw an arrow from the point toward that mass.

Calculate Each Field

Use:

g = GM/r²

for each mass.

Combine the Fields

Choose one direction as positive.

Then add the signed values.

The sign of the final answer determines the direction.

---

Worked Example with Two Masses

Mass A:

6.0 × 10¹¹ kg

is 4.0 m to the left of point P.

Mass B:

3.0 × 10¹¹ kg

is 2.0 m to the right of P.

Use:

G = 6.67 × 10⁻¹¹ N·m²/kg²

Field Due to A

gA = GM/r²

= (6.67 × 10⁻¹¹)(6.0 × 10¹¹)/(4.0)²

= 40.02/16

≈ 2.50 N/kg left

Field Due to B

gB = (6.67 × 10⁻¹¹)(3.0 × 10¹¹)/(2.0)²

= 20.01/4

≈ 5.00 N/kg right

Therefore:

g_net = 5.00 − 2.50

= 2.50 N/kg right

Although Mass A is larger, Mass B produces the stronger field because it is much closer.

---

Three Masses in a Line

Superposition works exactly the same way with more than two masses.

Suppose three masses produce fields at point P:

g₁ = 6 N/kg right

g₂ = 4 N/kg left

g₃ = 3 N/kg right

Choose right as positive:

g_net = +6 − 4 + 3

= 5 N/kg right

Each contribution is calculated independently before being combined.

---

Fields at Right Angles

Gravitational fields do not always lie along one straight line.

Suppose:

g₁ = 3 N/kg east

and:

g₂ = 4 N/kg north

These fields are perpendicular.

They cannot simply be added as:

3 + 4 = 7.

Instead, use vector addition.

Magnitude:

g_net = √(3² + 4²)

= √25

= 5 N/kg

Direction:

tan θ = 4/3

θ ≈ 53° north of east

---

Using Components

For more complicated two-dimensional systems, resolve each gravitational field vector into components.

For a field g at angle θ:

gₓ = g cos θ

gᵧ = g sin θ

Then add all horizontal components:

g_net,x = g₁x + g₂x + g₃x + ...

and all vertical components:

g_net,y = g₁y + g₂y + g₃y + ...

Finally:

g_net = √(g_net,x² + g_net,y²)

Direction can be found using:

tan θ = g_net,y/g_net,x

with the final quadrant determined from the signs of the components.

---

Symmetry Can Simplify Problems

Symmetry often makes gravitational field problems much easier.

Suppose two identical masses are positioned the same distance to the left and right of point P.

Their fields at P have:

• equal magnitudes
• opposite directions

Therefore:

g_net = 0

No detailed numerical calculation is necessary.

Always look for symmetry before beginning lengthy calculations.

---

Three Equal Masses in a Symmetric Arrangement

Imagine three identical masses placed at the corners of an equilateral triangle.

At the exact centre:

• each mass produces a gravitational field of equal magnitude
• the three field vectors are symmetrically spaced
• their vector sum is zero

Therefore:

g_net = 0

This is another example of fields cancelling through symmetry.

The individual fields are not zero.

Their vector sum is zero.

---

Four Equal Masses

Suppose four equal masses are placed at the corners of a square.

At the exact centre:

• each mass produces the same field strength
• opposite contributions cancel

Therefore:

g_net = 0

Symmetry allows us to predict the result before calculating.

---

Breaking the Symmetry

Now imagine one of the four masses is removed.

The remaining fields no longer cancel perfectly.

There will be a net gravitational field.

Its direction can often be predicted qualitatively before calculating.

This is an important physics skill:

predict the direction first, then calculate the magnitude.

A calculated answer that points in an obviously unreasonable direction is a warning that an error has occurred.

---

Superposition in the Earth–Moon System

Both Earth and the Moon create gravitational fields.

Between them:

• Earth's field points toward Earth
• the Moon's field points toward the Moon

At some location between them, the two fields can have equal magnitude and opposite direction.

The location is closer to the Moon because the Moon is much less massive than Earth.

This provides a real astronomical example of gravitational field superposition.

The simple two-body neutral-point model should not be confused with the full rotating-system analysis used for Lagrange points, which also involves orbital motion.

---

Superposition Does Not Change the Individual Fields

Suppose Earth produces a field of:

4 N/kg

at a certain location.

Another mass is introduced and produces:

1 N/kg in the opposite direction.

Earth's contribution does not suddenly become:

3 N/kg.

Earth still contributes:

4 N/kg

The other object contributes:

1 N/kg

The net field is:

3 N/kg

Superposition combines the fields without changing the individual contributions.

---

Net Field Versus Individual Field

It is useful to distinguish:

individual field

from:

net field

An individual field is the contribution from one source mass.

The net field is the vector sum of all contributions.

For example:

gEarth = 7 N/kg downward

gMoon = 0.03 N/kg upward

Then:

g_net = 7 − 0.03

= 6.97 N/kg downward

Both individual fields still exist.

---

Field Strength and Force

Once the net gravitational field is known, the gravitational force on an object can be calculated using:

F_net = mg_net

For example:

A 20 kg object is placed where:

g_net = 3 N/kg left

Then:

F_net = 20 × 3

= 60 N left

This is often easier than separately calculating every gravitational force and then adding them.

---

Example: From Multiple Fields to Force

At a location:

Mass A produces:

4 N/kg east

Mass B produces:

1.5 N/kg west

Therefore:

g_net = 4 − 1.5

= 2.5 N/kg east

A 6 kg object placed there experiences:

F = mg

= 6 × 2.5

= 15 N east

---

Field Diagrams and Superposition

When interpreting a field diagram involving several masses, look for:

• which masses are present
• which way the field lines point
• regions where lines are densely packed
• regions where the field is weaker
• symmetry
• possible neutral points
• how the field direction changes from one region to another

The field pattern represents the combined effect of all masses.

---

A Complete Superposition Strategy

For any simple gravitational field problem:

Sketch the system.

Mark all masses and the point of interest.

Draw the individual field directions.

Every arrow points from the point of interest toward the source mass.

Calculate distances carefully.

Measure r from the centre of each source mass.

Calculate each field.

Use:

g = GM/r²

Resolve vectors if necessary.

Use components when fields are not collinear.

Add the vectors.

Find the net field.

State magnitude and direction.

A vector answer without direction is incomplete.

Check the result.

The final direction should make physical sense.

---

Common Mistakes

Adding field magnitudes without considering direction

Gravitational field strength is a vector.

Assuming the larger mass always produces the stronger field

Distance also matters because:

g ∝ M/r²

A smaller nearby mass can produce a stronger field than a larger distant mass.

Assuming the neutral point is always halfway

This is true only for two equal masses in the simple isolated two-body arrangement.

Placing the neutral point closer to the larger mass

For unequal masses, the balance point between them is closer to the smaller mass.

Thinking zero net field means no gravitational fields exist

Individual fields can be non-zero while their vector sum is zero.

Adding perpendicular fields arithmetically

Fields at right angles require vector addition.

Forgetting that gravity points toward the source mass

Determine direction before assigning positive or negative signs.

Using altitude instead of distance from the centre

In:

g = GM/r²

r is the distance from the centre of the source mass.

Calculating force when the question asks for field strength

Field strength is measured in N/kg; force is measured in N.

---

Key Terms

Superposition — the principle that the net gravitational field is the vector sum of the fields produced by individual masses.

Gravitational field strength (g) — gravitational force per unit mass at a point.

Individual field — the gravitational field contribution produced by one source mass.

Net gravitational field — the vector sum of all gravitational field contributions at a point.

Vector — a quantity with both magnitude and direction.

Resultant — the single vector representing the combined effect of several vectors.

Source mass — a mass producing a gravitational field.

Test mass — a small mass used conceptually to determine the direction and strength of a gravitational field.

Neutral point — a point where gravitational field vectors cancel so that the net gravitational field is zero.

Balance point — another term often used for a location where opposing field contributions have equal magnitude.

Symmetry — a geometric arrangement that can cause equal field components to cancel.

Component — the part of a vector acting along a chosen direction, such as horizontal or vertical.

Collinear fields — field vectors acting along the same straight line.

Superposition of vectors — combining vectors while accounting for both their magnitudes and directions.

---

Key Takeaways

• Every mass produces its own gravitational field.
• When several masses are present, their gravitational fields exist simultaneously.
• The principle of superposition states that the net field is the vector sum of the individual fields.
• Gravitational field strength is a vector, so direction must always be considered.
• Fields pointing in the same direction add.
• Fields pointing in opposite directions subtract.
• Fields at angles must be combined using vector methods.
• Each source field can be calculated using g = GM/r².
• The net field can be zero even when several non-zero gravitational fields are present.
• For two equal masses, the neutral point between them is at the midpoint.
• For two unequal masses, the neutral point between them lies closer to the smaller mass.
• At a balance point, opposing gravitational field strengths are equal in magnitude.
• A larger mass does not necessarily produce the stronger field at a particular point because distance also affects field strength.
• Symmetry can simplify multi-mass gravitational systems dramatically.
• Once the net field is known, the gravitational force on a mass can be calculated using F = mg_net.
• A good superposition solution should always state both the magnitude and direction of the net gravitational field.

---

Check Your Understanding

1. State the principle of superposition for gravitational fields.
2. Why must gravitational field strengths be added as vectors rather than simply as numbers?
3. At a point P, one mass produces a field of 8 N/kg east and another produces 3 N/kg east. Determine the net field.
4. At a point P, one mass produces 12 N/kg west and another produces 7 N/kg east. Determine the magnitude and direction of the net field.
5. Explain why the gravitational fields produced by two masses point in opposite directions at a point located between the masses.
6. Two identical masses are separated by 80 m. Determine the location of the neutral point between them.
7. Explain why the neutral point between two unequal masses is closer to the smaller mass.
8. Two masses are 9M and M and are separated by 60 m. Determine the location of the neutral point between them.
9. At point P, two perpendicular gravitational fields have strengths 6 N/kg east and 8 N/kg north. Calculate the magnitude of the net gravitational field.
10. Three masses produce fields at P of 7 N/kg right, 4 N/kg left, and 2 N/kg right. Determine the net field.
11. Explain how four equal masses positioned at the corners of a square can produce a net gravitational field of zero at the centre.
12. At a particular point, a larger mass is farther away while a smaller mass is much closer. Explain why you cannot determine which produces the stronger gravitational field from their masses alone.
13. A 5 kg object is placed at a point where the net gravitational field is 4.2 N/kg south. Calculate the net gravitational force acting on the object.
14. Explain the difference between saying "the gravitational field is zero" and "the net gravitational field is zero because two fields cancel."
15. Two planets of unequal mass are separated in space. Describe a complete method for determining the position of the point between them where the gravitational fields balance.

The object's velocity changes by approximately:

9.8 m/s every second

while the near-surface gravitational field can be treated as uniform.

---

Measuring g Using Free Fall

Suppose an object is released from rest.

For constant acceleration:

d = 1/2 gt²

where:

• d = vertical distance fallen
• g = acceleration due to gravity
• t = fall time

Rearranging:

g = 2d/t²

Therefore, if distance and time are measured, g can be calculated.

---

Worked Example: Free-Fall Measurement

A ball is dropped from rest through a distance of:

1.80 m

The measured fall time is:

0.61 s

Calculate g.

g = 2d/t²

g = 2(1.80)/(0.61)²

g = 3.60/0.3721

g ≈ 9.67 m/s²

The experimental value is reasonably close to the accepted near-surface value of approximately:

9.8 m/s²

---

Why Experimental Values May Differ

A classroom experiment might produce:

9.4 m/s²

or:

10.1 m/s²

rather than exactly 9.8 m/s².

Possible reasons include:

• timing uncertainty
• inaccurate height measurement
• air resistance
• inconsistent release
• limited precision of equipment
• reaction time when using a stopwatch

A measured value does not need to exactly equal the accepted value to be useful.

Scientists evaluate the quality and uncertainty of measurements.

---

Improving a Free-Fall Experiment

A hand-operated stopwatch is not ideal because human reaction time may be significant compared with a short fall time.

Better methods could use:

• light gates
• motion sensors
• electronic timers
• high-speed video

Repeated trials can also improve reliability.

---

Using Several Measurements

Suppose five trials produce:

9.6 m/s²
9.9 m/s²
9.7 m/s²
10.0 m/s²
9.8 m/s²

The mean is:

(9.6 + 9.9 + 9.7 + 10.0 + 9.8)/5

= 49.0/5

= 9.8 m/s²

Repeating measurements helps reveal variation and reduces the influence of an unusual individual result when an appropriate average is used.

---

Measuring g with a Pendulum

A simple pendulum can also be used to estimate gravitational field strength.

For small oscillations:

T = 2π√(L/g)

where:

• T = period in seconds
• L = pendulum length in metres
• g = gravitational field strength in m/s²

Rearranging:

g = 4π²L/T²

By measuring the pendulum's length and period, scientists can estimate the local value of g.

---

Improving Pendulum Measurements

Timing a single swing produces relatively large percentage uncertainty.

Instead, measure the time for many oscillations.

For example:

20 oscillations take:

36.0 s

Then:

T = 36.0/20

= 1.80 s

This provides a more reliable estimate of the period than trying to time one oscillation directly.

---

Calculating Fields Around Planets and Moons

Gravitational field strength outside a spherical mass can be calculated using:

g = GM/r²

where:

• G = 6.67 × 10⁻¹¹ N·m²/kg²
• M = mass of the planet or moon
• r = distance from its centre

This equation allows gravitational field strength to be calculated even when direct measurement is difficult.

---

Surface Gravitational Field Strength

At the surface of a planet:

r = R

where R is the planet's radius.

Therefore:

g_surface = GM/R²

Surface gravitational field strength depends on both:

• mass of the planet
• radius of the planet

A more massive planet does not automatically have stronger surface gravity if it is also much larger.

---

Worked Example: Earth

Use approximate values:

Earth's mass:

M = 5.97 × 10²⁴ kg

Earth's radius:

R = 6.37 × 10⁶ m

Then:

g = GM/R²

g = (6.67 × 10⁻¹¹)(5.97 × 10²⁴)/(6.37 × 10⁶)²

g ≈ 9.8 N/kg

This agrees closely with measurements made near Earth's surface.

---

Worked Example: The Moon

Use approximate values:

Moon's mass:

M = 7.35 × 10²² kg

Moon's radius:

R = 1.74 × 10⁶ m

Then:

g = (6.67 × 10⁻¹¹)(7.35 × 10²²)/(1.74 × 10⁶)²

g ≈ 1.62 N/kg

The Moon's surface gravitational field is therefore much weaker than Earth's.

---

Comparing Earth and Moon

Using approximate surface values:

Ratio:

9.8/1.62 ≈ 6.05

Earth's surface gravitational field is therefore approximately:

6 times stronger

than the Moon's.

This explains why an astronaut weighs much less on the Moon.

---

Mass Does Not Change

Suppose an astronaut has a mass of:

72 kg

On Earth:

F = mg

= 72 × 9.8

= 705.6 N

On the Moon:

F = 72 × 1.62

≈ 116.6 N

The astronaut's:

mass remains 72 kg

but their:

weight changes

because gravitational field strength is different.

---

Calculating g Above a Planet

For an object above the surface:

r = R + h

where:

• R = planetary radius
• h = altitude

Then:

g = GM/(R + h)²

A common mistake is using h alone as the distance.

Gravitational calculations require distance from the centre of the planet.

---

Worked Example: Above Earth's Surface

Suppose a spacecraft is:

400 km

above Earth's surface.

Convert altitude:

400 km = 4.00 × 10⁵ m

Earth's radius is approximately:

6.37 × 10⁶ m

Therefore:

r = 6.37 × 10⁶ + 4.00 × 10⁵

r = 6.77 × 10⁶ m

Using:

g = GM/r²

g ≈ 8.7 N/kg

Gravity is weaker than at Earth's surface, but it is still substantial.

---

Astronauts in Orbit Are Still in a Strong Gravitational Field

A common misconception is that astronauts float because there is almost no gravity in orbit.

At an altitude of a few hundred kilometres, Earth's gravitational field is still quite strong.

Astronauts appear weightless because:

the spacecraft and astronauts are falling together around Earth.

They are in continuous free fall.

---

Comparing Field Strengths Using Ratios

Sometimes the mass of the planet and G do not need to be used.

Since:

g ∝ 1/r²

for the same planet:

g₂/g₁ = r₁²/r₂²

This makes comparisons much quicker.

---

Worked Example: Twice the Distance

Suppose:

g = 12 N/kg

at distance r.

At distance:

2r

g₂ = 12 × (r/2r)²

= 12 × 1/4

= 3 N/kg

So doubling the distance from the centre reduces the field strength to one-quarter.

---

Worked Example: Three Times the Distance

Suppose:

g₁ = 18 N/kg

At:

r₂ = 3r₁

then:

g₂ = 18/3²

= 18/9

= 2 N/kg

The field strength has decreased by a factor of 9.

---

Comparing Different Planets

For two planets:

g₁ = GM₁/R₁²

and:

g₂ = GM₂/R₂²

Dividing gives:

g₂/g₁ = (M₂/M₁)(R₁/R₂)²

This allows planets to be compared using relative masses and radii.

---

Example: Twice the Mass, Same Radius

Planet B has:

2 times Earth's mass

and:

the same radius as Earth

Then:

gB/gEarth = 2

Therefore:

gB = 2gEarth

If:

gEarth = 9.8 N/kg

then:

gB = 19.6 N/kg

---

Example: Same Mass, Twice the Radius

Planet C has:

the same mass as Earth

but:

twice Earth's radius

Then:

gC/gEarth = 1/2²

= 1/4

Therefore:

gC = 9.8/4

= 2.45 N/kg

Doubling radius has a major effect because radius is squared.

---

Example: Four Times the Mass and Twice the Radius

Planet D has:

4 times Earth's mass

and:

2 times Earth's radius

Then:

gD/gEarth = 4/2²

= 4/4

= 1

Therefore:

gD = gEarth

The two planets would have the same surface gravitational field strength despite having very different masses and sizes.

---

Interpreting a Table of Gravitational Data

Consider this hypothetical data:

From the table we can conclude:

• Location A has the strongest gravitational field.
• Location D has the weakest.
• A 1 kg object at A would experience 12 N of gravitational force.
• A 1 kg object at D would experience 1.5 N.
• An object would have greater weight at A than at D.

However, we cannot determine why the fields differ without additional information about the masses and distances involved.

---

Interpreting Force and Mass Data

Suppose measurements produce:

Calculate:

g = F/m

Using any row:

g = 3.7/1.0

= 3.7 N/kg

or:

g = 14.8/4.0

= 3.7 N/kg

The consistent ratio shows that the measurements were made in a field of approximately:

3.7 N/kg

---

Interpreting Free-Fall Data

Suppose an experiment measures the velocity of a falling object:

Acceleration is:

a = Δv/Δt

Using:

Δv = 19.6 m/s

Δt = 2.0 s

a = 19.6/2.0

= 9.8 m/s²

Therefore:

g = 9.8 m/s² = 9.8 N/kg

---

Velocity-Time Graphs and g

For an object in free fall with negligible air resistance, the gradient of a velocity-time graph gives the acceleration.

Therefore:

gradient = g

if the chosen positive direction makes downward acceleration positive.

If upward is defined as positive, the gradient would instead be:

−g

The sign depends on the coordinate system.

The magnitude of the gravitational field strength remains positive.

---

Distance-Time Data

If an object is dropped from rest in an approximately uniform gravitational field:

d = 1/2 gt²

Therefore:

d ∝ t²

A graph of:

d against t²

should be approximately linear.

Its gradient is:

g/2

Therefore:

g = 2 × gradient

This provides another experimental method for determining gravitational field strength.

---

Field Strength Versus Distance

For a spherical mass:

g = GM/r²

Therefore, a graph of g against r is curved.

As r increases:

• g decreases rapidly at first
• the curve gradually becomes flatter
• g approaches zero at very large distances

The field does not suddenly stop at a particular distance.

---

Linearizing Gravitational Data

Because:

g = GM/r²

we can write:

g = GM(1/r²)

Therefore, if we plot:

g against 1/r²

we expect a straight line for an ideal spherical source.

The gradient is:

GM

This is useful because a curved inverse-square relationship becomes a linear graph.

If G is known, the mass M can then be calculated from the gradient.

---

Using Data to Identify an Inverse-Square Relationship

Suppose measurements give:

Notice:

At 2r:

16 → 4

The field becomes one-quarter.

At 4r:

16 → 1

The field becomes one-sixteenth.

This pattern strongly suggests:

g ∝ 1/r²

---

Percentage Comparison

Suppose Location A has:

gA = 9.8 N/kg

and Location B has:

gB = 3.7 N/kg

To express B as a percentage of A:

percentage = (3.7/9.8) × 100

≈ 37.8%

So Location B's gravitational field strength is approximately:

38% of Location A's

This can make comparisons easier to interpret.

---

Comparing Weight in Different Fields

Because:

F = mg

an object's weight is directly proportional to g.

Suppose:

gA = 10 N/kg

gB = 4 N/kg

Then:

FB/FA = 4/10

= 0.4

An object at B weighs only:

40% as much

as it does at A.

Its mass remains unchanged.

---

Field Strength Is Location-Dependent

Gravitational field strength is not a property of the test object.

It is a property of the gravitational field at a particular location.

At the same location:

• a 1 kg object experiences the same g as a 10 kg object
• their gravitational forces are different
• their free-fall accelerations are the same if other forces are negligible

This distinction is fundamental when interpreting gravitational data.

---

Why g Varies Across Earth

The familiar:

g ≈ 9.8 N/kg

is an approximation.

The measured value varies slightly with location because of factors including:

• altitude
• latitude
• Earth's rotation
• Earth's slightly non-spherical shape
• local variations in geological density

Highly sensitive instruments can detect these small differences.

---

Gravimeters

A gravimeter is an instrument designed to measure small variations in gravitational acceleration.

Gravimeters can be used in:

• geophysical surveys
• geological studies
• monitoring volcanic regions
• studying underground density variations
• precision gravitational research

If the density beneath a location differs from its surroundings, extremely precise gravity measurements may reveal a small difference in the local field.

---

Comparing Measurement and Calculation

There are two broad approaches to determining gravitational field strength.

Measurement

Observe gravity's effect directly.

Examples:

• measure weight of a known mass
• measure free-fall acceleration
• use a pendulum
• use a gravimeter

Calculation

Use information about the source mass and distance:

g = GM/r²

Both approaches should produce compatible results when the assumptions and measurements are appropriate.

---

Evaluating Gravitational Field Data

When interpreting experimental data, ask:

Are the units correct?

g should normally be expressed in:

N/kg

or:

m/s²

Are the results consistent?

Repeated measurements should usually be reasonably close.

Are there outliers?

Investigate unusual measurements rather than automatically removing them.

Does the trend make physical sense?

Increasing distance from a planet should decrease g.

Does the pattern fit an inverse-square relationship?

Doubling distance from the centre should reduce g to approximately one-quarter for an isolated spherical source.

What uncertainties are present?

Consider measurement precision and experimental limitations.

---

Worked Data Analysis

Suppose a probe records:

Check the inverse-square prediction.

At 2R:

20/2² = 5.0 N/kg

At 3R:

20/3² ≈ 2.22 N/kg

At 4R:

20/4² = 1.25 N/kg

The measurements closely match:

g ∝ 1/r²

Therefore, the data are consistent with an inverse-square gravitational field.

---

Choosing the Best Method

Different situations require different approaches.

Known mass and measured weight?

Use:

g = F/m

Free-fall data?

Use acceleration:

g = a

Known planetary mass and distance?

Use:

g = GM/r²

Comparing two distances from the same planet?

Use:

g₂/g₁ = r₁²/r₂²

Comparing planetary surfaces?

Use:

g = GM/R²

Selecting the appropriate relationship is often the most important first step.

---

Common Mistakes

Confusing g with gravitational force

g is field strength measured in N/kg or m/s².

Force is measured in N.

Thinking g depends on the test object's mass

At a particular location, gravitational field strength is independent of the test mass.

Confusing mass and weight

Mass is measured in kg.

Weight is measured in N.

Using altitude as r

For:

g = GM/r²

r is measured from the centre of the source mass.

Forgetting to square r

Gravity follows an inverse-square relationship.

Thinking astronauts in orbit experience zero gravity

Earth's gravitational field remains substantial at typical orbital altitudes.

Assuming every experimental value must equal exactly 9.8

Measurements contain uncertainty, and g also varies slightly with location.

Ignoring direction when acceleration data are signed

Depending on the coordinate system, gravitational acceleration may be written as +g or −g.

Assuming a larger planet must have stronger surface gravity

Both mass and radius determine surface gravitational field strength.

---

Key Terms

Gravitational field strength (g) — gravitational force per unit mass at a location.

Acceleration due to gravity — the acceleration experienced by a freely falling object when gravity is the only significant force.

Newton per kilogram (N/kg) — the SI unit of gravitational field strength.

Metre per second squared (m/s²) — the SI unit of acceleration, numerically equivalent to N/kg for gravitational field strength.

Free fall — motion in which gravity is the only significant force.

Weight — gravitational force acting on a mass.

Gravimeter — an instrument used to measure gravitational acceleration or small variations in gravitational field strength.

Pendulum — an oscillating system whose period can be used to estimate local gravitational field strength under appropriate conditions.

Inverse-square relationship — a relationship in which a quantity varies as 1/r².

Planetary radius (R) — distance from the centre of a planet to its surface.

Altitude — height above a planetary surface.

Experimental uncertainty — the range of uncertainty associated with a measurement.

Gradient — rate of change represented by the slope of a graph.

Linearization — transforming variables so a non-linear relationship can be represented by a straight-line graph.

---

Key Takeaways

• Gravitational field strength can be measured experimentally or calculated mathematically.
• One direct method is g = F/m, using the measured weight of a known mass.
• A graph of weight against mass has gradient equal to gravitational field strength.
• Gravitational field strength is numerically equal to free-fall acceleration when gravity is the only significant force.
• 1 N/kg = 1 m/s².
• Near Earth's surface, g ≈ 9.8 N/kg = 9.8 m/s².
• Free-fall measurements can be used to determine g.
• Pendulum measurements can also be used to estimate local gravitational field strength.
• Outside a spherical mass, g = GM/r².
• At a planet's surface, g = GM/R².
• Surface gravitational field strength depends on both planetary mass and radius.
• The Moon's surface gravitational field is much weaker than Earth's.
• An object's mass remains constant when it moves between gravitational fields, but its weight changes.
• Gravitational field strength decreases according to an inverse-square relationship as distance from a spherical mass increases.
• Data involving force, acceleration, distance, or planetary properties can all provide information about gravitational fields.
• Measurements should be evaluated for uncertainty, consistency, trends, and possible experimental error.

---

Check Your Understanding

1. Define gravitational field strength and state its two equivalent units.
2. A 3.0 kg mass has a measured weight of 29.4 N. Calculate the gravitational field strength.
3. A 12 kg object is placed where g = 4.5 N/kg. Calculate its weight.
4. Explain why 1 N/kg is equivalent to 1 m/s².
5. Explain why gravitational field strength is equal to free-fall acceleration when gravity is the only significant force.
6. A ball is dropped from rest and falls 2.0 m in 0.64 s. Use g = 2d/t² to estimate the local gravitational field strength.
7. Explain why using a light gate would generally produce a better measurement of g than using a handheld stopwatch.
8. A graph of weight against mass has a gradient of 3.7 N/kg. What does this tell you about the gravitational field?
9. A planet has mass 8.0 × 10²³ kg and radius 3.0 × 10⁶ m. Calculate its surface gravitational field strength using G = 6.67 × 10⁻¹¹ N·m²/kg².
10. A planet produces a field strength of 32 N/kg at distance r. What is the field strength at 2r? What about 4r?
11. A spacecraft is 600 km above a planet whose radius is 4,500 km. Determine the value of r that should be used in g = GM/r².
12. Planet B has the same mass as Planet A but twice its radius. Compare their surface gravitational field strengths.
13. Planet C has four times the mass and twice the radius of Planet A. Compare their surface gravitational field strengths.
14. A probe measures gravitational field strengths of 16.0, 4.0, 1.78, and 1.0 N/kg at distances r, 2r, 3r, and 4r. Explain how these data provide evidence for an inverse-square relationship.
15. Describe two different experimental methods for determining gravitational field strength and explain one important source of uncertainty for each method.

Applications of Gravitational Field Strength

Gravitational field strength affects everything from a person's weight on another planet to the motion of satellites, moons, planets, and spacecraft. Once the gravitational field strength at a location is known, we can predict the gravitational force acting on an object and analyze how that object will move.

The basic relationship is:

F = mg

where:

• F = gravitational force or weight in N
• m = mass in kg
• g = gravitational field strength in N/kg

This means that an object's weight depends on the gravitational field it is in, while its mass does not.

---

Mass and Weight Are Different

Mass measures an object's inertia and is measured in kilograms.

Weight is the gravitational force acting on that mass and is measured in newtons.

An astronaut with a mass of 70 kg has:

70 kg of mass on Earth

70 kg of mass on the Moon

70 kg of mass on Mars

The astronaut's mass does not change simply because their location changes.

Their weight does change because gravitational field strength changes.

---

How Field Strength Determines Weight

Weight is calculated using:

F = mg

Suppose a person has a mass of:

60 kg

Near Earth's surface:

g ≈ 9.8 N/kg

Therefore:

F = 60 × 9.8

= 588 N

If the same person moves to a location where:

g = 4.0 N/kg

then:

F = 60 × 4.0

= 240 N

The person's mass remains 60 kg, but their weight decreases.

---

Weight Is Proportional to Field Strength

For a fixed mass:

F ∝ g

This means that if gravitational field strength doubles:

weight doubles.

If gravitational field strength becomes half as large:

weight becomes half as large.

If:

g₂ = 3g₁

then:

F₂ = 3F₁

provided the object's mass remains unchanged.

---

Comparing Weight Without Recalculating Everything

Because:

F = mg

for the same object:

F₂/F₁ = g₂/g₁

This gives a useful shortcut.

Suppose an object weighs:

800 N

where:

g = 10 N/kg

It moves somewhere where:

g = 2.5 N/kg

Then:

F₂/800 = 2.5/10

F₂ = 800 × 0.25

F₂ = 200 N

The new gravitational field is one-quarter as strong, so the object's weight is one-quarter as large.

---

Weight on Earth and the Moon

Near Earth's surface:

g ≈ 9.8 N/kg

Near the Moon's surface:

g ≈ 1.62 N/kg

Consider an astronaut with mass:

75 kg

On Earth:

F = 75 × 9.8

= 735 N

On the Moon:

F = 75 × 1.62

≈ 122 N

The astronaut's Moon weight is only about one-sixth of their Earth weight.

---

Why Astronauts Can Jump Higher on the Moon

A person on the Moon has the same:

• mass
• muscles
• basic ability to push against the ground

but the gravitational field pulling them downward is much weaker.

Therefore:

• their weight is smaller
• downward acceleration is smaller
• they remain airborne longer
• they can jump higher

The Moon's lower gravitational field strength changes motion as well as weight.

---

Comparing Different Worlds

Approximate surface gravitational field strengths vary considerably throughout the Solar System.

These values can be used to compare how much an object would weigh at the surfaces or reference levels of these bodies.

For gas giants such as Jupiter and Neptune, "surface gravity" refers to a conventional atmospheric reference level rather than a solid surface.

---

Worked Example: Weight on Mars

A rover has mass:

900 kg

Mars has approximately:

g = 3.7 N/kg

The rover's weight is:

F = mg

F = 900 × 3.7

= 3330 N

The rover still has a mass of 900 kg.

Its gravitational force is much smaller than it would be on Earth.

---

The Same Rover on Earth

On Earth:

F = 900 × 9.8

= 8820 N

Therefore:

Earth weight = 8820 N

Mars weight = 3330 N

Difference:

8820 − 3330

= 5490 N

This difference affects engineering decisions such as:

• structural loads
• suspension systems
• traction
• landing systems
• movement across the surface

---

Weight on Jupiter

Suppose a 100 kg object could be placed at Jupiter's conventional reference level where:

g ≈ 24.8 N/kg

Then:

F = 100 × 24.8

= 2480 N

On Earth:

F = 100 × 9.8

= 980 N

The gravitational force would therefore be approximately:

2480/980 ≈ 2.53

times its Earth weight.

Again, Jupiter does not have a solid surface like Earth, so this comparison refers to a defined atmospheric level.

---

Why Large Planets Do Not Always Have Extremely Strong Surface Gravity

Surface gravitational field strength is:

g = GM/R²

where:

• M = planetary mass
• R = planetary radius

Greater mass increases g.

Greater radius decreases g at the surface.

Therefore, surface gravity depends on both mass and size.

---

Worked Example: A Larger Planet

Planet B has:

4 times Earth's mass

and:

2 times Earth's radius

Compare its surface gravitational field strength with Earth's.

Using:

g ∝ M/R²

gB/gEarth = 4/2²

= 4/4

= 1

Therefore:

gB = gEarth

Despite having four times Earth's mass, the larger radius exactly offsets the increased mass in this simplified comparison.

---

Field Strength Above a Planet

Gravitational field strength decreases as distance from a planet increases.

For a spherical planet:

g = GM/r²

where r is measured from the planet's centre.

Therefore, spacecraft travelling away from a planet experience progressively weaker gravitational fields.

---

Altitude and Distance from the Centre

Suppose a spacecraft is at altitude h above a planet of radius R.

The distance used in the gravitational equation is:

r = R + h

Therefore:

g = GM/(R + h)²

Do not use altitude alone.

---

Worked Example: Gravity at Twice Earth's Radius

Suppose a spacecraft is at:

r = 2R

where R is Earth's radius.

At Earth's surface:

g ≈ 9.8 N/kg

Because:

g ∝ 1/r²

at 2R:

g = 9.8/2²

= 9.8/4

= 2.45 N/kg

The spacecraft is still in Earth's gravitational field, but the field is one-quarter as strong as at the surface.

---

Weight Changes with Altitude

Suppose a satellite has mass:

1000 kg

At Earth's surface:

F = 1000 × 9.8

= 9800 N

At a distance of 2R from Earth's centre:

g = 2.45 N/kg

Therefore:

F = 1000 × 2.45

= 2450 N

The satellite's mass remains 1000 kg, but the gravitational force acting on it decreases with distance.

---

Gravity Does Not Suddenly Stop in Space

A common misconception is that there is no gravity once a spacecraft leaves Earth's atmosphere.

Earth's gravitational field extends far into space.

As distance increases:

g decreases

according to:

g ∝ 1/r²

but it does not suddenly become zero.

The Moon itself remains strongly influenced by Earth's gravitational field despite being far beyond Earth's atmosphere.

---

Spacecraft in Orbit

A satellite orbiting Earth is continuously affected by Earth's gravitational field.

Gravity provides the inward acceleration needed to curve the spacecraft's path.

Without Earth's gravitational field, the satellite would move approximately along a straight-line path rather than remaining in orbit.

---

Gravity Provides Centripetal Acceleration

For a circular orbit:

gravitational acceleration = centripetal acceleration

The gravitational field points toward the centre of the planet.

The spacecraft's velocity is approximately tangent to the orbit.

Gravity continually changes the direction of the spacecraft's velocity.

This produces the curved orbital path.

---

Orbital Speed

For a circular orbit around a spherical body:

gravitational acceleration is:

g = GM/r²

centripetal acceleration is:

a = v²/r

For a circular gravitational orbit:

GM/r² = v²/r

Therefore:

v = √(GM/r)

This shows that orbital speed depends on:

• mass of the central body
• orbital radius

A spacecraft in a larger circular orbit moves more slowly than one in a smaller circular orbit around the same body.

---

Worked Example: Comparing Orbital Speeds

Suppose Satellite A orbits at radius:

r

Satellite B orbits at:

4r

Since:

v ∝ 1/√r

then:

vB/vA = √(r/4r)

= 1/2

Therefore:

Satellite B's circular orbital speed is half that of Satellite A.

---

Why Astronauts Appear Weightless in Orbit

Astronauts aboard an orbiting spacecraft are still strongly affected by gravity.

They appear weightless because:

the astronauts and spacecraft are falling together.

The spacecraft continuously falls toward Earth while its sideways velocity carries it around Earth.

Everything inside follows essentially the same free-fall motion.

This creates the sensation commonly called weightlessness or microgravity.

---

Apparent Weight Versus Gravitational Force

An astronaut in orbit can have a substantial gravitational force acting on them while having almost no apparent weight.

This distinction is important.

Gravitational force:

still acts.

Support force from the spacecraft floor:

can be nearly zero.

What we commonly experience as "weight" while standing on the ground is closely associated with the support force exerted by the ground.

During free fall, this support force disappears.

---

Field Strength and Spacecraft Trajectories

Spacecraft trajectories depend on gravitational fields.

Mission planners must consider the fields produced by:

• planets
• moons
• the Sun
• sometimes other astronomical bodies

As a spacecraft moves, both the magnitude and direction of the net gravitational field can change.

This changes the spacecraft's:

• velocity
• direction
• orbital path
• energy

Accurate gravitational modelling is therefore essential for space missions.

---

Multiple Gravitational Fields

A spacecraft travelling through the Solar System is rarely affected by only one gravitational field.

For example, a spacecraft travelling near the Moon experiences gravitational fields from:

• Earth
• Moon
• Sun

The net field is determined using superposition:

g_net = g₁ + g₂ + g₃ + ...

Because gravitational field strength is a vector, directions must be included.

---

Gravity Assist Manoeuvres

Spacecraft can use planetary gravitational fields to alter their trajectories.

This technique is called a gravity assist or gravitational slingshot.

A spacecraft approaches a moving planet and follows a curved path through the planet's gravitational field.

In a Sun-centered reference frame, the interaction can change the spacecraft's:

• speed
• direction
• orbital energy

Gravity assists can reduce the amount of propellant required for long-distance missions.

---

Landing on Another World

Gravitational field strength is critical when designing landing systems.

A spacecraft descending toward a planet or moon experiences gravitational acceleration.

Engineers must account for:

• local g
• spacecraft mass
• descent velocity
• atmospheric drag, if present
• thrust
• landing structure strength

A landing system designed for the Moon may not be suitable for Earth because Earth's gravitational field is much stronger and Earth also has a substantial atmosphere.

---

Required Hovering Thrust

Suppose a spacecraft has mass:

2000 kg

and is hovering where:

g = 3.7 N/kg

Its weight is:

F = mg

= 2000 × 3.7

= 7400 N

To hover without accelerating vertically, the engines must provide approximately:

7400 N upward

assuming no other vertical forces.

---

Comparing Hovering on Earth and Mars

For the same 2000 kg spacecraft:

On Earth:

F = 2000 × 9.8

= 19,600 N

On Mars:

F = 2000 × 3.7

= 7400 N

Therefore, considerably less upward thrust is needed to hover on Mars.

However, actual spacecraft operations also depend on atmospheric conditions, engine performance, fuel, and other engineering constraints.

---

Launching from Different Worlds

Gravitational field strength affects the difficulty of launching spacecraft.

A stronger gravitational field generally means that more energy is required to move away from the body.

However, escape depends on the gravitational field over the entire path, not merely the surface value of g.

For a spherical body, escape speed is:

v_escape = √(2GM/R)

Therefore, both planetary mass and radius matter.

---

Escape Velocity

Escape velocity is the minimum initial speed required in an idealized no-propulsion, no-drag model for an object to escape from a body's gravitational field without falling back.

For Earth, the familiar value is approximately:

11.2 km/s

This does not mean rockets must instantly reach this speed at Earth's surface. Real rockets provide thrust over time and follow more complicated trajectories.

Escape velocity is mainly a useful way of describing the depth of a body's gravitational potential.

---

Gravitational Fields and Moons

Moons remain in orbit because of the gravitational fields of their planets.

For example, Earth's gravitational field continuously accelerates the Moon toward Earth.

The Moon's sideways velocity prevents it from simply falling directly into Earth.

Instead, it continually falls around Earth.

The same basic principle explains artificial satellite orbits.

---

Gravitational Fields and Planetary Orbits

The Sun produces an enormous gravitational field because of its large mass.

This field governs the large-scale motion of planets throughout the Solar System.

Each planet is continually accelerated toward the Sun while also moving sideways.

The result is an orbit.

Gravitational fields therefore provide the structure that organizes planetary systems.

---

Gravitational Fields and Stars

Gravity plays a central role in the formation and evolution of stars.

Large clouds of gas and dust can begin collapsing under their own gravitational attraction.

As matter becomes concentrated:

• density increases
• gravitational effects become stronger
• temperature rises
• pressure increases

Eventually, conditions can become sufficient for nuclear fusion to begin.

Gravity therefore plays an essential role in star formation.

---

Gravity and Stellar Stability

Inside a stable star, two major effects oppose one another.

Gravity pulls inward.

Pressure associated with the hot stellar material and energy production pushes outward.

A stable star can exist when these effects are approximately balanced.

This state is called hydrostatic equilibrium.

If this balance changes dramatically, the star's structure and evolution can change.

---

Gravity and Galaxies

Gravitational fields also operate on scales much larger than individual stars.

Gravity influences:

• star clusters
• galaxies
• galaxy groups
• galaxy clusters

Stars within galaxies move under the combined gravitational influence of the matter in those galaxies.

Measurements of these motions are also important evidence used by astronomers when studying the distribution of matter, including dark matter.

---

Gravitational Fields and Black Holes

A black hole produces an extremely strong gravitational field near it because a large amount of mass-energy is concentrated into a very compact region.

Far from a spherical object, its external gravitational influence depends mainly on its mass and distance.

If the Sun could somehow be replaced by a black hole with exactly the same mass, Earth's orbit would not suddenly change merely because the central object was a black hole.

The extreme effects arise when objects approach very close to the compact mass.

---

Field Strength and Tidal Effects

Gravitational field strength can vary from one side of an object to the other.

This difference in gravitational field is responsible for tidal effects.

For example, the Moon's gravitational field is slightly stronger on the side of Earth closer to the Moon than on the far side.

These differences contribute to Earth's ocean tides.

Tidal effects depend on how rapidly the gravitational field changes with distance, not simply on the field strength at one point.

---

Why Tides Matter in Astronomy

Tidal gravitational effects can:

• produce ocean tides
• stretch astronomical bodies
• affect moons
• heat the interiors of some moons
• influence orbital evolution
• contribute to tidal locking

The Moon is tidally locked to Earth, which is why approximately the same lunar hemisphere always faces Earth.

Gravitational fields can therefore affect not only where objects move, but also how they rotate and evolve.

---

Practical Example: Designing a Lunar Vehicle

Suppose a lunar rover has mass:

500 kg

Moon:

g = 1.62 N/kg

Its weight is:

F = 500 × 1.62

= 810 N

On Earth:

F = 500 × 9.8

= 4900 N

This difference affects the rover's:

• suspension
• traction
• wheel loading
• stability
• jumping or bouncing behaviour

However, its inertia remains associated with its 500 kg mass.

A rover does not become six times easier to accelerate horizontally simply because its weight is approximately six times smaller.

---

Low Gravity Does Not Mean Low Inertia

This is an important practical distinction.

Imagine pushing a large spacecraft on the Moon.

Its weight is much lower than on Earth.

But its mass is unchanged.

According to:

F = ma

the same horizontal force produces the same acceleration if other forces are negligible.

Lower gravity changes weight, not inertia.

This is particularly important when moving large objects in low-gravity environments.

---

Practical Example: Weighing Samples

Suppose an astronaut collects a rock with mass:

8.0 kg

On Earth:

F = 8.0 × 9.8

= 78.4 N

On the Moon:

F = 8.0 × 1.62

≈ 13.0 N

A force-based scale calibrated for Earth would therefore give misleading readings if used on the Moon without adjustment.

A balance that compares masses can still determine mass because mass itself has not changed.

---

Practical Example: Unknown Field Strength

A 15 kg instrument has a measured gravitational force of:

55.5 N

Calculate the gravitational field strength.

g = F/m

g = 55.5/15

= 3.7 N/kg

This value is close to the surface gravitational field strength associated with Mars.

---

Practical Example: Unknown Mass

An object weighs:

324 N

on a moon where:

g = 1.8 N/kg

Find its mass.

m = F/g

m = 324/1.8

= 180 kg

The object's mass would remain 180 kg wherever it travels.

---

Practical Example: Comparing Two Locations

A 50 kg object weighs:

490 N

at Location A.

At Location B it weighs:

185 N

At A:

gA = 490/50

= 9.8 N/kg

At B:

gB = 185/50

= 3.7 N/kg

Therefore:

gB/gA = 3.7/9.8

≈ 0.378

Location B's gravitational field is approximately:

38% as strong as Location A's.

---

Field Strength Data and Astronomy

Astronomers can use the motion of objects to learn about gravitational fields.

For example, observations of:

• planetary orbits
• moon orbits
• satellite motion
• binary stars
• stars orbiting galactic centres

can provide information about the masses creating those gravitational fields.

In this way, gravity can reveal properties of objects that cannot be measured directly.

---

Inferring Mass from an Orbit

For a circular orbit:

GM/r² = v²/r

Rearranging:

M = v²r/G

Therefore, if astronomers measure:

• orbital radius
• orbital speed

they can estimate the mass responsible for the orbit.

This is an extremely important application of gravitational physics.

---

Example: Discovering Unseen Mass

Suppose stars in a galaxy move faster than expected based only on the visible matter.

Their motion indicates that the gravitational field is stronger than the visible matter alone appears able to produce.

This discrepancy is one major line of evidence supporting the presence of additional unseen mass, commonly described as dark matter.

Astronomers therefore use gravitational effects as a way of detecting matter indirectly.

---

Why Gravitational Fields Matter in Astronomy

Gravity operates across enormous distances and is the dominant interaction governing the large-scale organization of ordinary astronomical systems.

Gravitational fields help explain:

• falling objects
• planetary weight
• moon orbits
• artificial satellites
• planetary orbits
• spacecraft trajectories
• star formation
• binary star motion
• galaxies
• tides
• compact objects
• large-scale distributions of matter

Without gravity, the universe would not have its familiar large-scale structure.

---

Choosing the Correct Calculation

A useful strategy is to identify what information the problem provides.

Mass and field strength known?

Use:

F = mg

Weight and mass known?

Use:

g = F/m

Source mass and distance known?

Use:

g = GM/r²

Comparing distances around the same spherical mass?

Use:

g₂/g₁ = r₁²/r₂²

Comparing planetary surfaces?

Use:

g = GM/R²

Circular orbital motion?

Use gravitational field concepts together with centripetal acceleration.

Selecting the appropriate relationship is often more important than memorizing many separate formulas.

---

Common Mistakes

Saying mass changes on another planet

Mass remains essentially unchanged. Weight changes because g changes.

Giving weight in kilograms

Weight is a force and is measured in newtons.

Assuming low weight means low inertia

Inertia depends on mass, not weight.

Thinking gravity disappears in orbit

Orbiting spacecraft remain under gravitational acceleration.

Thinking astronauts float because Earth's field is zero

Astronauts and their spacecraft are in continuous free fall.

Using altitude instead of distance from the centre

For:

g = GM/r²

use the distance from the centre of the source mass.

Assuming larger planets always have stronger surface gravity

Radius as well as mass affects surface field strength.

Thinking gravity only matters near planets

Gravity governs systems ranging from falling objects to galaxies.

Confusing field strength with gravitational force

Field strength is measured in N/kg.

Force is measured in N.

Treating gravity assist as free creation of energy

In a gravity assist, energy and momentum are exchanged within the larger spacecraft-planet system; energy is not created.

---

Key Terms

Gravitational field strength (g) — gravitational force per unit mass at a location.

Weight — gravitational force acting on an object.

Mass — a measure of an object's inertia, measured in kilograms.

Surface gravity — gravitational field strength at the surface or defined reference level of an astronomical body.

Orbital motion — motion of one object around another under gravitational influence.

Centripetal acceleration — inward acceleration required for circular motion.

Free fall — motion in which gravity is the only significant force.

Microgravity — a condition of very small apparent weight commonly experienced in orbit.

Orbital speed — the speed required for a particular orbit.

Escape velocity — the minimum initial speed required in an idealized model to escape a body's gravitational influence without further propulsion.

Gravity assist — use of a moving astronomical body's gravitational field to alter a spacecraft's trajectory and Sun-centered speed.

Superposition — combining gravitational field vectors from multiple masses.

Tidal effect — an effect produced by differences in gravitational field strength across an extended object.

Hydrostatic equilibrium — balance between inward gravity and outward pressure in a star or other self-gravitating body.

Dark matter — matter inferred primarily through gravitational effects but not observed through ordinary electromagnetic radiation.

Inertia — resistance to changes in motion, determined by mass.

---

Key Takeaways

• Weight depends directly on gravitational field strength through F = mg.
• Mass remains essentially constant when an object moves between planets or moons.
• An object weighs less in a weaker gravitational field and more in a stronger field.
• Weight comparisons can be made directly from ratios of gravitational field strength.
• Surface gravitational field strength depends on both the mass and radius of a planet.
• Gravitational field strength decreases as distance from a spherical mass increases.
• Spacecraft in orbit are still strongly affected by gravity.
• Astronauts experience microgravity because they and their spacecraft are in continuous free fall.
• Gravity provides the inward acceleration necessary for orbital motion.
• Gravitational fields must be considered when designing landers, rovers, satellites, and spacecraft.
• Lower gravity reduces weight but does not reduce mass or inertia.
• Multiple gravitational fields combine according to superposition.
• Gravity assists use planetary motion and gravitational interaction to alter spacecraft trajectories.
• Gravitational fields control the motion of moons, planets, stars, and many larger astronomical systems.
• Astronomers can use orbital motion to infer the masses of objects that may be difficult or impossible to measure directly.
• Gravitational fields are therefore essential both for practical spaceflight and for understanding the structure and evolution of the universe.

---

Check Your Understanding

1. Explain the difference between mass and weight.
2. A 65 kg astronaut stands on Earth where g = 9.8 N/kg. Calculate the astronaut's weight.
3. The same astronaut travels to the Moon where g = 1.62 N/kg. Calculate the astronaut's weight and state what happens to their mass.
4. A 1200 kg rover operates on Mars where g = 3.7 N/kg. Calculate its weight.
5. An object weighs 980 N on Earth. Calculate its mass and then determine its weight somewhere where g = 2.0 N/kg.
6. Planet B has twice Earth's mass and the same radius. Compare its surface gravitational field strength with Earth's.
7. Planet C has the same mass as Earth but three times Earth's radius. What fraction of Earth's surface gravitational field strength would Planet C have?
8. A spacecraft moves from distance r to distance 3r from a planet's centre. By what factor does the gravitational field strength change?
9. Explain why astronauts aboard an orbiting spacecraft appear weightless even though Earth's gravitational field is still acting strongly on them.
10. A 2500 kg spacecraft hovers where g = 4.0 N/kg. Calculate the upward thrust required to maintain constant altitude, ignoring other forces.
11. Explain why a vehicle on the Moon has much less weight than on Earth but still has the same inertia.
12. Describe how gravity allows a satellite to remain in orbit rather than moving in a straight line through space.
13. Explain how a gravity assist can be useful during an interplanetary spacecraft mission.
14. Astronomers observe a star moving in an orbit around an object they cannot see directly. Explain how the star's motion can provide evidence about the unseen object's mass.
15. Explain why gravitational fields are important at three different scales: for a person or spacecraft, for a planetary system, and for stars or galaxies.