Basics of Gravitational Fields
4. Inverse-Square Law
Learning Outcomes
- I can explain the inverse-square relationship in gravitation.
- I can predict how gravitational force changes with distance.
- I can calculate force changes when distance is doubled or halved.
- I can interpret graphs showing inverse-square relationships.
- I can apply the inverse-square law to planetary and satellite systems.
Inverse-Square Law
Gravity becomes weaker as objects move farther apart, but it does not decrease in a simple one-to-one way. If the distance between two objects doubles, the gravitational force does not become half as large — it becomes one-quarter as large.
This relationship is known as the inverse-square law.
Learning Targets
- I can explain the inverse-square relationship in gravitation.
- I can predict how gravitational force changes with distance.
- I can calculate force changes when distance is doubled or halved.
- I can interpret graphs showing inverse-square relationships.
- I can apply the inverse-square law to planetary and satellite systems.
1. What Is an Inverse-Square Relationship?
Newton's Law of Universal Gravitation is:
F = Gm₁m₂ / r²
The important part for this topic is:
F ∝ 1/r²
This means that gravitational force is inversely proportional to the square of the distance between the centres of two objects.
As distance increases:
Gravitational force decreases.
As distance decreases:
Gravitational force increases.
The word square is especially important because distance appears as r², not simply r.
2. Why "Inverse-Square"?
There are two parts to the name.
Inverse means that when one quantity increases, the other decreases.
Square means that the distance is multiplied by itself.
For example:
2² = 4
3² = 9
4² = 16
5² = 25
Therefore, if the distance becomes twice as large, the gravitational force becomes 1/4 as large.
If the distance becomes three times as large, the force becomes 1/9 as large.
If the distance becomes four times as large, the force becomes 1/16 as large.
3. The Basic Pattern
The inverse-square relationship produces a very useful pattern:
| Distance | Gravitational Force |
|---|---|
| r | F |
| 2r | F/4 |
| 3r | F/9 |
| 4r | F/16 |
| 5r | F/25 |
So:
Distance × 2 → Force ÷ 4
Distance × 3 → Force ÷ 9
Distance × 4 → Force ÷ 16
This pattern allows us to solve many gravitational problems without using the full gravitation equation.
4. Doubling the Distance
Suppose two objects attract each other with a gravitational force of 100 N.
Their distance is doubled.
Because:
2² = 4
the gravitational force becomes:
100 N ÷ 4 = 25 N
Therefore:
Original force = 100 N
Distance doubled → New force = 25 N
The force is now one-quarter of the original force.
5. Tripling the Distance
Suppose the original gravitational force is:
180 N
The distance is tripled.
Because:
3² = 9
the force becomes:
180 N ÷ 9 = 20 N
Therefore:
Distance × 3 → Force ÷ 9
The new gravitational force is:
20 N
6. Halving the Distance
The inverse-square law also works when objects move closer together.
Suppose the distance is halved.
The new distance is:
r/2
Because the distance is squared, the force changes by:
2² = 4
But because the objects are moving closer rather than farther apart, the force becomes four times greater.
Therefore:
Distance ÷ 2 → Force × 4
For example:
Original force:
50 N
Distance is halved:
50 N × 4 = 200 N
New gravitational force:
200 N
7. What If the Distance Becomes One-Third?
Suppose the distance becomes one-third of its original value.
The objects are now three times closer.
The gravitational force becomes:
3² = 9 times greater
Therefore:
Distance ÷ 3 → Force × 9
If the original force was:
20 N
then:
20 N × 9 = 180 N
The new force is:
180 N
8. A Useful Rule
When the distance changes by a factor, square that factor to determine how the gravitational force changes.
Moving farther apart
Distance × n → Force ÷ n²
Moving closer together
Distance ÷ n → Force × n²
For example:
| Change in Distance | Change in Force |
|---|---|
| × 2 | ÷ 4 |
| × 3 | ÷ 9 |
| × 4 | ÷ 16 |
| ÷ 2 | × 4 |
| ÷ 3 | × 9 |
| ÷ 4 | × 16 |
This table is worth remembering.
9. Why Distance Has Such a Large Effect
The square in the equation means that gravitational force decreases very rapidly at first as distance increases.
Imagine two objects moving from:
1 m apart → 2 m apart
The force drops to:
1/4
Move them to:
3 m apart
The force is now:
1/9
At:
10 m apart
the force is only:
1/100
of the force at 1 m.
So even a relatively modest increase in distance can produce a large decrease in gravitational attraction.
10. The Inverse-Square Graph
If gravitational force is plotted against distance, the graph is curved, not a straight line.
The general shape looks like this:
The graph:
- Starts high when distance is small.
- Falls steeply at first.
- Becomes flatter as distance increases.
- Approaches zero but does not suddenly become zero.
This curved shape is characteristic of an inverse-square relationship.
11. Interpreting an Inverse-Square Graph
Suppose a graph shows gravitational force against distance.
At r, the force is:
F
At 2r, look for:
F/4
At 3r, look for:
F/9
At 4r, look for:
F/16
The graph should therefore decrease very quickly near the beginning and then gradually flatten.
A straight downward-sloping line would not represent an inverse-square relationship.
12. Worked Example — Reading the Pattern
Suppose the gravitational force at a distance of 1 m is:
160 N
What would the force be at 2 m?
Distance doubles:
160 N ÷ 4 = 40 N
At 3 m:
160 N ÷ 9 ≈ 17.8 N
At 4 m:
160 N ÷ 16 = 10 N
So:
| Distance | Force |
|---|---|
| 1 m | 160 N |
| 2 m | 40 N |
| 3 m | 17.8 N |
| 4 m | 10 N |
Notice how rapidly the force decreases.
13. Using Ratios
For more complicated changes, we can compare the original and new distances.
A useful relationship is:
New force = Original force × (Original distance / New distance)²
For example, suppose:
Original force = 90 N
Original distance = 2 m
New distance = 6 m
The distance has increased by a factor of:
6 ÷ 2 = 3
Therefore:
New force = 90 N ÷ 3²
New force = 90 N ÷ 9
New force = 10 N
This method is especially useful when the original distance is not 1.
14. Worked Example — 4 m to 8 m
Two objects are initially 4 m apart and exert a gravitational force of 60 N.
They move to 8 m apart.
The distance has doubled:
8 ÷ 4 = 2
Therefore:
New force = 60 N ÷ 2²
New force = 60 N ÷ 4
New force = 15 N
15. Worked Example — 6 m to 3 m
Two objects are initially 6 m apart.
Their gravitational attraction is:
25 N
They move to 3 m apart.
The distance has been halved.
Therefore:
New force = 25 N × 4
New force = 100 N
Moving the objects twice as close produces four times the gravitational force.
16. Worked Example — 12 m to 4 m
Two objects exert a gravitational force of 8 N when they are 12 m apart.
They move to 4 m apart.
The new distance is:
12 ÷ 4 = 3 times smaller
Therefore, the force becomes:
3² = 9 times greater
So:
8 N × 9 = 72 N
The new force is:
72 N
17. Distance Means Centre to Centre
When applying the inverse-square law to planets, moons and satellites, distance is measured from the centre of one object to the centre of the other.
This is especially important near a planet.
Suppose a satellite is 500 km above Earth's surface.
Its distance from Earth's centre is not 500 km.
You must also include Earth's radius.
Approximately:
Earth radius = 6370 km
Therefore:
Distance from Earth's centre = 6370 km + 500 km
Distance = 6870 km
This centre-to-centre distance is the value used in gravitational calculations.
18. Satellites and the Inverse-Square Law
Satellites closer to Earth experience stronger gravitational attraction than satellites farther away.
As a satellite moves farther from Earth's centre:
Distance increases → gravitational force decreases
However, gravity does not suddenly disappear in space.
Even satellites thousands of kilometres above Earth still experience Earth's gravitational attraction.
In fact, gravity is what keeps satellites in orbit.
19. Does Gravity Become Zero?
According to Newton's gravitational model, gravitational force becomes smaller and smaller as distance increases.
However, the inverse-square relationship does not make the force suddenly become exactly zero at some particular distance.
Instead, it approaches zero as distance becomes extremely large.
This is why astronomical objects can gravitationally influence other objects over enormous distances.
20. The Moon and Earth's Gravity
The Moon is much farther from Earth's centre than objects on Earth's surface.
Earth's radius is about:
6370 km
The average Earth-Moon distance is about:
384,400 km
The Moon is therefore roughly:
60 Earth radii
from Earth's centre.
Because gravity follows an inverse-square relationship, Earth's gravitational field at the Moon's distance is much weaker than at Earth's surface.
The distance factor is approximately:
60² = 3600
So Earth's gravitational influence per kilogram at the Moon's distance is roughly 1/3600 of its surface value.
Despite this reduction, Earth's gravity is still sufficient to play the central role in keeping the Moon in orbit.
21. Planets and the Sun
The same inverse-square relationship applies to planets orbiting the Sun.
A planet farther from the Sun experiences a weaker gravitational attraction than the same mass would experience closer to the Sun.
For a given object:
Greater distance from Sun → weaker gravitational force
Smaller distance from Sun → stronger gravitational force
This relationship is an important part of understanding planetary motion.
However, remember that the gravitational force also depends on the mass of the planet, so we cannot compare the actual force on two different planets using distance alone.
22. Example — A Spacecraft Moving Away from Earth
Imagine a spacecraft experiencing a gravitational force of:
6400 N
at a particular distance from Earth's centre.
If it moves to twice that distance:
Force = 6400 N ÷ 4
Force = 1600 N
If it moves to four times the original distance:
Force = 6400 N ÷ 16
Force = 400 N
If it moves to eight times the original distance:
Force = 6400 N ÷ 64
Force = 100 N
The force decreases rapidly, but it does not suddenly disappear.
23. Inverse-Square Relationships Beyond Gravity
Inverse-square relationships appear in several areas of physics.
Examples include:
- Gravitational field strength
- Electric field strength around a point charge
- Light intensity from a point-like source
- Radiation intensity from a point-like source
The reason is often connected to something spreading outward through three-dimensional space.
Imagine energy spreading outward from a source.
At a greater distance, the same influence is spread over a much larger spherical area.
The surface area of a sphere increases with:
r²
This geometric idea helps explain why many physical quantities follow inverse-square relationships.
24. A Useful Visual Model
Imagine spheres surrounding an object.
At distance r, the influence is spread across a certain spherical area.
At distance 2r, the spherical area becomes:
4 times larger
At distance 3r:
9 times larger
At distance 4r:
16 times larger
This produces the familiar pattern:
1, 1/4, 1/9, 1/16...
This geometric pattern is at the heart of the inverse-square law.
25. Combining Mass and Distance Changes
Sometimes both mass and distance change.
Suppose:
- One mass doubles.
- The distance doubles.
Doubling the mass causes:
Force × 2
Doubling the distance causes:
Force ÷ 4
Combined:
2 ÷ 4 = 1/2
Therefore, the new gravitational force is half the original force.
26. Worked Example — Several Changes
Two objects initially attract each other with:
F = 120 N
Now:
- Both masses double.
- Distance triples.
Effect of masses:
2 × 2 = 4
So:
Force × 4
Effect of distance:
3² = 9
So:
Force ÷ 9
Overall:
New force = 120 N × 4 ÷ 9
New force ≈ 53.3 N
Even though both masses increased, the larger separation caused the gravitational force to decrease overall.
27. Common Mistakes
Mistake 1 — Thinking doubling distance halves the force
Incorrect:
Distance × 2 → Force ÷ 2
Correct:
Distance × 2 → Force ÷ 4
because distance is squared.
Mistake 2 — Forgetting the square
The relationship is:
F ∝ 1/r²
not:
F ∝ 1/r
Mistake 3 — Applying the rule backwards
If distance is halved, force increases.
Distance ÷ 2 → Force × 4
Mistake 4 — Measuring from the surface
For planets and satellites, gravitational distance is measured from the centre of the planet, not simply from its surface.
Mistake 5 — Thinking gravity stops beyond the atmosphere
Earth's gravity extends far beyond the atmosphere.
Satellites orbit precisely because Earth's gravity is still acting on them.
Mistake 6 — Expecting a straight-line graph
An inverse-square graph is curved.
It falls rapidly at small distances and becomes progressively flatter at large distances.
28. Key Terms
Inverse-square law – a relationship in which one quantity varies inversely with the square of another.
Gravitational force – the attractive force between objects with mass.
Distance (r) – centre-to-centre separation between the interacting objects.
Inverse proportionality – a relationship in which one quantity decreases as another increases.
Orbit – the curved path of one object around another under gravitational influence.
Satellite – an object that orbits another object.
Gravitational field – a region in which an object with mass experiences gravitational force.
29. Check Your Understanding
1. What does it mean to say gravitational force follows an inverse-square law?
2. What happens to gravitational force if distance doubles?
3. What happens if distance triples?
4. What happens if distance is halved?
5. What happens if distance becomes one-third as large?
6. Two objects attract each other with 80 N of force. Their distance doubles. What is the new force?
7. Two objects attract each other with 30 N of force. Their distance is halved. What is the new force?
8. Two objects are initially 10 m apart and attract with a force of 45 N. They move to 30 m apart. Calculate the new force.
9. Describe the shape of a graph of gravitational force against distance.
10. Why must Earth's radius be considered when calculating the gravitational force on a satellite above Earth's surface?
11. Explain why astronauts in orbit are still affected by Earth's gravity.
12. Both masses double while the distance doubles. What happens to the gravitational force?
Key Takeaways
- Gravitational force follows an inverse-square relationship with distance.
- The relationship can be written as F ∝ 1/r².
- Increasing distance causes gravitational force to decrease rapidly.
- Distance × 2 → Force ÷ 4.
- Distance × 3 → Force ÷ 9.
- Distance × 4 → Force ÷ 16.
- Distance ÷ 2 → Force × 4.
- The graph of gravitational force against distance is a curve, not a straight line.
- Distance in gravitational calculations is measured centre to centre.
- Satellites farther from a planet experience weaker gravitational attraction.
- Gravity does not suddenly disappear in space; it becomes progressively weaker with distance.
- The inverse-square law helps explain gravitational interactions involving planets, moons, satellites and spacecraft.
Did You Know?
Astronauts aboard an orbiting spacecraft are often described as being in "zero gravity," but this is misleading. Earth's gravity is still acting strongly enough to keep both the astronauts and their spacecraft in orbit. They appear weightless because they and their spacecraft are falling together around Earth.