What is the Inverse-Square Law?

One of the most important features of gravity is that its strength decreases rapidly as the distance between two objects increases.

This relationship is called the inverse-square law.

The inverse-square law states:

The gravitational force between two objects is inversely proportional to the square of the distance between their centres.

This means that even a small increase in distance can produce a large decrease in gravitational force.

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The Mathematics of the Inverse-Square Law

Newton's Law of Universal Gravitation is:

Notice that the distance r appears in the denominator as r².

This means:

$F\propto\frac{1}{r^2}$

The force does not decrease in direct proportion to distance—it decreases with the square of the distance.

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How Distance Affects Gravity

Suppose the distance between two objects changes while their masses remain constant.

If the distance doubles

$r\rightarrow2r$

The force becomes:

$\frac{1}{2^2} = \frac14$

The gravitational force becomes one-quarter of its original value.

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If the distance triples

$r\rightarrow3r$

The force becomes:

$\frac19$91​

Only one-ninth of the original force remains.

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If the distance becomes four times larger

$r\rightarrow4r$

The force becomes:

$\frac1{16}$161​

Only one-sixteenth of the original force remains.

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If the distance is halved

$r\rightarrow\frac12r$

The force becomes:

$4F$4F

The gravitational force becomes four times stronger.

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Summary Table

This demonstrates how strongly gravity depends on distance.

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Why Does Gravity Decrease So Quickly?

Imagine a light bulb shining equally in all directions.

As you move farther away:

• The same amount of light spreads over a larger area.
• The light appears dimmer.

Gravity behaves in a similar way.

As the distance increases, the gravitational influence spreads over a larger spherical area.

Since the surface area of a sphere increases as:

$A=4\pi r^2$

the strength of gravity decreases with the square of the distance.

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Worked Example 1

Two planets exert a gravitational force of 800 N on each other.

If the distance between them doubles, what is the new force?

Step 1

Doubling the distance gives:

$\frac14$41​

of the original force.

Step 2

Calculate:

$800\times\frac14=200\text{ N}$

Answer

$\boxed{200\text{ N}}$200 N​

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Worked Example 2

A satellite experiences a gravitational force of 120 N.

It moves to half its original distance from Earth.

What is the new force?

Step 1

Halving the distance increases the force by:

$4$4

Step 2

Calculate:

$120\times4=480\text{ N}$

Answer

$\boxed{480\text{ N}}$480 N​

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Interpreting Graphs

A graph of gravitational force against distance has a distinctive shape.

It is not a straight line.

Instead:

• The force decreases rapidly at small distances.
• The curve gradually flattens as distance increases.
• The graph approaches zero but never actually reaches it.

This type of curve is called an inverse-square curve.

Unlike a linear graph, equal increases in distance do not produce equal decreases in force.

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Applications in the Solar System

The inverse-square law explains why:

Planets farther from the Sun experience weaker gravity.

For example:

• Mercury experiences a much stronger gravitational pull from the Sun than Neptune.

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Satellites

As a satellite moves farther from Earth:

• Earth's gravitational pull decreases.
• The satellite moves more slowly in its orbit.

This is why satellites in higher orbits take longer to complete one revolution.

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Moons

The same law explains why moons orbiting far from their planets experience weaker gravitational forces than nearby moons.

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Why the Inverse-Square Law Matters

Scientists use the inverse-square law to:

• Predict planetary orbits.
• Calculate satellite trajectories.
• Design space missions.
• Study binary star systems.
• Understand galaxy interactions.

Without the inverse-square law, modern astronomy and space exploration would not be possible.

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Common Mistakes

Students often confuse an inverse relationship with an inverse-square relationship.

Incorrect thinking

"If the distance doubles, the force halves."

❌ This is wrong.

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Correct thinking

"If the distance doubles, the force becomes one-quarter."

✔ Because the distance is squared.

Always remember:

Square the distance first.

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Key Terms

Inverse-Square Law — A relationship in which a quantity decreases in proportion to the square of the distance.

Inverse Proportion — A relationship in which one quantity decreases as another increases.

Relative Force — The force expressed as a fraction or multiple of another force.

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Key Takeaways

• Gravitational force follows the inverse-square law.
• Doubling the distance reduces the force to one-quarter.
• Halving the distance increases the force by four times.
• The inverse-square law produces a curved graph rather than a straight line.
• This relationship explains the motion of planets, moons, and satellites.
• The inverse-square law is fundamental to astronomy, astrophysics, and space exploration.

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Suggested Images

Suggested placement:

1. After "How Distance Affects Gravity" – Diagram comparing two masses at distances r, 2r, and 3r, with arrows showing the decreasing gravitational force.
2. After "Why Does Gravity Decrease So Quickly?" – Illustration of expanding spheres (or light spreading from a bulb) to visualize why intensity and gravitational influence decrease with$r^2$r2.
3. After "Applications in the Solar System" – Diagram of the Solar System highlighting that planets farther from the Sun experience weaker gravitational attraction.
4. After "Satellites" – Comparison of a low-Earth orbit satellite and a high-altitude satellite, illustrating weaker gravity and longer orbital periods at greater distances.