Gravitational Constant

The gravitational constant, represented by the symbol G, is a fundamental constant of nature used in Newton's Law of Universal Gravitation. It provides the scale that determines how strong the gravitational attraction between two masses is.

Its extremely small value helps explain something we experience every day: although every object with mass attracts every other object, gravitational forces between ordinary objects are usually far too small for us to notice.

Learning Targets

• I can define the gravitational constant, G.
• I can explain the role of G in Newton's Law of Gravitation.
• I can identify the units of the gravitational constant.
• I can use G in gravitational force calculations.
• I can explain why the value of G makes gravity relatively weak compared to other forces.

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1. What Is the Gravitational Constant?

The gravitational constant, G, is the constant of proportionality used in Newton's Law of Universal Gravitation.

Its approximate value is:

G = 6.67 × 10⁻¹¹ N m²/kg²

This is a universal constant. This means that its value is the same when applying Newtonian gravity anywhere in the universe.

G is not the same as the gravitational field strength, g.

We will examine that important distinction later.

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2. G in Newton's Law of Gravitation

Newton's Law of Universal Gravitation is:

F = Gm₁m₂ / r²

Where:

• F = gravitational force in newtons (N)
• G = gravitational constant
• m₁ = mass of object 1 in kilograms (kg)
• m₂ = mass of object 2 in kilograms (kg)
• r = distance between the centres of the objects in metres (m)

G connects the masses and distance in the equation to an actual gravitational force.

Without G, the expression:

m₁m₂ / r²

would tell us how mass and distance affect the force, but it would not give the correct force in newtons.

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3. Why Do We Need G?

Suppose two objects each have a mass of 1 kg and their centres are 1 m apart.

Newton's equation becomes:

F = G × 1 × 1 / 1²

Therefore:

F = G

So their gravitational attraction is:

F = 6.67 × 10⁻¹¹ N

That is an extraordinarily small force.

Written as an ordinary decimal:

F = 0.0000000000667 N

This demonstrates why you do not notice your books, pencils, chairs and other nearby objects gravitationally attracting one another.

The force exists — it is simply extremely small.

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4. Understanding Scientific Notation

Because G is so small, scientific notation is much more convenient.

G = 6.67 × 10⁻¹¹ N m²/kg²

The exponent −11 tells us that the decimal point moves 11 places to the left.

Therefore:

6.67 × 10⁻¹¹ = 0.0000000000667

Scientific notation allows scientists to work efficiently with extremely large and extremely small quantities.

Other examples include:

10⁻³ = 0.001

10⁻⁶ = 0.000001

10⁻⁹ = 0.000000001

10⁻¹¹ = 0.00000000001

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5. Units of G

The units of the gravitational constant are:

N m²/kg²

This can be read as:

newton metre squared per kilogram squared

Why does G have these unusual units?

Look again at:

F = Gm₁m₂ / r²

The masses contribute units of:

kg × kg = kg²

Distance squared contributes:

m²

G must have units that cause the final answer to come out in newtons.

Therefore:

G = 6.67 × 10⁻¹¹ N m²/kg²

The units are an important part of the constant and should be included when stating its value.

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6. G Is a Universal Constant

The word universal is important.

G does not change depending on whether the objects are:

• On Earth
• On the Moon
• Near the Sun
• In deep space
• Inside another solar system

The same value of G is used.

What does change are the:

• Masses of the objects
• Distances between the objects

These changing quantities produce different gravitational forces.

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7. G Is Not the Same as g

This is one of the most common sources of confusion.

G and g represent different quantities.

Capital G is essentially the same everywhere.

Lowercase g depends on the gravitational field at a particular location.

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8. Worked Example — Two Small Objects

Two objects have masses:

m₁ = 10 kg

m₂ = 20 kg

Their centres are 2 m apart.

Calculate the gravitational force.

Use:

F = Gm₁m₂ / r²

and:

G = 6.67 × 10⁻¹¹ N m²/kg²

Substitute:

F = (6.67 × 10⁻¹¹ × 10 × 20) / 2²

First calculate:

10 × 20 = 200

and:

2² = 4

Therefore:

F = (6.67 × 10⁻¹¹ × 200) / 4

F = 3.335 × 10⁻⁹ N

Rounded to three significant figures:

F = 3.34 × 10⁻⁹ N

This force is extremely small.

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9. Worked Example — Larger Masses

Now consider two objects with masses:

m₁ = 1000 kg

m₂ = 2000 kg

Distance:

r = 5 m

Use:

F = Gm₁m₂ / r²

Substitute:

F = (6.67 × 10⁻¹¹ × 1000 × 2000) / 5²

Calculate the mass product:

1000 × 2000 = 2,000,000

Calculate the distance squared:

5² = 25

Therefore:

F = (6.67 × 10⁻¹¹ × 2,000,000) / 25

F ≈ 5.34 × 10⁻⁶ N

Even with masses of thousands of kilograms, the gravitational force at this distance is still tiny.

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10. Why Don't We Notice Gravity Between Everyday Objects?

Every object around you produces a gravitational field.

Your desk attracts you.

Your computer attracts you.

You attract the walls.

Two students standing beside each other attract one another gravitationally.

Yet none of these forces are noticeable.

The main reason is the tiny value of:

G = 6.67 × 10⁻¹¹ N m²/kg²

For everyday-sized masses, multiplying by G produces an extremely small force.

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11. Why Is Earth's Gravity Noticeable?

If G is so small, why does Earth pull us downward so strongly?

The answer is Earth's enormous mass.

Earth's mass is approximately:

5.97 × 10²⁴ kg

Newton's equation includes the product of the two masses:

m₁ × m₂

Earth's enormous mass compensates for the very small value of G.

As a result, the gravitational attraction between Earth and objects near its surface is easily noticeable.

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12. Gravity and Astronomical Objects

Gravity becomes especially important on astronomical scales because planets, stars and moons have enormous masses.

Examples include:

• Earth attracting the Moon
• The Moon attracting Earth
• The Sun attracting Earth
• Earth attracting artificial satellites
• Stars interacting gravitationally within galaxies

Although G is tiny, the masses involved can be enormous.

For example, the Sun's mass is approximately:

1.99 × 10³⁰ kg

Multiplying G by astronomical masses can therefore produce extremely large gravitational forces.

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13. Worked Example — Earth and a Person

Suppose a person has a mass of:

60 kg

Earth has a mass of approximately:

5.97 × 10²⁴ kg

The person is approximately one Earth radius from Earth's centre:

r = 6.37 × 10⁶ m

Use:

F = Gm₁m₂ / r²

Substitute:

F = (6.67 × 10⁻¹¹ × 5.97 × 10²⁴ × 60) / (6.37 × 10⁶)²

This gives approximately:

F ≈ 589 N

That is close to the person's weight calculated using:

W = mg

W = 60 × 9.8 = 588 N

The small difference comes from rounding the values used.

This shows that the familiar equation W = mg is connected to Newton's more general law of gravitation.

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14. Measuring G

Because G is so small, measuring it is difficult.

One of the most famous early experiments related to determining G was performed by Henry Cavendish in 1798.

He used an extremely sensitive device called a torsion balance.

The apparatus used:

• Small lead spheres
• Large lead spheres
• A lightweight rod
• A thin suspension wire

The gravitational attraction between the large and small spheres caused the rod to rotate slightly.

The effect was tiny, so the apparatus had to be extremely sensitive.

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15. The Cavendish Experiment

The basic idea was surprisingly simple.

Large masses were placed near smaller masses.

Gravity caused the masses to attract each other.

The gravitational force twisted the suspension wire slightly.

By measuring this tiny rotation and knowing the masses and distances involved, the gravitational interaction could be quantified.

This work made it possible to determine Earth's density and, together with other known relationships, the value now expressed through G.

The experiment dramatically demonstrated that gravity acts not only between planets and stars but also between ordinary laboratory objects.

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16. Why Is Gravity Considered Weak?

There are four fundamental interactions commonly described in physics:

• Gravitational interaction
• Electromagnetic interaction
• Strong nuclear interaction
• Weak nuclear interaction

Gravity is extraordinarily weak compared with the electromagnetic interaction at the scale of individual particles.

For example, two charged particles can exert an electromagnetic force on each other that is vastly stronger than their gravitational attraction.

The tiny value of G reflects the weakness of gravitational interactions between ordinary-sized masses.

However, gravity becomes dominant on astronomical scales because:

• Mass is always gravitationally attractive in ordinary situations.
• Astronomical bodies contain enormous amounts of mass.
• Positive and negative electric charges can cancel overall, while gravitational mass does not cancel in the same everyday way.

This allows gravity to dominate the large-scale structure of planets, stars and galaxies.

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17. Gravity Is Weak but Important

Calling gravity "weak" can seem strange.

After all, gravity:

• Holds us on Earth
• Keeps the Moon in orbit
• Keeps planets around the Sun
• Helps form stars
• Influences galaxies

The key is scale.

Between two small objects:

Gravity is extremely weak.

Between two astronomical objects:

Gravity can be enormous because their masses are enormous.

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18. Using G Correctly in Calculations

When using Newton's gravitational equation, remember:

1. Use kilograms for mass.

2. Use metres for distance.

3. Measure distance from centre to centre.

4. Square the distance.

5. Use G = 6.67 × 10⁻¹¹ N m²/kg².

6. Give gravitational force in newtons.

A useful checklist is:

kg → metres → square r → substitute G → calculate → N

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19. Calculator Skills

Calculations involving G normally require scientific notation.

Suppose you need to enter:

6.67 × 10⁻¹¹

Most scientific calculators have an EXP, EE, or ×10ˣ button.

Depending on the calculator, you might enter:

6.67 EXP −11

You should not type:

6.67 × 10 × −11

because that represents a completely different calculation.

Correct use of scientific notation is especially important in gravitational calculations.

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20. Worked Example — Check Your Calculator

Calculate:

G × 500 × 800

Use:

G = 6.67 × 10⁻¹¹

First:

500 × 800 = 400,000

Then:

6.67 × 10⁻¹¹ × 400,000

Result:

2.668 × 10⁻⁵

If the objects were 1 m apart, the gravitational force would therefore be:

2.668 × 10⁻⁵ N

If your calculator gives a value thousands or millions of times larger, check how you entered the exponent.

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21. Changing G?

In standard Newtonian gravitational calculations, G is a constant.

If gravitational force changes, it is because:

• One or both masses changed, or
• The distance between the objects changed.

We do not change G to explain different gravitational forces.

For example:

Earth and Moon:

G = 6.67 × 10⁻¹¹ N m²/kg²

Sun and Earth:

G = 6.67 × 10⁻¹¹ N m²/kg²

Two laboratory masses:

G = 6.67 × 10⁻¹¹ N m²/kg²

The same constant applies.

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22. Real-World Applications of G

The gravitational constant is used when calculating gravitational interactions involving:

Planets – forces between planets and stars.

Moons – gravitational interactions with their planets.

Satellites – gravitational forces keeping spacecraft in orbit.

Space missions – predicting the motion of spacecraft.

Astronomy – determining masses and gravitational interactions of astronomical bodies.

Laboratory physics – measuring extremely small gravitational attractions.

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23. Common Mistakes

Mistake 1 — Confusing G and g

G = universal gravitational constant

g = gravitational field strength

They are not interchangeable.

Mistake 2 — Forgetting the negative exponent

G is:

6.67 × 10⁻¹¹

not:

6.67 × 10¹¹

The difference is enormous.

Mistake 3 — Leaving out the units

The complete value is:

G = 6.67 × 10⁻¹¹ N m²/kg²

Mistake 4 — Changing G on another planet

G remains the same.

The gravitational field strength g changes from planet to planet.

Mistake 5 — Assuming a tiny G means gravity is always tiny

Astronomical objects have enormous masses, so their gravitational forces can still be very large.

Mistake 6 — Entering scientific notation incorrectly

Always check the exponent carefully when using a calculator.

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24. Key Terms

Gravitational constant (G) – the universal constant that sets the scale of gravitational attraction in Newton's Law of Gravitation.

Universal constant – a physical quantity whose value is the same wherever the law applies.

Gravitational force – the attractive force between objects with mass.

Newton (N) – the SI unit of force.

Scientific notation – a method of expressing very large or very small numbers using powers of ten.

Torsion balance – a sensitive apparatus that can measure very small forces through the twisting of a wire.

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

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25. Check Your Understanding

1. What does the symbol G represent?

2. State the approximate value of G.

3. What are the units of G?

4. Write Newton's Law of Universal Gravitation.

5. Explain the role of G in the equation.

6. Is G larger on Earth than on the Moon? Explain.

7. Explain the difference between G and g.

8. Why is the gravitational attraction between two classroom objects difficult to detect?

9. Why can gravity still produce enormous forces between planets and stars even though G is very small?

10. Two 1 kg objects are 1 m apart. Approximately what gravitational force do they exert on each other?

11. Why was a very sensitive apparatus needed to measure gravitational attraction between laboratory masses?

12. A student uses G = 6.67 × 10¹¹ in a calculation. Identify the error.

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Key Takeaways

• The gravitational constant is represented by G.
• Its approximate value is 6.67 × 10⁻¹¹ N m²/kg².
• G appears in Newton's Law of Universal Gravitation: F = Gm₁m₂ / r².
• G converts the relationship between mass and distance into the correct gravitational force.
• G is universal and does not change from planet to planet.
• Capital G should not be confused with lowercase g, the gravitational field strength.
• The extremely small value of G means gravitational forces between ordinary objects are usually extremely weak.
• Gravity becomes important on astronomical scales because planets, stars and other astronomical objects have enormous masses.
• G can be used to calculate gravitational forces ranging from tiny laboratory attractions to forces between planets and stars.

Did You Know?

The gravitational constant remains one of the more difficult fundamental constants to measure with very high precision. The gravitational attraction between laboratory-sized objects is so tiny that experiments must be carefully protected from vibrations, temperature changes and other disturbances that could overwhelm the gravitational effect being measured.