2. Newton's Law of Gravitation

Learning Outcome

I can state Newton's Law of Universal Gravitation.
I can identify the variables that affect gravitational force.
I can explain how mass influences gravitational attraction.
I can calculate gravitational force between two objects.
I can apply Newton's Law of Gravitation to real-world situations.

Key Topics:

Mathematical formulation: $ F = G \frac{m_1m_2}{r^2} $
Units and dimensions of G

Newton's Great Discovery

Before the 17th century, people knew that objects fell to Earth, but they did not understand why planets stayed in orbit around the Sun.

In 1687, Sir Isaac Newton proposed that the same force responsible for an apple falling from a tree also keeps the Moon in orbit around Earth and the planets in orbit around the Sun.

He called this force gravitation and described it with one of the most important equations in physics—Newton's Law of Universal Gravitation.

The word universal means that the law applies everywhere in the universe, from tiny objects on Earth to galaxies billions of light-years away.

Newton's Law of Universal Gravitation

Newton's Law states:

Every object in the universe attracts every other object with a gravitational force.

The strength of this force depends on:

The masses of the two objects.
The distance between their centres.

The mathematical relationship is:

F = G \frac{m_1 m_2}{r^2}

where:

F = gravitational force (N)
G = universal gravitational constant
m₁ = mass of the first object (kg)
m₂ = mass of the second object (kg)
r = distance between the centres of the two objects (m)

The gravitational constant is:

G=6.67\times10^{-11}\text{ N·m}^2\text{/kg}^2

$G=6.67×10−11 N\cdotpm2/kg2$

This very small value explains why gravitational forces between everyday objects are usually too weak to notice.

Variables that Affect Gravitational Force

There are only two factors that determine the strength of gravitational attraction.

1. Mass

Larger masses produce stronger gravitational forces.

If either mass increases:

The gravitational force increases.

If one mass doubles:

The force doubles.

If both masses double:

The force becomes four times larger.

2. Distance

Gravity becomes weaker as objects move farther apart.

However, gravity does not decrease in a simple linear way.

Instead, it follows the inverse-square law.

This means:

If the distance doubles:

The force becomes one-quarter as large.

If the distance triples:

The force becomes one-ninth as large.

If the distance becomes four times larger:

The force becomes one-sixteenth as large.

Distance Change	New Gravitational Force
Double distance	1/4 of original
Triple distance	1/9 of original
Four times farther	1/16 of original
Half the distance	4 times greater

Notice that distance has a much greater effect than mass.

The Universal Gravitational Constant

The constant G is the same everywhere in the universe.

Its small value shows that gravity is an extremely weak force between small objects.

For example:

Two students standing 1 metre apart attract one another gravitationally—but the force is so tiny that it is completely overwhelmed by friction and other everyday forces.

Gravity only becomes dominant when enormous masses such as planets and stars are involved.

Equal and Opposite Forces

Newton's Law of Gravitation follows Newton's Third Law of Motion.

If Earth pulls on the Moon with a certain force:

The Moon pulls on Earth with exactly the same force.

The forces are equal in magnitude but act on different objects.

The reason Earth moves much less is that it has far greater mass.

Worked Example

Question

Calculate the gravitational force between two objects.

Object A:

Mass = 10 kg

Object B:

Mass = 20 kg

Distance between centres = 2.0 m

Step 1

Write the equation.

F=G\frac{m_1m_2}{r^2}

$F = G r 2 m 1 m 2$

Step 2

Substitute the values.

F=6.67\times10^{-11} \frac{(10)(20)}{2^2}

$F = 6.67 \times 1 0 -11 22 (10) (20)$

Step 3

Calculate.

F=6.67\times10^{-11} \times50

$F = 6.67 \times 1 0 -11 \times 50$

F=3.34\times10^{-9}\text{ N}

$F = 3.34 \times 1 0 -9 N$

Answer:

The gravitational force is approximately:

\boxed{3.3\times10^{-9}\text{ N}}

3.3×10−9 N

This extremely small force illustrates why gravity between everyday objects is difficult to detect.

Real-World Applications

Newton's Law of Gravitation is used to predict and understand many important phenomena.

Examples include:

Planetary Motion

Calculating the forces keeping planets in orbit around the Sun.

Moon Orbits

Predicting the Moon's motion around Earth.

Artificial Satellites

Designing satellite orbits for:

GPS
Weather forecasting
Communications
Earth observation

Space Missions

Planning spacecraft trajectories to the Moon, Mars, and beyond.

Binary Stars

Calculating how pairs of stars orbit one another.

Galaxies

Studying how galaxies rotate and interact through gravity.

Why Gravity Controls the Solar System

The Sun contains about 99.8% of the Solar System's total mass.

Because gravitational force increases with mass:

The Sun exerts an enormous gravitational pull on every planet.

Each planet moves forward while simultaneously being pulled toward the Sun.

This continuous inward pull causes the planets to travel in stable orbits rather than moving away into space.

Limitations of Newton's Theory

Newton's Law accurately predicts the motion of most objects in the Solar System.

However, for:

Extremely strong gravitational fields,
Objects moving near the speed of light,

Albert Einstein's General Theory of Relativity provides more accurate predictions.

Even so, Newton's Law remains an excellent approximation for almost all engineering and everyday astronomical calculations.

Key Terms

Universal Gravitation — The attractive force between any two objects with mass.

Gravitational Constant (G) — A constant that determines the strength of gravitational attraction.

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

Centre of Mass — The point at which an object's mass is considered to be concentrated for gravitational calculations.

Key Takeaways

Every pair of objects with mass attracts one another.
Gravitational force depends on mass and distance.
Larger masses produce stronger gravitational forces.
Gravitational force decreases with the square of the distance between objects.
Gravity between everyday objects is extremely small because the gravitational constant is very small.
Newton's Law of Gravitation explains the motion of planets, moons, satellites, and many other astronomical systems.

Suggested Images

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Suggested placement:

After "Newton's Great Discovery" – Illustration of Isaac Newton and the famous falling apple, introducing the historical context.
After "Newton's Law of Universal Gravitation" – Diagram of two masses with attractive force arrows acting along the line joining their centres.
After "Distance" – Inverse-square law illustration comparing the gravitational force at different separations.
After "Why Gravity Controls the Solar System" – Solar System diagram emphasizing the Sun's dominant gravitational influence on the planets.