What is Newton's law of gravity? Revision Notes for AQA A-Level Physics

What is Newton's law of gravity?

Historical development

In 1684, three members of the Royal Society—Christopher Wren, Robert Hooke, and Edmond Halley—discussed whether a gravitational force that decreased with the square of distance could explain planetary orbits around the Sun and the Moon's orbit around Earth. Unable to develop mathematical proof themselves, Halley approached Isaac Newton for assistance.

Newton had studied Galileo's projectile motion experiments, which showed that objects follow parabolic paths. Newton developed a theoretical scenario to explain the Moon's motion by treating it as a projectile.

infoNote

Newton's Cannon Thought Experiment

Newton imagined a cannon positioned on top of an extremely high mountain, firing cannon balls horizontally with progressively greater amounts of gunpowder. As the cannon balls travelled faster, they would travel further before hitting the ground. At a sufficiently high speed, the curvature of Earth would need to be considered. Eventually, at high enough speed, the falling cannon ball would never reach the ground but would orbit Earth continuously.

The key insight: The Moon behaves similarly—constantly falling towards Earth but never reaching the surface due to Earth's curvature. Newton's diagram of projectile motion from p. 6 of the Second Edition... | Download Scientific Diagram Newton's drawing of his cannon ball thought experiment

Through comparing the motion of a falling object at Earth's surface with the Moon's orbital motion, Newton demonstrated mathematically that gravitational force varies inversely with the square of distance from Earth's centre:

$\text{gravitational force} \propto \frac{1}{(\text{distance})^2}$

This fundamental relationship became known as the inverse square law.

Newton's law of universal gravitation

Newton concluded that gravity acts as an attractive force between all masses. This universal force not only pulls objects towards Earth's surface but also maintains the Moon in orbit around Earth and planets in orbit around the Sun. The force depends on both the distance between masses and the mass of each body.

chatImportant

Newton's Law of Universal Gravitation

Any two point masses attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of their separation.

In each of these examples, Newton’s law can be used to calculate the gravitational force of attraction that the objects exert on each other.

This relationship is expressed by the equation:

$F = \frac{Gm\_1m\_2}{r^2}$

where:

$F$ = gravitational force (N)
$m\_1$ and $m\_2$ = masses of the two objects (kg)
$r$ = distance between the centres of the two masses (m)
$G$ = gravitational constant

The gravitational constant

The constant of proportionality $G$ is called the gravitational constant and has the value:

$G = 6.67 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2}$

Newton presented his law of gravity alongside his laws of motion in Philosophiae Naturalis Principia Mathematica, first published in 1687.

infoNote

The Cavendish Experiment

The value of $G$ was determined experimentally by British scientist Henry Cavendish. He constructed an apparatus with two small lead spheres attached to the ends of a suspended rod, free to rotate. He positioned two larger lead spheres so that gravitational attraction between the large and small spheres caused the suspended rod to oscillate.

By measuring the oscillation, Cavendish determined the gravitational force between the spheres. Knowing this force and the masses involved, he calculated Earth's average density, obtaining a value within 1% of the currently accepted value. The gravitational constant was determined from Cavendish's experimental measurements.

Point masses

chatImportant

Point Masses Concept

Newton's law refers to point masses—spherically symmetrical objects of uniform density whose gravitational effect acts as if all their mass is concentrated at their centre.

This approximation applies to planets, stars, and other spherical bodies when calculating gravitational forces between them.

Worked example: Earth-Moon system

The Earth-Moon system demonstrates the inverse square law in action.

lightbulbExample

Worked Example: Verifying the Inverse Square Law with the Earth-Moon System

Given:

Acceleration due to gravity at Earth's surface: $g = 9.81 \text{ m s}^{-2}$
Earth's radius: $R = 6.38 \times 10^6 \text{ m}$
Mean Earth-Moon distance: $r = 3.84 \times 10^8 \text{ m}$
Moon's orbital period: $T = 27.5 \text{ days}$

Analysis:

The Moon is kept in orbit by gravitational force, which provides the centripetal acceleration. The gravitational acceleration experienced by the Moon can be compared with the surface value to verify the inverse square relationship.

Step 1: Calculate the ratio of distances

$\frac{r}{R} = \frac{3.84 \times 10^8}{6.38 \times 10^6} \approx 60.2$

Step 2: Apply the inverse square law

According to the inverse square law, the gravitational acceleration at the Moon's orbital distance should be approximately $(60.2)^2 \approx 3600$ times weaker than at Earth's surface.

Conclusion:

This prediction matches the observed centripetal acceleration of the Moon in its orbit, confirming that gravitational force decreases with the square of distance.

Remember!

bookmarkSummary

Key Points to Remember:

Newton's law of universal gravitation: Any two point masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of their separation: $F = \frac{Gm\_1m\_2}{r^2}$
Inverse square law: Gravitational force decreases with the square of distance—doubling the distance reduces the force to one quarter
Gravitational constant: $G = 6.67 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2}$ is a universal constant determined experimentally by Cavendish
Point masses: Spherical objects can be treated as if all their mass is concentrated at their centre when calculating gravitational forces
Universal application: The same gravitational law governs falling objects on Earth, the Moon's orbit, and planetary motion around the Sun

What is Newton's law of gravity? (AQA A-Level Physics): Revision Notes