Newton's Laws and Universal Gravitation

The three laws of motion, the inverse-square law and Newton's Moon test

Woolsthorpe Manor, 1666

Plague closes Cambridge. A twenty-three-year-old undergraduate is sent home to his mother's farm in Lincolnshire, where he will spend the next eighteen months with nothing to do. Every scholar before him took one thing for granted: the heavens and the Earth run on different laws. Down here, things fall. Up there, things move in perfect circles for reasons of divine geometry. Two separate worlds, two separate rulebooks. Aristotle said so, and two thousand years of scholarship agreed. Sitting in the garden, Isaac Newton asked a question so simple it had become invisible: What if it's one rulebook? What if the force that pulls the apple down is the same force that holds the Moon in the sky — just weaker, because the Moon is further away? It's an outrageous claim. But it's a testable one, and by the end of this lesson you'll run the test yourself, exactly as he did.

Watch first · presented by Shoaib Aziz

Every number in this film is one the lab at the end of the lesson computes for itself — the 60.27 distance ratio, both accelerations and the 99.2% agreement.

Nature and Nature's laws lay hid in night: God said, Let Newton be! and all was light.

– Alexander Pope, epitaph intended for Sir Isaac Newton, 1730

Before we can test anything, we need Newton's three laws — not as trivia to memorise, but as the three claims that make the test possible.

💡First Law — Inertia

An object in motion stays in motion, in a straight line, at constant speed, unless a force acts on it. This is the law that makes orbits mysterious. A satellite doesn't want to go in a circle. Left alone it would fly off tangentially into interstellar space in a perfectly straight line. Something must be continuously bending it.

\vec{F} = m\vec{a}

Second Law — force produces acceleration in proportion to mass. Acceleration means any change in velocity, including a change of direction at constant speed.

💡Third Law — action and reaction

For every action there is an equal and opposite reaction. Earth pulls the satellite down with force F; the satellite pulls Earth up with exactly the same F. This sounds like a technicality. It isn't. It's the reason the two-body problem is genuinely a two-body problem, and it's why in Unit 2 we'll have to be careful about what 'the centre' even means. For now, Earth is

10^{22}

times more massive than a satellite, so its acceleration is negligible and we'll quietly ignore it.

Now the law that ties them together. Newton proposed that every mass attracts every other mass, with a force that weakens as the square of the distance:

\vec{F}_g = -\frac{G M m}{r^2}\,\hat{r}

Universal gravitation. G = 6.674×10⁻¹¹ N·m²/kg². The minus sign and r̂ together say: always attractive, always along the line joining the centres.

The inverse square is the part that looks arbitrary. Why

r^2

and not

r

, or

r^3

? It isn't arbitrary at all — it's geometry. Picture the gravitational influence spreading out from a mass like light from a bulb. The same total amount of influence has to cover a sphere, and a sphere's surface area grows as

r^2

. Go twice as far and the same influence is smeared over four times the area.

Figure 1.1

The inverse-square law is not an arbitrary exponent — it is what happens when a fixed quantity has to cover the surface of an expanding sphere.

💡One constant instead of two

In practice nobody uses

G

and

M

separately.

G

is one of the worst-measured constants in physics (we know it to about 5 digits), and Earth's mass is inferred from

G

anyway. But the product

GM

can be measured superbly by watching how things orbit — to eleven digits. So astrodynamics uses a single combined constant, the gravitational parameter

\mu

\mu_{\oplus} = GM_{\oplus} = 398{,}600.4418\ \text{km}^3/\text{s}^2

Earth's gravitational parameter. You will use this number in almost every calculation for the rest of the course. Learn it.

Now we have everything we need for Newton's test. Here is the logic, and it is beautiful.

The Moon sits about 60 Earth radii from Earth's centre. If gravity really does fall off as

1/r^2

, then gravity out at the Moon must be

1/60^2 = 1/3600

of its strength at Earth's surface. Surface gravity is 9.81 m/s². So the prediction is:

a_{\text{Moon, predicted}} = \frac{9.81}{60^2} = 0.00272\ \text{m/s}^2

What the inverse-square law demands, using nothing but surface gravity and a distance ratio.

That's the prediction. Now the measurement — and crucially, it uses completely independent data: the Moon's orbital period (27.3 days) and its distance (384,400 km). Anything moving in a circle of radius

r

at speed

v

is accelerating toward the centre at

v^2/r

. Nothing about gravity enters this calculation at all.

📐Worked Example: measuring the Moon's acceleration

Orbital circumference:

2\pi r = 2\pi(384{,}400) = 2.415\times10^6

km Period:

T = 27.32

days

= 2.360\times10^6

s Speed:

v = 2.415\times10^6 / 2.360\times10^6 = 1.023

km/s Centripetal acceleration:

a = v^2/r = (1.023)^2 / 384{,}400 = 2.72\times10^{-6}

km/s²

a = \mathbf{0.00272\ m/s^2}

The prediction was 0.00272 m/s². The measurement is 0.00272 m/s².

Figure 1.2

Newton's Moon test. Two accelerations, computed from entirely independent data, differing by a factor of 3607 — which is 60², to within the accuracy of 17th-century astronomy. The garden and the sky obey one law.

🕰Twenty years before publication

Newton did this calculation in 1666 and then said nothing about it for two decades. The usual story is modesty; the likelier truth is that he was stuck on a genuine mathematical problem. His law says the force depends on the distance between the centres. But the Earth isn't a point — it's a vast ball of rock, and every piece of it pulls with its own force in its own direction. Proving that a sphere pulls exactly as though all its mass were concentrated at the centre took Newton until roughly 1685, and required him to invent integral calculus to do it. That proof is why we can write

\mu/r^2

and mean it. In Unit 7 we'll discover what happens when the body isn't a perfect sphere — and it turns out to matter enormously.

↓But there is a problem

So the Moon is being pulled toward Earth, right now, at 0.00272 m/s². It is genuinely, continuously falling. It has been falling for four and a half billion years. So why has it never landed?