The gravitational constant

Q: What is the value of the gravitational constant?

6.67430 × 10⁻¹¹ N m²/kg². G is one of the few constants not fixed by definition. CODATA averages the best results and widens the error bar to cover their disagreements, which gives about 22 parts per million.

How hard does gravity pull, and how do you weigh the Earth?

G = 6.674\,30 \times 10^{-11}\ \text{N\,m}^2/\text{kg}^2

Measured, to ±0.00015 × 10⁻¹¹ (about 22 parts per million) 13 min read, with an experiment to run

Gravity holds you to the floor and the Moon to the Earth, yet between two everyday things it is almost nothing. The gravitational constant, $G$ , sets how strong that pull is.

It is the weakest and the worst-known of the big constants. We know it to only about five digits, and different labs still disagree. Measuring it also told us, for the first time, what the Earth weighs.

1 · Play first

Twist a wire with two lead balls

In 1798 Henry Cavendish hung a rod with two small lead balls from a thin wire, and set two big lead balls beside them. The tiny pull of the big balls twists the wire by a fraction of a degree, and the wire's slow swing tells you how stiff it is. A mirror on the wire throws a spot of light onto a scale, so the twist shows up in millimetres.

Time the swing, mark where the spot comes to rest, then swap the big balls to the other side and mark the new rest point. The shift between the two turns into $G$ .

Mark the spot each time it stops and turns back. Two or more marks time the swing; the middle of two marks is where it would come to rest.

Time speed-up (a real swing takes 7 min)×60Change the universe: gravity×1.0

Spot now: 0.00 mm
Swing timed: mark 2 turns
Rest point A: not set
Rest point B: not set
Shift A to B: –

Time the swing, set rest point A, swap the balls and set rest point B. Your value of G appears here.

The strip shows where the light spot has been. Grey lines mark the two rest points and the middle.

What to notice

The twist is tiny, so Cavendish read it with a telescope from outside the room to avoid warming the air.
A slower swing means a softer wire, so the same pull twists it further.
Turn up gravity with the slider and the spot shifts further, while the swing takes just as long.

2 · The equation, piece by piece

The wire's twist balances the pull. Its swing time $T$ tells you how stiff it is, so you never need to know the wire's stiffness directly. Tap any piece.

\htmlClass{t0}{G} = \frac{\htmlClass{t1}{4\pi^2}\,\htmlClass{t2}{L}\,\htmlClass{t3}{d^2}\,\htmlClass{t4}{\theta}}{\htmlClass{t5}{M}\,\htmlClass{t6}{T^2}}

G

The gravitational constant

What we want, in N m²/kg².

Cavendish's result works out to $G \approx 6.74\times10^{-11}$ , within about 1% of today's value, from one of the most delicate experiments of the 1700s.

3 · How we found it

1Newton's law without a number

In 1687 Isaac Newton wrote that every mass pulls every other: the force grows with both masses and falls with the square of the distance. But he had no way to know how strong the pull was in plain units.

Astronomers could measure $G$ times the mass of the Sun or Earth, but never $G$ alone. Without it, nobody knew what the Earth weighed.

2Weighing the world

John Michell designed a torsion balance but died in 1793. Henry Cavendish rebuilt it and in 1798 measured the density of the Earth as about 5.45 times water. Later scientists turned that into $G$ .

Once $G$ was known, $g = GM/R^2$ gave the Earth's mass: about $6\times10^{24}$ kg.

3Still arguing today

Modern labs use torsion balances, falling atoms and swinging pendulums. Their best results disagree by more than their own error bars, so the official value carries a bigger uncertainty than any one result claims.

1687
Newton publishes the law of gravity in the Principia.
1774
Maskelyne measures a mountain's pull on a plumb line in Scotland.
1798
Cavendish's torsion balance weighs the Earth.
1873
Cornu and Baille repeat the experiment and the symbol G comes into use.
2014
An atom-interferometry measurement of $G$ is published.
2018
Two Chinese torsion-balance methods give the most precise results yet, which still disagree slightly.

4 · How the value is pinned down

$G$ is one of the few constants not fixed by definition. CODATA averages the best results and widens the error bar to cover their disagreements, which gives about 22 parts per million.

Gravity is so weak that a lab's walls, people walking by and tiny temperature changes all matter. That is why it remains the hardest constant to pin down.

5 · What if it were different?

If G were bigger

Stars would squeeze harder and burn out much faster, perhaps too fast for planets to settle around them.

If G were smaller

Gas clouds might never collapse into stars at all, and the universe would stay dark and spread out.

Why it matters to you

Knowing $G$ lets us work out the mass of the Earth, the Moon and the planets that space missions steer around.

6 · Try it at home

Weigh the Earth with a pendulum

You cannot measure $G$ at home, but you can do the other half of Cavendish's job: measure $g$ with a pendulum and combine it with today's $G$ to weigh the Earth.

You need

A string about 1 m long
A small heavy weight, such as a bunch of keys or a nut
A stopwatch or phone timer
A tape measure

Do this

Tie the weight to the string and hang it from a door frame. Measure from the pivot to the middle of the weight.
Pull it aside a little (no more than a hand width) and let go.
Time 20 full swings, there and back, and divide by 20 to get one swing time $T$ .
Work out $g = 4\pi^2 L / T^2$ . You should get close to 9.8 m/s².

The maths

With Earth's radius $R = 6{,}371$ km, the Earth's mass is $M = gR^2/G$ . With $g = 9.8$ that is $9.8 \times (6.371\times10^6)^2 / 6.674\times10^{-11} \approx 6.0\times10^{24}$ kg.

Stay safe: Keep the swing small and make sure nobody walks into the weight.

7 · Common mix-ups

MythGravity is a strong force.

ActuallyIt is by far the weakest. A small magnet can lift a paper clip against the whole Earth's pull.

Myth $G$ and $g$ are the same thing.

Actually $g$ is the pull at Earth's surface (9.8 m/s²). $G$ is the universal number that sets how strong gravity is everywhere.

MythCavendish measured $G$ directly.

ActuallyHe reported the Earth's density. $G$ was worked out from his data later.

8 · References

CODATA value: Newtonian constant of gravitation, NIST, 2018
Experiments to determine the density of the Earth, H. Cavendish, Philosophical Transactions 88, 1798
Measurements of the gravitational constant using two independent methods, Q. Li et al., Nature 560, 2018
Precision measurement of the Newtonian gravitational constant using cold atoms, G. Rosi et al., Nature 510, 2014

Next constantThe Planck constant: Why does light come in tiny packets, and how big is one?