Measuring the Constants¶
The Formulas¶
Gravitational acceleration (at Earth's surface):
Gravitational constant:
The connection between them:
Where \(M\) is Earth's mass and \(R\) is Earth's radius.
What They Mean¶
These two numbers encode how gravity works. \(g\) tells you how fast things fall near Earth's surface. \(G\) tells you the fundamental strength of gravity anywhere in the universe.
We take them for granted — every physics textbook lists them as known facts. But someone had to measure them first, with no modern instruments, no electronics, no lasers. Just logic, patience, and absurdly clever experimental design.
Measuring \(g\): Galileo and the Problem of Speed¶
The Challenge (1600s)¶
Drop a rock from a building. How long does it take to hit the ground? About 1-2 seconds. Now try to time that with your heartbeat (the best "clock" available in 1600). Impossible. Objects fall too fast to measure with the tools of the era.
Worse, Aristotle had taught for 2,000 years that heavier objects fall faster. A stone falls faster than a feather — everyone can see that. To show that gravity accelerates all objects equally (and air resistance is what makes the feather slow), you'd need precise timing. Which you don't have.
Galileo's Insight: Dilute Gravity¶
Galileo Galilei couldn't slow down time. So he slowed down gravity.
He built long, smooth wooden ramps (inclined planes) and rolled polished bronze balls down them. By tilting the ramp at a shallow angle, he reduced the component of gravity along the ramp. A 5-degree ramp gives roughly \(g \cdot \sin(5°) \approx 0.85\) m/s² — about 12 times slower than freefall.
Now the ball moved slowly enough to be timed.
The Water Clock¶
Galileo's timing device was beautifully simple: a bucket of water with a small hole in the bottom, dripping into a cup. He started the drip when the ball started rolling and stopped it when the ball reached the end of the ramp. Then he weighed the water collected.
More water = more time. Weight is much easier to measure precisely than duration.
The Discovery: \(d \propto t^2\)¶
Galileo rolled the ball down different lengths of ramp and measured the time for each:
| Distance (\(d\)) | Time (relative) | \(d/t^2\) |
|---|---|---|
| 1 unit | 1 | 1 |
| 4 units | 2 | 1 |
| 9 units | 3 | 1 |
| 16 units | 4 | 1 |
Distance was proportional to the square of the time. This is the hallmark of constant acceleration:
The ball didn't reach a terminal speed and coast — it kept speeding up at a constant rate. And crucially, the rate didn't depend on the ball's weight. A heavier ball took the same time as a lighter one (once air resistance was negligible).
From Ramp to Freefall¶
On a ramp angled at \(\theta\), the acceleration along the ramp is \(a = g \sin\theta\). Galileo measured \(a\) for different angles and extrapolated to \(\theta = 90°\) (straight down):
At \(\theta = 90°\), \(\sin\theta = 1\), so \(a = g\). He found \(g \approx 9.8\) m/s² — remarkably close to the modern value, using nothing but a wooden ramp, bronze balls, and a dripping bucket.
What This Proved¶
- All objects fall at the same rate (in vacuum). Aristotle was wrong.
- Gravity produces constant acceleration, not constant speed.
- The acceleration of free fall is about 9.8 m/s² at Earth's surface.
This was one of the most important experiments in history. It showed that the universe obeys mathematical laws that can be discovered through careful measurement.
Measuring \(G\): Cavendish and the Weight of the Earth¶
The Challenge (1798)¶
Newton's law of gravitation was famous by the late 1700s:
But there was a problem: \(G\) was unknown. Without it, you could predict relative motions (planet A orbits twice as fast as planet B) but not absolute forces. You couldn't calculate the mass of the Earth, the Sun, or anything else.
To find \(G\), you needed to measure the gravitational force between two objects of known mass. But gravity is absurdly weak. The gravitational attraction between two 1 kg masses placed 1 meter apart is:
That's about \(10^{-11}\) newtons. For comparison, the weight of a single grain of sand is about \(10^{-5}\) N. The force you need to measure is a million times smaller than a grain of sand.
Cavendish's Apparatus¶
Henry Cavendish was one of the wealthiest and most reclusive scientists in history. He communicated with his servants by written notes and fled if a stranger appeared at his door. But he was a meticulous experimenter.
In 1798, he set up an experiment designed by the geologist John Michell (who died before completing it). Here's what Cavendish built:
- A torsion balance: A 6-foot wooden rod suspended horizontally by a thin metal wire, with a 1.6 lb (0.73 kg) lead ball on each end
- Two massive lead spheres: Each weighing 350 lb (159 kg), placed near the small balls
- A sealed shed: The entire apparatus was enclosed to eliminate air currents
- A telescope: Cavendish observed the tiny deflection of the rod through a small hole in the wall, so his body heat wouldn't disturb the measurement
How It Worked¶
The large lead spheres attracted the small ones gravitationally, causing the rod to twist by a tiny angle. The wire resisted the twist like a spring — Cavendish had calibrated its stiffness precisely.
By measuring the deflection angle (using a beam of light reflected from a mirror on the rod), he could calculate the gravitational force. Since he knew the masses (\(m_1, m_2\)) and the distance (\(r\)), he could solve for \(G\):
The Result¶
Cavendish found \(G \approx 6.74 \times 10^{-11}\) N·m²/kg² — within about 1% of the modern value of \(6.674 \times 10^{-11}\).
But he didn't report it as "\(G\)." He reported the density of the Earth: about 5.48 times denser than water (modern value: 5.51). Here's how:
We know \(g = GM/R^2\) at Earth's surface. Rearranging:
Using \(g = 9.81\) m/s², \(R = 6.371 \times 10^6\) m, and his measured \(G\):
Dividing by Earth's volume gives the average density. Cavendish had weighed the Earth.
Why This Matters¶
Rocks on the surface have a density of about 2.5-3 g/cm³. But Cavendish found Earth's average density was 5.5 g/cm³ — almost double. This meant the Earth's interior must be made of something much denser than surface rocks. We now know: an iron-nickel core. Cavendish's measurement, made in a shed with lead balls and a wire, told us what the inside of the planet looks like.
Variables Explained¶
| Symbol | Name | Unit | Description |
|---|---|---|---|
| \(g\) | Gravitational acceleration | m/s² | How fast things accelerate in freefall near Earth |
| \(G\) | Gravitational constant | N·m²/kg² | The universal strength of gravity |
| \(M\) | Earth's mass | kg | \(\approx 5.97 \times 10^{24}\) |
| \(R\) | Earth's radius | m | \(\approx 6.371 \times 10^{6}\) |
Why \(G\) Is So Small¶
Gravity is the weakest of the four fundamental forces — by an absurd margin. The electric force between two protons is \(10^{36}\) times stronger than the gravitational force between them.
Why does gravity feel strong to us? Because the Earth is enormous (\(6 \times 10^{24}\) kg) and electric charges nearly perfectly cancel in everyday matter. Gravity is the last force standing at astronomical scales — not because it's strong, but because it's the only one that doesn't cancel itself out.
Common Mistakes¶
- Confusing \(g\) and \(G\): \(g\) is local (depends on where you are — different on the Moon). \(G\) is universal (same everywhere in the universe).
- Thinking \(g\) is exact: \(g = 9.81\) m/s² is an approximation for Earth's surface. It varies from 9.78 at the equator to 9.83 at the poles, and decreases with altitude.
- Assuming Galileo dropped things from the Tower of Pisa: This is almost certainly a myth. His actual experiments used inclined planes — much more precise and controllable than dropping objects.
- Thinking Cavendish measured \(G\) directly: He actually measured Earth's density. The value of \(G\) was extracted from his data later by other scientists.
Related Formulas¶
- Newton's Laws — the framework that \(g\) and \(G\) plug into
- Gravitational Potential Energy — uses \(g\) directly: \(U = mgh\)
- Coulomb's Law — Coulomb used a similar torsion balance to measure electric forces
History¶
- ~350 BCE — Aristotle teaches that heavier objects fall faster
- 1589 — Galileo (allegedly) drops objects from the Tower of Pisa; almost certainly apocryphal
- 1604 — Galileo discovers \(d \propto t^2\) using inclined planes and water clocks
- 1687 — Newton publishes universal gravitation (\(F = GMm/r^2\)) but cannot determine \(G\)
- 1783 — John Michell designs the torsion balance experiment but dies before completing it
- 1798 — Cavendish performs the experiment, determines Earth's density (and implicitly \(G\))
- 2014 — Most precise measurement of \(G\) to date: \(6.67408 \times 10^{-11}\) N·m²/kg² (still only known to 4 significant figures — making it the least precisely known fundamental constant)
References¶
- Galilei, G. (1638). Discourses and Mathematical Demonstrations Relating to Two New Sciences
- Cavendish, H. (1798). "Experiments to Determine the Density of the Earth." Philosophical Transactions of the Royal Society
- Clotfelter, B. E. "The Cavendish Experiment as Cavendish Knew It." American Journal of Physics, 1987.
- Feynman, R. P. The Feynman Lectures on Physics, Vol. 1, Ch. 7.