How Galileo measured g — inclined planes, water clocks, and pendulums

Drop a stone and you find g ≈ 9.8 m/s². That number was first obtained by Galileo (~1600) — about 90 years before Newton wrote down the law of gravity. But how? In an era with no precise clocks, how do you time a 1-second fall with 0.1-second accuracy?

Galileo's answer is a masterpiece of experimental design: rather than fight gravity's speed, he slowed gravity down.

The technical problem of Galileo's era

Around 1600, the best timing tools were:

• Tower mechanical clocks: accurate to ~15 minutes per day
• Water clock (clepsydra): better, but still error a few percent
• Heartbeat: ~1 beat per second, ~10% error
• Pendulum: exactly what Galileo was studying — so he couldn't use it as a reference

A free-fall drop from 20 m takes about 2 seconds. To see the kinematics you'd need sub-0.1-second precision. Not available.

Galileo's solution — the inclined plane

Brilliant idea: instead of a vertical drop, roll the ball on a gently inclined plane. It accelerates far more slowly. At 5° tilt, the acceleration is g × sin(5°) ≈ 0.85 m/s² — a tenth of g. Now speeds and distances are measurable with the tools he had.

The ball moves with s ∝ t² — in the first second it goes a little, in the second second three times as far, then five times, then seven. That is Galileo's "odd-numbers rule" — the signature of constant acceleration.

Galileo's experiment — the actual details

Apparatus (from the account in Two New Sciences, 1638):

• Wooden board ~12 arms (~5.5 m) long, with a straight groove smoothed by parchment to minimize friction
• Bronze ball ~1 inch diameter, polished
• Water clock: a large water-filled vessel with a thin tube at the bottom draining into a small vessel. Whenever they released the ball, water flowed; when the ball stopped, water stopped. The mass of water collected = the time elapsed.

Galileo's water-clock precision: about 1/10 second — which, at 5-10° tilts, made multi-second measurements possible.

Galileo's finding — "distance is proportional to time squared"

Galileo released the ball from the top and measured time to reach various positions:

Distance (m)	Time (s)	Time²
0.25	~1.0	~1.0
1.0	~2.0	~4.0
2.25	~3.0	~9.0
4.0	~4.0	~16.0

The ratio s/t² is constant! That is:

\[ s = \frac{1}{2} a t^2 \]

with a = const. This was Galileo's foundational discovery: motion on an inclined plane has constant acceleration — and by extension, so does free fall.

From inclined-plane geometry:

\[ a = g \sin\theta \]

So if you measure a from the experiment and θ with a protractor, you get g:

\[ g = \frac{a}{\sin\theta} \]

Galileo's numbers — actual precision

Modern reproductions of Galileo's experiment with 17th-century tools yield g to about 5-10% precision. The true value:

\[ g_\text{true} = 9.81\ \mathrm{m/s^2} \]

Galileo's best measurement (from his own writings): between 9.4 and 10.2 m/s². Within his tool's error bounds. Remarkable for an era without digital stopwatches.

The pendulum — an independent check

Galileo noticed that a pendulum's period is nearly independent of its swing amplitude (for small angles) — a critical insight for later timing. For a simple pendulum:

\[ T = 2\pi \sqrt{\frac{L}{g}} \]

So if you have L and T:

\[ g = \frac{4\pi^2 L}{T^2} \]

Simple example: a pendulum of length 1 m has period T ≈ 2.006 s. Compute g:

\[ g = \frac{4\pi^2 \cdot 1}{2.006^2} \approx \frac{39.48}{4.024} \approx 9.81\ \mathrm{m/s^2} \]

Galileo knew this relation qualitatively but Huygens (~1656) first derived the full formula. With a pendulum, g precision reached below 1% — a foundation Cavendish later built on (see Gravity Article 2).

Why not "the balls from the Tower of Pisa"?

The famous story that Galileo dropped two balls of different mass from the Tower of Pisa and saw them land together is a legend. In Two New Sciences, Galileo only mentions once that "I've heard there is a slight difference" — nowhere does he claim to have performed this experiment. This legend was written by his student Viviani in 1654 (after Galileo's death).

Galileo's actual experiment was the inclined plane — which was more accurate, repeatable, and yielded actual kinematic laws. Dropping from a tower is dramatic but scientifically unhelpful: air currents dominate small mass differences.

Comparison with modern precision

Today:

• Superconducting gravimeters: precision 10⁻¹², sensitive to lunar tides
• Cold-atom gravimeters: matter-wave interferometry, 10⁻⁹ precision
• Modern inclined-plane: with photocell timing and low-friction tracks, ~0.1% precision achievable by students
• Simple pendulum: with period-averaging over hundreds of swings, ~0.01% possible

Galileo's precision (~5-10%) seems modest today, but for his era and tools, remarkable. More importantly: methodologically. Galileo showed how controlled experiments could quantify laws of nature.

What you should be able to do

After this article, you should be able to:

• Explain why Galileo couldn't directly time free fall (17th-century clock limitations)
• Understand the inclined-plane trick as "slowing gravity": a = g sin θ
• Interpret Galileo's "odd-numbers rule": s = ½ a t² means successive distances in equal time intervals have ratios 1:3:5:7:...
• Extract g from a simple pendulum given T and L
• Explain why the "tower of Pisa" story is a legend and what Galileo actually did

Connection to the rest of the series

• Gravity Article 1 (Newton → 1/r²): Newton used Galileo's g in his "Moon test" — this article shows where that g came from
• Gravity Article 2 (Cavendish G): Cavendish needed g for Earth's density — which Galileo had provided
• §2.5 Free fall (application): all free-fall equations rely on a = -g, which Galileo first measured

📚 Primary source: Galileo Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze (1638). English translation (free): Two New Sciences. 📖 Modern reconstruction: Stillman Drake, Galileo At Work (1978). 📖 A good video: Harvard Natural Sciences Lecture Demonstrations — reconstruction with 17th-century tools.