The Pendulum: Teaching Galileo's Discovery and Why Period Doesn't Depend on Weight

Sometime around 1583, Galileo Galilei sat in the cathedral of Pisa watching a lamp swinging on a chain. He timed it against his own pulse and noticed something odd: regardless of whether the lamp was swinging in wide arcs or narrow ones, each complete swing took the same amount of time. This was isochronism — “equal time” — and it was wrong in his understanding but right in a useful approximation. A pendulum of fixed length swings at nearly constant period for small angles, and this principle, when incorporated into clocks in 1656 by Christiaan Huygens, gave humanity its first accurate mechanical timepieces. The same physics applies to your child’s string and washer, swinging on a doorframe, right now.

Key Takeaways

• Isochronism: For small swing angles (less than ~15°), a pendulum’s period is nearly independent of amplitude (how far it swings). Galileo’s observation was approximately correct.
• The period formula: T = 2π√(L/g), where T is period in seconds, L is pendulum length in meters, and g is gravitational acceleration (9.81 m/s² on Earth’s surface).
• Critical insight: Period depends on LENGTH and GRAVITY — not on the mass of the bob. A one-kilogram weight and a one-gram weight on identical strings swing in identical time.
• Simple harmonic motion (SHM): For small angles, a pendulum approximates SHM — a restoring force proportional to displacement, generating sinusoidal oscillation. At larger angles, the motion deviates from pure SHM.
• Real applications: Pendulum clocks, seismographs (detecting earthquake waves), Foucault pendulums (demonstrating Earth’s rotation), and gravitational measurement (measuring local g with high precision).

Galileo’s Discovery and Its Historical Impact

Galileo didn’t just observe the swinging lamp — he devised a way to test his observation. He built pendulums of different lengths and masses, using his pulse as a timing device (the heart was the best available clock). His crucial finding: two pendulums of the same length but different bob masses swung in the same time. This directly contradicted Aristotelian physics, which held that heavier objects behaved fundamentally differently from lighter ones.

Galileo couldn’t explain why this was true — that required Newton’s laws of motion, developed after Galileo’s death. But he documented the regularity precisely enough that Christiaan Huygens, 73 years later, used it to create the first successful pendulum clock in 1656. Huygens’s clock was accurate to about 10 seconds per day — extraordinary precision for its era. The previous best timekeeping devices (water clocks, verge escapement clocks) drifted by 15 minutes or more per day.

This matters because longitude at sea required accurate time. Latitude is easy to find from the Sun’s angle; longitude requires knowing exactly what time it is at a reference point (Greenwich) while observing local noon. Every 4 minutes of error = 1 degree of longitude error = 111 km at the equator. Shipwrecks on poorly mapped coastlines killed thousands of sailors annually. The pendulum clock, and later John Harrison’s marine chronometer, solved this with physics.

The Physics: Simple Harmonic Motion

The Restoring Force

A pendulum bob displaced from equilibrium experiences a restoring force pulling it back toward center. For a simple pendulum (massless string, point mass bob, no air resistance):

F_restoring = −mg sin(θ)

For small angles (θ < 15°), sin(θ) ≈ θ (in radians), so:

F_restoring ≈ −mg(θ) = −(mg/L) × x

Where x is the horizontal displacement from equilibrium. This is Hooke’s Law form — force proportional to displacement — defining simple harmonic motion with spring constant k_eff = mg/L.

The Period Formula

For SHM, the period (time for one complete oscillation) is:

T = 2π√(m/k_eff) = 2π√(m/(mg/L)) = 2π√(L/g)

The masses cancel — confirming that period is independent of bob mass. This is why Galileo’s intuition was correct.

Numerical examples:

• L = 0.25 m: T = 2π√(0.25/9.81) = 2π × 0.1595 = 1.00 s
• L = 1.0 m: T = 2π√(1.0/9.81) = 2π × 0.3193 = 2.01 s
• L = 0.063 m (~6.3 cm): T = 2π√(0.063/9.81) = 0.5 s

A 25 cm pendulum has almost exactly a 1-second period — convenient for making a clock that ticks once per second.

Where the Formula Breaks Down

The sin(θ) ≈ θ approximation fails for larger angles:

Angle (degrees)	True period / Small-angle period	Error
5°	1.0005	0.05%
10°	1.002	0.2%
15°	1.004	0.4%
30°	1.017	1.7%
45°	1.040	4.0%
90°	1.180	18%

For practical clock-making, keeping swings below 10° maintains accuracy better than 0.2%.

Energy in a Pendulum

The pendulum is a perfect example of energy conservation:

• At the top of the swing (momentarily stopped): All energy is potential energy. PE = mgh (where h = height above lowest point).
• At the bottom of the swing (maximum speed): All energy is kinetic energy. KE = ½mv².
• At intermediate positions: Energy is split between PE and KE.

Setting PE = KE: mgh = ½mv², so v_max = √(2gh). The mass cancels again.

Air resistance and friction gradually remove energy (the amplitude decreases), which is why real pendulums need an escapement mechanism to add a small energy pulse on each swing — exactly what a clock’s mainspring or weight provides.

Real-World Applications

Seismographs: Early seismographs used massive pendulum bobs suspended from frames — when the ground moved, the frame moved but the pendulum bob’s inertia kept it momentarily stationary, recording the difference on a paper drum. Modern seismographs use electromagnetic sensors, but the same physical principle applies.

Foucault Pendulum: In 1851, Léon Foucault suspended a 67 kg iron ball on a 67-meter wire from the Panthéon in Paris and let it swing for hours. The plane of oscillation appeared to rotate — but it was actually the Earth rotating beneath the pendulum. At the Paris latitude, the apparent rotation rate is about 11.3° per hour. This was the first direct physical demonstration that Earth rotates.

Measuring Local Gravity: Because T = 2π√(L/g), measuring period and length with precision lets you calculate g at any location on Earth. Gravity varies by latitude (Earth’s rotation reduces effective gravity near equator) and altitude. Geologists use precision gravimeters (based on pendulum or spring principles) to detect underground density variations — useful for finding oil, ore deposits, and volcanic chambers.

Location	Measured g (m/s²)	Reason for Difference
North Pole	9.832	Closer to Earth’s center; no rotational correction
Equator (sea level)	9.780	Farther from center; maximum rotational correction
Mount Everest summit	9.764	Highest altitude; farthest from center
Denver, CO	9.796	High altitude correction
New York City	9.803	Mid-latitude, sea level

How to Teach Your Kid About Pendulums

Ages 5–8: Swing a Washer, Count Swings

Materials: String (50 cm and 100 cm pieces), steel washers (2 of any size, 1 heavy washer), tape measure, timer.

Tie a washer to the 50 cm string. Hold the string at the top and set it swinging (small arc, about 5 cm). Count how many complete swings (back AND forth = 1) in 30 seconds. Then swap for the heavy washer — same length. Count again. Should be the same!

Now use the 100 cm string with the same washer. Count swings in 30 seconds. Should be fewer (longer pendulum, longer period).

What you’re demonstrating: Mass doesn’t change the period. Length does.

The question to ask: “What do you think would happen if you put TEN washers on the string instead of one — would it swing faster, slower, or the same?”

Ages 9–12: Test Length vs. Period, Compare to Formula

Materials: String (5 measured lengths: 10 cm, 25 cm, 50 cm, 75 cm, 100 cm), same weight bob (steel washer or nut), timer, ruler, graph paper.

For each length:

1. Set pendulum swinging at small amplitude (<10°).
2. Time 10 complete oscillations (this reduces timing error vs. timing 1).
3. Period T = total time ÷ 10.
4. Record length and period.

Calculate predicted period using T = 2π√(L/g) for each length.

Graph: Plot L (x-axis) vs. T² (y-axis). The formula predicts T² = (4π²/g) × L — a straight line with slope 4π²/g ≈ 4.03 s²/m. Measure the slope of your best-fit line. Calculate g from your slope: g = 4π²/slope.

Compare your measured g to 9.81 m/s². Typical student error: 2–5%.

Also test: Keep length constant at 50 cm. Test bob masses of 10g, 50g, 100g, 200g. Period should remain constant — this is the key confirmation.

The question to ask: “Your measured g was [value]. If you repeated this experiment on the Moon (g = 1.62 m/s²), what period would you measure for a 50 cm pendulum — and what length would you need on the Moon for a 1-second period?”

Ages 13+: Measure g with High Precision and Design a Pendulum Clock

Materials: String (minimum 1 m), heavy bob (100–200g for less air resistance effect), precise timing (phone camera at 240fps for precision timing), steel ruler (measure to mm), spreadsheet.

Precision measurement protocol:

1. Measure string length to nearest mm. Measure from pivot to center of mass of bob.
2. Time 50 complete oscillations using the phone camera for frame-accurate timing.
3. Calculate g with propagated uncertainty: Δg/g = √((ΔT/T)² + (ΔL/L)²). For 1 mm length error in 100 cm: ΔL/L = 0.1%; for 0.1s timing error in 50 oscillations: ΔT/T = 0.1%. Combined error: ~0.14%. Your measured g should be within 0.15 m/s² of the accepted local value.

Design a pendulum clock escapement: A pendulum clock requires:

1. A pendulum that swings at a known period
2. An escapement — a mechanism that allows one tooth of a gear to advance per swing
3. A weight or spring to supply energy

Build the simplest version: a pendulum (25 cm for 1-second period) with a folded cardboard lever that catches and releases a notched cardboard wheel each half-swing. The wheel advances by one tooth per full oscillation. Add enough teeth to make the wheel complete one rotation per minute (60 teeth → one rotation per 60 seconds → seconds hand).

The question to ask: “Your pendulum clock gains 30 seconds per day. Using the period formula, calculate how much you need to shorten the pendulum to bring it within 5 seconds per day accuracy — and explain why this correction changes the period.”

What to Watch For Over 3 Months

• Week 1: Does your child try different masses spontaneously after being told mass doesn’t matter? Testing the claim (rather than just accepting it) is genuine scientific behavior.
• Month 1: Do they notice that swinging the pendulum in very large arcs makes it slightly slower? Discovering the breakdown of the small-angle approximation is a sophisticated observation.
• Month 2: Are they curious about how their phone knows the time so precisely? (GPS satellites, atomic clocks, NTP — each a deeper layer of timekeeping physics.)
• Month 3: The highest outcome — they try to build a pendulum clock, fail several times due to the escapement mechanism, and learn more from the failures than from reading about pendulums ever would have taught them.

Frequently Asked Questions

If mass doesn’t matter, why do grandfather clocks use heavy bobs? Heavy bobs are used not to affect the period, but for two practical reasons: (1) a heavy bob has more momentum and resists being stopped by friction and air resistance — the clock runs more reliably; (2) a heavy bob is less affected by air currents in the room. The period is still determined only by length and g.

What’s the difference between a pendulum’s period and its frequency? Period (T) is the time for one complete oscillation, measured in seconds. Frequency (f) is oscillations per second, measured in Hertz: f = 1/T. A pendulum with T = 2 s has frequency f = 0.5 Hz. These are inverse relationships — shorter period means higher frequency.

Why doesn’t a pendulum swing forever? Energy is removed on each swing by air resistance (the pendulum pushes through air, which is a viscous force) and by friction at the pivot. Without a mechanism to continuously add energy (a wound spring, hanging weight, or electric motor), the oscillation amplitude decreases exponentially over time — this is called damping. The period itself barely changes as the amplitude decreases; the clock just eventually stops.

How accurate were the first pendulum clocks compared to modern devices? Huygens’s 1656 pendulum clock was accurate to ~10 seconds per day. Later refinements (temperature-compensating pendulums using Invar steel; vacuum enclosures to eliminate air resistance) brought pendulum clocks to ~0.01 seconds per day. A modern GPS-disciplined atomic clock is accurate to ~1 nanosecond (0.000000001 seconds) per day — about 10 billion times more accurate.

---

About the author

Ricky Flores is the founder of HiWave Makers and an electrical engineer with 15+ years of experience building consumer technology at Apple, Samsung, and Texas Instruments. He writes about how kids learn to build, think, and create in a tech-saturated world. Read more at hiwavemakers.com.

Sources

1. American Association of Physics Teachers (AAPT). (2023). “Simple harmonic motion and the pendulum: Guided inquiry for secondary physics.” The Physics Teacher, 61(3), 184–190.
2. Drake, S. (1978). Galileo at Work: His Scientific Biography. University of Chicago Press. Chapter 2: The pendulum discovery.
3. Huygens, C. (1673). Horologium Oscillatorium. Paris. (Original work on pendulum clocks and cycloid paths.)
4. Fowles, G. R., & Cassiday, G. L. (2005). Analytical Mechanics (7th ed.). Thomson Brooks/Cole. Chapter 3: Oscillations — simple harmonic motion.
5. National Institute of Standards and Technology (NIST). (2024). “Time and frequency: History of timekeeping.” https://www.nist.gov/pml/time-and-frequency-division
6. NSTA National Science Teaching Association. (2022). “Pendulum physics: An inquiry-based investigation for middle school.” Science Scope, 45(2), 16–23.