What if a pendulum clock went to the Moon?

Some old clocks have a long arm that swings. It is called a pendulum.

Tick, tock! Each swing moves the clock along. What if the clock went to the Moon?

A pendulum is a weight hanging from a rod or string that swings back and forth. In a pendulum clock, the gears count the swings and move the hands forward a little each time. The clock is only right if every swing takes exactly the right time.

Gravity on the Moon is much weaker than on Earth. So what would happen to a pendulum clock carried there? Would it keep good time, run fast, or run slow? Swing a pendulum on both worlds in the model and find out.

The OpenStax physics text notes that pendulums have crucial uses in clocks and that, for small swings, a simple pendulum's period depends only on its length and on the acceleration due to gravity. That makes a pendulum clock a gravity-dependent instrument.

This lab takes a clock regulated by a 1 m pendulum, swinging a small 5° arc, from Earth (g = 9.81 m/s²) to the Moon (g = 1.62 m/s², NASA). The question: how does its period change, how wrong would the clock be after an hour, and what single change would put it right?

On Earth, the pendulum swings over and back. It takes about two seconds.

The clock counts every swing. So the hands move at just the right speed.

In the model, a pendulum 1 meter long, pulled a little to one side and let go, takes about 2.0 seconds to swing over and back again. That is about 30 swings every minute.

The heaviness of the bob does not matter: OpenStax explains that only the length and the pull of gravity change the time of a swing. And for small swings, how far you pull it hardly matters either. That steadiness is what makes a pendulum good at keeping time.

With L = 1 m, g = 9.81 m/s² and a 5° release, the model's exact period is 2.007 s (the small-swing formula gives 2.006 s), or 29.9 swings per minute. Released at 20°, the period grows only to 2.022 s, in line with OpenStax's note that the period is nearly independent of amplitude below about 15°.

The bob's mass cancels from the equation of motion, θ″ = −(g/L) sin θ, so a heavier bob gives exactly the same period. Only L and g remain.

On the Moon, gravity pulls much more gently.

So the pendulum swings slowly. Each swing takes about five seconds! The clock falls behind.

Choose the Moon. The pendulum is the same, but the Moon's pull is about one sixth of Earth's. Now each swing over and back takes about 4.9 seconds, nearly two and a half times as long, and the pendulum makes only about 12 swings a minute.

The clock still moves its hands a little for every swing, so it runs slow. In the model, when one real hour has passed, the clock shows only about 24 minutes. To fix it, make the pendulum shorter: about 16.5 centimeters long, roughly one sixth of the length, swings in about 2 seconds on the Moon.

At g = 1.62 m/s² the model gives T = 4.939 s, or 12.1 swings per minute: 2.461 times the Earth period, exactly √(9.81/1.62). A clock geared for 2.007 s swings would advance 60 × 2.007/4.939 ≈ 24.4 minutes per real hour, or about 9.75 hours per day, losing roughly 14.25 hours daily.

Keeping the period means keeping L/g fixed, so L must scale with g: 1 m × 1.62/9.81 = 0.165 m. The model confirms T = 2.006 s for a 0.165 m pendulum on the Moon. The bob also moves more slowly: 0.11 m/s at the bottom, against 0.27 m/s on Earth.

Gravity is what pulls the pendulum back down to the middle.

A strong pull makes quick swings. A gentle pull makes slow swings.

When a pendulum is pulled to the side, gravity pulls it back toward the bottom. It overshoots, and gravity pulls it back again. On the Moon that pull is weaker, so the bob speeds up more gently and each swing takes longer.

A shorter pendulum has a shorter path to travel, so it swings faster. That is why a shorter pendulum can make up for weaker gravity. Our model is a simple pendulum: a small weight on a light, stiff rod, with no air or friction unless you add them.

The restoring force is the component of weight along the arc, −mg sin θ; dividing by mass leaves an angular acceleration of −(g/L) sin θ. For small angles this is simple harmonic motion with T = 2π√(L/g), so T scales as g^(−1/2): one sixth of the gravity means √6.06 ≈ 2.46 times the period.

Model limits: a point bob on a weightless rod (a real clock pendulum is a physical pendulum whose period depends on its mass distribution), linear damping if switched on, and no clock mechanism. The model also ignores how a real clock would cope with Moon dust, temperature swings or being carried there.

The first pendulum clock was made more than 350 years ago.

Today, the best clocks use atoms instead. Scientists are even planning a special time for the Moon.

NIST, the U.S. measurement lab, explains that Galileo studied swinging pendulums as early as 1582, and that Christiaan Huygens built the first pendulum clock in 1656. It was wrong by less than a minute a day, better than any clock before it. Later pendulum clocks were steady to a hundredth of a second a day.

Today, the most precise clocks count the vibrations of atoms, not swings. NIST explains that atomic clocks on the Moon would tick about 56 millionths of a second per day faster than on Earth, because of Einstein's ideas about gravity and time. That tiny difference could still upset the timing needed to land spacecraft.

According to NIST's history of timekeeping, Galileo studied pendulum motion from 1582 and sketched a pendulum clock he never built; Huygens made the first one in 1656, with an error under 1 minute a day, later under 10 s. George Graham reached about 1 s a day in 1721 by compensating for temperature, and Siegmund Riefler's nearly free pendulum of 1889 reached a hundredth of a second a day.

The Moon raises a far subtler timing problem. NIST explains that atomic clocks on the lunar surface would tick faster than Earth's by about 56 microseconds per day: in relativity, clocks tick slightly faster where gravity is weaker, and even this small drift could disrupt the timing needed for spacecraft landings. In 2024 NIST physicists Neil Ashby and Bijunath Patla published a relativistic framework for estimating clock rates on the Moon, which NIST describes as a first step toward a lunar coordinate time and a GPS-like navigation system, supporting NASA's Artemis program.