Pendulum Period Calculator
Calculate the oscillation period and frequency of a simple pendulum from its length. Compare pendulum behavior on Earth, the Moon, Mars, and Jupiter, and animate the swing in real time. No signup, runs entirely in your browser.
⏱ 6 min read · Complete guide below
How to Use the Pendulum Calculator
- 1Enter the pendulum length in meters — measured from the pivot to the center of the bob.
- 2Choose the gravity preset — Earth, Moon, Mars, or Jupiter — or enter a custom g.
- 3Read the period and frequency from T = 2π√(L/g), and watch the animated swing update to match.
- 4Keep swing angles small — the formula holds within 1% up to about 15° of amplitude.
Worked Example: Designing a Grandfather Clock Pendulum
A traditional grandfather clock ticks once per second, meaning each half-swing takes 1 s and the full period is T = 2 s. Rearranging the formula to solve for length: L = gT² / 4π² = 9.81 × 4 / 39.48 ≈ 0.994 m. That is why grandfather clocks are the height they are — the physics fixes the pendulum at almost exactly one meter, and the cabinet is built around it.
Take that same 0.994 m pendulum to the Moon, where g = 1.62 m/s², and the period becomes T = 2π√(0.994 / 1.62) ≈ 4.9 s — the clock would run almost 2.5× slow. Notice mass never entered either calculation: a brass bob and a lead bob of the same length keep identical time. Only two things set the period, length and local gravity, which is exactly what the calculator's two inputs reflect.
Why Only Length and Gravity Matter
The pendulum formula T = 2π√(L/g) contains a genuinely surprising fact: the period depends on only two things — the length of the pendulum and the local strength of gravity — and on nothing else. The mass of the bob does not appear, and neither does the amplitude, as long as the swing is small. Galileo noticed this “isochronism” as a teenager reportedly watching a lamp swing in a cathedral, and it is the property that made the pendulum the heart of accurate timekeeping for nearly three centuries. Mass cancels out because gravity pulls harder on a heavier bob but that same heavier bob resists acceleration in exactly equal measure — the two effects offset perfectly, leaving the timing unchanged.
The Small-Angle Approximation
There is a catch worth knowing. The neat formula assumes the swing angle is small, because it relies on the approximation that sin(θ) is nearly equal to θ for small angles. In practice this holds beautifully up to about 15°, where the error is under 1%. Push the amplitude much higher and the real period grows slightly longer than the formula predicts, because a wide swing spends proportionally more time near the turning points. This is why precision clock pendulums are designed to swing through only a narrow arc — it keeps them firmly in the regime where the simple formula, and this calculator, are accurate.
Pendulums as a Tool for Measuring Gravity
Because the period ties length and gravity together, a pendulum can be run in reverse to measure g. Rearranging the formula gives g = 4π²L/T², so timing the swing of a pendulum of known length yields the local gravitational acceleration — a classic and remarkably accurate physics-lab experiment. This is also why the calculator's planet presets are so illustrative: the very same pendulum keeps different time on the Moon, Mars, and Jupiter purely because g differs. Switching the gravity preset and watching the animated swing speed change makes the abstract formula tangible, showing at a glance how a weaker or stronger gravitational field stretches or compresses the rhythm of the swing.
Pendulum Physics Tips
Grandfather clocks
A traditional grandfather clock uses a 1-meter pendulum tuned to a 2-second period (1 second per half-swing). This means the tick-tock marks whole seconds precisely.
Measuring g with a pendulum
Rearrange the formula: g = 4π²L/T². Measure the period of a pendulum of known length to estimate local gravitational acceleration — a classic physics lab method.
Double the length
To double the period, you need to quadruple the length, because T ∝ √L. A 4-meter pendulum has exactly twice the period of a 1-meter pendulum.
Temperature sensitivity
Real clock pendulums expand slightly with heat, lengthening the period. Precision clocks use invar (invariant steel) pendulums to minimize this thermal drift.
Frequently Asked Questions
What is the formula for pendulum period?
T = 2π√(L/g), where T is the period in seconds, L is the pendulum length in meters, and g is gravitational acceleration (9.81 m/s² on Earth). This formula is valid for small angles (less than ~15°).
Why does mass not affect the period?
The gravitational force on a pendulum bob is proportional to its mass, but so is its inertia. These cancel out exactly, making the period independent of mass — a result first observed by Galileo.
What length gives a 1-second period?
On Earth with g = 9.81 m/s², a pendulum with T = 1 s has length L = g/(4π²) ≈ 0.2485 m (about 24.9 cm). A 2-second period (grandfather clock standard) requires L ≈ 0.994 m.
How would a pendulum behave on the Moon?
On the Moon (g = 1.62 m/s²), the same 1-meter pendulum would have a period of 2π√(1/1.62) ≈ 4.94 seconds — about 2.46 times slower than on Earth.
What is the small angle approximation?
The formula T = 2π√(L/g) assumes the angle is small enough that sin(θ) ≈ θ. For angles up to ~15° the error is under 1%. For larger amplitudes, the period increases slightly beyond this formula's prediction.
Is my data stored?
No. All calculations run locally in your browser. No data is sent to any server.
How do I measure gravity (g) with a pendulum?
Rearrange the period formula to solve for gravity: g = 4π²L/T². Set up a pendulum of a known length, time how long a number of full swings take (timing many swings and dividing gives a more accurate period), and plug the length and measured period into the formula. This classic experiment reliably produces a value close to 9.81 m/s² on Earth and is a staple of introductory physics labs.
Does a longer pendulum swing faster or slower?
Slower. The period is proportional to the square root of the length, so a longer pendulum takes more time per swing. Because of the square-root relationship, to double the period you must quadruple the length: a 4-metre pendulum has exactly twice the period of a 1-metre one. This is why tall grandfather clocks, with their roughly one-metre pendulums, tick at a stately two-second period.
What happens to the period at large swing angles?
The simple formula T = 2π√(L/g) assumes small angles. As the amplitude grows beyond about 15°, the real period becomes slightly longer than the formula predicts, because a wide swing spends more time near its turning points. The effect is small for moderate angles but grows for very large ones. For accurate timekeeping, real pendulum clocks deliberately use a narrow swing to stay within the small-angle regime.
Why does a pendulum keep the same time regardless of the bob's weight?
Because gravity and inertia scale together with mass and cancel out. A heavier bob feels a proportionally stronger gravitational pull, but it also resists changes in motion proportionally more, so the two effects offset exactly and the period is unchanged. This mass-independence, first noted by Galileo, is what made the pendulum such a reliable basis for clocks — you can swap the bob without retuning the timing.