Pendulum Period Calculator
Enter any two of the three variables — period, string length, and gravitational acceleration — and this calculator solves for the missing one, showing the formula, substitution, and working.
Pendulum Period Calculator Tool
Fill in exactly 2 of the 3 fields below and leave the other blank — the calculator solves for the missing value automatically.
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Step-by-Step Solution
How This Calculator Works
A simple pendulum is a mass (the "bob") hanging from a fixed point by a string or rod that is free to swing back and forth. Its period — the time for one complete swing out and back — depends only on the length of the string and the strength of gravity, not on the mass of the bob.
Enter any two of the three variables (period, length, gravity) and leave the third blank. The calculator rearranges T = 2π√(L/g) to solve for the missing value, showing the algebra and arithmetic step by step.
The Formula
T is the period in seconds, L is the length from pivot to the bob's centre of mass in metres, and g is local gravitational acceleration in m/s². This is the small-angle approximation — accurate to within 1% for swings under about 23°.
Rearranged Forms
The calculator uses whichever rearrangement matches the variable you left blank.
What Affects a Pendulum's Period
- String length. Doubling the length increases the period by a factor of √2 (about 41%). Period scales with the square root of length, not linearly.
- Gravity. On the Moon (g ≈ 1.6 m/s²), a pendulum swings about 2.5× slower than on Earth. Use the planet dropdown to see this instantly.
- Amplitude (at large angles). The formula assumes small swings. Beyond about 15°, the real period is slightly longer than the formula predicts — the discrepancy grows with amplitude.
Mass does not appear in the formula. A 1 kg bob and a 100 kg bob on identical strings have exactly the same period — the gravitational force and the inertia both scale with mass, so mass cancels.
Real-World Pendulum Periods
These values are calculated from T = 2π√(L/g) using g = 9.807 m/s² (Earth):
The Small-Angle Approximation
The formula T = 2π√(L/g) is derived by approximating sin θ ≈ θ, which is only valid for small angles measured in radians. As the swing angle increases, the true period gets progressively longer:
| Amplitude | Period Error |
|---|---|
| 5° | +0.05% |
| 15° | +0.43% |
| 23° | +1.0% |
| 30° | +1.7% |
| 45° | +4.0% |
| 60° | +7.3% |
| 90° | +18% |
For most classroom and real-world pendulums, the amplitude is well under 15° and the formula is effectively exact. The Pendulum Simulation uses the full non-linear equation, so you can see the divergence at large angles for yourself.
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Frequently Asked Questions
What is the formula for the period of a simple pendulum?
The period of a simple pendulum is T = 2π√(L/g), where L is the length of the string (measured to the centre of mass of the bob) and g is the local gravitational acceleration. This formula assumes small swing angles — typically under about 15°.
Does the mass of the pendulum bob affect its period?
No. The mass of the bob cancels out completely in the derivation. A heavy lead bob and a light wooden bob on strings of the same length, swinging through the same small angle, have exactly the same period. This is a consequence of the equivalence of gravitational and inertial mass.
Why does the formula only work for small angles?
The derivation assumes sin θ ≈ θ (in radians), which is only accurate for small angles. At larger amplitudes — above about 15° — the true period is longer than T = 2π√(L/g) predicts. At 30° the error is about 1.7%; at 60° it exceeds 7%. The exact solution involves an elliptic integral with no closed-form expression.
How do I find g using a pendulum?
Measure the string length L and time several complete swings to get an accurate period T. Then rearrange the formula to g = 4π²L/T². This is one of the classic physics experiments for determining local gravitational acceleration, and this calculator's third variable does exactly that calculation for you.
What counts as the "length" of a pendulum?
The length L is the distance from the pivot point to the centre of mass of the bob — not to the bottom of the bob. For a small, dense bob on a long string, the difference is negligible. For a large bob or a physical (compound) pendulum, the effective length must account for the mass distribution.
Why is a grandfather clock pendulum exactly 1 metre long?
A 1-metre pendulum on Earth has a period very close to 2 seconds (one second per half-swing, or "tick"). This convenient relationship made it the standard for mechanical clocks. In fact, the metre was once proposed to be defined as the length of a seconds pendulum — though that idea was ultimately abandoned because g varies slightly with location.
The Scientists Behind the Pendulum
Who Built the First Pendulum Clock
Christiaan Huygens (1629–1695)
Huygens invented the pendulum clock in 1656 and derived the formula T = 2π√(L/g), turning Galileo's pendulum observations into the most accurate timekeeper for nearly 300 years.
Full biography coming soonWho Discovered Pendulum Isochronism
Galileo Galilei (1564–1642)
Around 1602, Galileo observed that a pendulum's period is nearly independent of its amplitude — a property called isochronism — laying the groundwork for pendulum-based timekeeping.
Read his full biography →