Pendulum Calculator
Calculate the period, frequency, and length of a simple pendulum. Find how pendulum length and gravity affect oscillation speed.
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How to use this calculator
T = period (seconds per oscillation), L = length of pendulum (m), g = gravitational acceleration (m/s²). The period is independent of mass and amplitude (for small angles < 15°).
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Enter the pendulum length in metres to calculate its period and frequency.
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Or enter a target period (in seconds) and the calculator finds the required length.
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Change the gravity value to simulate other planets (Moon = 1.62, Mars = 3.72).
Frequently asked questions
Does "period" mean one swing or a round trip?
A full cycle: out and back to the starting point. The formula T = 2π√(L ÷ g) gives that time, and the result also lists frequency in hertz, angular frequency in rad/s and oscillations per minute. A 1 m pendulum has T = 2.0061 s and 29.91 oscillations per minute, but passes the bottom of its swing twice per period.
How do I find the length for a given period?
Put the period in the T field and 0 in the length field. The tool inverts the formula to L = g (T ÷ 2π)². A 2 s period needs 0.9940 m and a 1 s period needs 0.2485 m at g = 9.81. If both fields are non-zero, the period wins and the length you typed is ignored.
Why is a 1-metre pendulum not exactly 2 seconds?
Because 2π√(1 ÷ 9.81) is 2.0061 s, a bit over 2. The "seconds pendulum" with a 2 s period is 0.9940 m on this model. The difference is small, but it is why a pendulum clock is regulated by raising or lowering its bob.
Do mass and amplitude change the answer?
Mass does not appear in the formula. Amplitude does matter in reality, though the tool ignores it. For a swing of angle θ in radians, the true period is longer than T by roughly θ² ÷ 16 as a fraction: about 0.4% at 15° and about 1.7% at 30°. Beyond about 20° use a correction.
How much does local gravity matter?
Slightly on Earth. Gravity is roughly 9.78 m/s² at the equator and 9.83 m/s² near the poles; for a 1 m pendulum that moves the period from 2.0091 s to 2.0040 s. On the Moon (1.62) the same length takes 4.9365 s, and on Mars (3.72) 3.2577 s.
What error messages can appear?
Gravity at or below zero gives "Gravity must be positive". Having 0 in both length and period gives "Enter either length or period". Negative lengths or periods are refused with a "must be positive" note.
Does it model a real swinging rod, a spring or a double pendulum?
No. It is the ideal simple pendulum: a point mass on a weightless, non-stretching string, swinging through a small angle with no friction. A physical pendulum (a rod, a clock bob with a heavy rod) has its period set by its moment of inertia, and a spring-mass system uses a different formula.
Pendulum calculator — period, frequency and length of a simple pendulum
Why only length and gravity matter
For small swings, T = 2π√(L ÷ g). The period grows with the square root of the length, so quadrupling the string only doubles the period, and it shrinks as gravity gets stronger. The bob’s weight cancels out of the equation, which is why a light and a heavy bob on the same string keep the same time. Gravity is the only environmental input, so the tool exposes it as an editable field instead of fixing it at Earth’s value.
Worked example: from length to period
A 1 m pendulum at 9.81 m/s² gives T = 2.0061 s, f = 0.4985 Hz, ω = 3.1321 rad/s and 29.91 oscillations per minute. Shortening it to 0.25 m cuts the period to 1.0030 s, nearly exactly half, since a quarter of the length means a square root factor of one half. Going the other way, the 4 m pendulum in the table takes 4.0121 s, roughly double the 1 m figure, which confirms the square-root relationship.
Worked example: from period to length
To build a pendulum with a 2 s round trip, type 2 in the period field and 0 for length. The tool gives 0.9940 m. For a 1 s period you need 0.2485 m, and the printed frequency is exactly 1.0000 Hz. The result is a useful rule of thumb for timing demonstrations, since a pendulum a little under a quarter of a metre long swings once per second.
Length versus period at g = 9.81
The tool’s results for a range of lengths on Earth.
| Length | Period | Oscillations per minute |
|---|---|---|
| 0.25 m | 1.0030 s | 59.82 |
| 0.5 m | 1.4185 s | 42.30 |
| 1 m | 2.0061 s | 29.91 |
| 2.5 m | 3.1719 s | 18.92 |
| 4 m | 4.0121 s | 14.95 |
The same 1 m pendulum on other bodies
Setting gravity to 1.62 m/s² produces a period of 4.9365 s, about two and a half times slower than on Earth. At 3.72 m/s² the period is 3.2577 s. Measuring a pendulum’s period is therefore a way of estimating g: rearranged, g = 4π²L ÷ T². With the 1 m example, T = 2.0061 s returns g = 9.81 m/s², and a timing error of 1% in T produces roughly a 2% error in g because of the square.
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