How this instrument works
The period is the time for one complete swing — out, back, and returned to the starting point, not just one pass through the bottom. For a simple pendulum swinging through a small angle, that time depends on only two things: how long the pendulum is and how strongly gravity pulls where it hangs. Neither the mass of the bob nor the width of the swing enters the formula, which is why a lead ball and a wooden bead on strings of identical length keep exactly the same beat.
The square root comes from the physics of restoring force. Displace the bob by a small angle and gravity pulls it back toward vertical with a force proportional to that angle — the signature of simple harmonic motion, whose angular frequency works out to √(g ⁄ L). Period is 2π divided by that frequency, which rearranges to T = 2π√(L ⁄ g). The approximation sin θ ≈ θ is what makes this tidy, and it holds closely enough for swings under roughly 20 degrees, which is why the formula quietly assumes you are not pushing the pendulum wildly off vertical.
Because period scales with the square root of length rather than length itself, doubling the length does not double the period — it stretches it by √2, about 41 percent longer. That single relationship is what let 17th-century clockmakers regulate a pendulum clock with a small adjustment nut under the bob instead of rebuilding the mechanism. The formula also assumes a rigid, weightless string and a bob small enough to treat as a point; a swinging rod or a person on a swing set needs its moment of inertia instead of length alone, since the mass is spread out rather than concentrated at the end.
- Enter the Pendulum length — the distance from the pivot to the bob's center of mass — in meters, or switch the unit menu to centimeters for small models.
- No separate gravity entry is needed: the instrument fixes g at standard gravity, 9.80665 m/s², the same constant built into the formula.
- Read the Oscillation period in seconds: the time for one full swing, out and back, not just one pass through the bottom of the arc.
- To hit a target period rather than guess, adjust the Pendulum length — doubling the desired period needs four times the length, since period only tracks length's square root.
Worked example — a 1-meter pendulum's swing
Take a pendulum exactly 1 meter long, measured from the pivot to the bob's center of mass. With standard gravity g = 9.80665 m/s², the formula gives T = 2π√(1 ⁄ 9.80665) = 2π × 0.319330 = 2.00640929259 seconds, which is what the Oscillation period field reports to six decimal places. Timed with a stopwatch, that reads as about 2.006 seconds for one full out-and-back swing.
That length is not far from the historic 'seconds pendulum,' the reference clockmakers and early metrologists used for centuries, defined so its full period equals exactly 2 seconds. Solving the same formula backward for length, T = 2 s requires L = g(T ⁄ 2π)² ≈ 0.9936 meters at standard gravity — a few centimeters shorter than the 1-meter pendulum above, which is why 18th-century proposals to define the meter itself from a seconds pendulum never quite lined up with the meter used today.
Questions
Does the pendulum's mass or bob material change the period?
No. The restoring force on the bob and its inertia both scale with mass, so mass cancels out of the equation entirely — a heavy iron bob and a light wooden one on strings of identical length keep exactly the same beat. What does matter is where the mass sits, since length is measured to the bob's center of mass; a bob with an unusually long shape shifts that effective length slightly.
Why doesn't the formula include the swing amplitude?
Because it assumes a small angle, under roughly 20 degrees, where sin θ ≈ θ closely enough that the motion behaves as simple harmonic and amplitude drops out of the math. Push the pendulum through a wide arc, say 90 degrees, and the true period runs about 18 percent longer than this formula predicts, which is why clock pendulums are kept swinging through a narrow arc on purpose.
How much does quadrupling the pendulum's length change the period?
It exactly doubles it, because period scales with the square root of length, not length itself. A 1-meter pendulum has a period near 2.006 seconds; stretch the same pendulum to 4 meters and it becomes about 4.013 seconds — precisely double, not quadruple. That square-root relationship is why a fast pendulum clock gets corrected with a small nut turn rather than a length guess.
What about very short pendulums, like those in metronomes?
A pendulum about 9.93 centimeters long has a period of roughly 0.632 seconds, a touch faster than one swing per second, which is close to the rate mechanical metronomes and some older seismometers use, since a period near half a second to a second fits a small case while staying slow enough to read by eye. Shrinking the length further keeps shortening the period, but only by the square root, so a case half that size does not swing twice as fast.
Does gravity change with location, and does that affect the result?
Yes, slightly. Measured gravity runs from about 9.780 m/s² at the equator to 9.832 m/s² at the poles, with altitude shifting it further still, because both Earth's rotation and distance from its center affect the pull. This instrument fixes g at the international standard value, 9.80665 m/s², the engineering reference rather than a reading at any one spot, so a real pendulum clock built near the equator runs measurably slower than one built near a pole.
What is the difference between period and frequency for a pendulum?
Period is the time for one full swing, in seconds; frequency is how many swings happen each second, in hertz, and the two are exact reciprocals: f = 1 ⁄ T. A 1-meter pendulum's 2.006-second period corresponds to a frequency of about 0.498 Hz — just under half an oscillation per second, or one swing roughly every two seconds. Which one to use is mostly about the tool in hand: a stopwatch reads naturally in seconds, a counted window of oscillations reads naturally in hertz.