Why length and gravity set the period, and mass does not
A pendulum keeps time because gravity supplies a restoring torque that grows with displacement. Pull the bob aside by an angle θ and the component of weight along the arc is mg·sinθ. For small angles sinθ ≈ θ, so the restoring force is proportional to the displacement — the same condition that makes a spring oscillate — and the motion is simple harmonic.
Write the equation of motion and the mass cancels. The restoring torque is proportional to m, and so is the inertia resisting it, so m divides out and never reappears. A lead bob and a cork bob on strings of equal length keep identical time in a vacuum. That cancellation is the same one behind the equivalence principle, and it is why a pendulum is a gravimeter rather than a scale.
What remains is ω = √(g/L) and therefore T = 2π√(L/g). The square root matters: to double the period you must quadruple the length. A 1 m pendulum beats about 2.006 s; a 4 m pendulum beats about 4.013 s, not 8. That is why long-case clocks are tall but not absurdly so, and why halving a pendulum only speeds it by a factor of 1.414.
The most-quoted case is the seconds pendulum, which takes one second for each one-way swing and therefore has a period of two seconds. Its length is L = gT²/4π² = g/π², which at standard gravity is 0.993621 m — famously just under a metre, and the near-coincidence that led the French Academy in 1791 to consider defining the metre as the seconds-pendulum length before choosing the meridian instead.
The small-angle approximation and what it costs you
The clean formula depends on replacing sinθ by θ, and that substitution is only exact at zero. The true period of a pendulum depends on amplitude, and it always comes out longer than the small-angle value, because sinθ < θ for every positive angle, so the real restoring torque is weaker than the linear model assumes.
The exact period involves a complete elliptic integral of the first kind, but the series expansion is more useful: T = T₀(1 + θ₀²/16 + 11θ₀⁴/3072 + …), with the amplitude in radians. This calculator uses those two correction terms. Compare the truncated series with the exact elliptic-integral value T/T₀ = (2/π)K(sin(θ₀/2)) and it is good to better than one part in ten thousand out to about 50° (1.049673 against 1.049781), and to roughly three parts in ten thousand at 60° (1.072845 against 1.073182). Past 90° you need the elliptic integral itself.
Put numbers on it. At 1° the correction is 0.0019% — about 1.6 seconds a day, which a precision clock cares about. At 5° it is 0.0476%, roughly 41 seconds a day. At 20° it is 0.767%, more than eleven minutes a day. At 45° the series gives 3.99%. Christiaan Huygens solved this in 1656–1673 by hanging the pendulum between cycloidal cheeks, which makes the path isochronous at any amplitude; modern clocks instead keep the amplitude small and constant, which is easier.
Two other departures from the ideal matter in practice. A real pendulum is a rigid body, not a point mass, so you should use the equivalent length L_eq = I/(m·d), where I is the moment of inertia about the pivot and d is the pivot-to-centre-of-mass distance. And a real bob loses energy to air drag and pivot friction, which decays the amplitude — helpful for timekeeping since the amplitude error falls with it, unhelpful because an escapement must then keep feeding energy in.
Worked example: measuring g in a school laboratory
You hang a brass bob on a thread. From the clamp to the centre of the bob measures 0.9500 m. You start a stopwatch as the bob passes the lowest point, count 50 complete swings, and stop at 97.72 s. The amplitude at the start is about 6°.
- Get the period. T = 97.72 s ÷ 50 = 1.9544 s. Timing many swings rather than one divides your reaction-time error by 50.
- Square it. T² = 1.9544² = 3.8197 s².
- Apply the formula. g = 4π²L / T² = 4 × 9.8696044 × 0.9500 ÷ 3.8197 = 37.5045 ÷ 3.8197 = 9.8187 m/s².
- Correct for amplitude. At 6° = 0.10472 rad, the factor is 1 + 0.10472²/16 = 1.000685. Your measured period is 0.0685% too long, so the small-angle period is 1.9544 ÷ 1.000685 = 1.95306 s, giving T² = 3.81444 and g = 37.5045 ÷ 3.81444 = 9.8322 m/s².
- Judge the result. That is within 0.3% of standard gravity, which is a good undergraduate result. Note that ignoring the amplitude correction pushed the answer down by 0.013 m/s², because a slow pendulum masquerades as weak gravity.
The dominant uncertainty here is the length, not the time. A 1 mm error in L out of 950 mm is 0.105%, which propagates directly into g. Locating the centre of mass of the bob to better than a millimetre is the hard part of the experiment.
Reading the result, and what counts as a sensible value
If you are solving for period, the answer should scale as the square root of length: check that quadrupling L doubles T. If it does not, you have mixed units somewhere — entering the length in centimetres while the formula expects metres inflates the period by a factor of 10.
If you are solving for local gravity, the plausible band is narrow. Earth-surface values run from about 9.764 m/s² on high equatorial mountains to about 9.834 m/s² at the poles, with sea-level values near 9.780 at the equator and 9.832 at the pole. Standard gravity, 9.80665 m/s², is a defined constant rather than a measurement, fixed by the CGPM in 1901 and roughly matching sea level at 45° latitude. Anything outside 9.70–9.90 points to a measurement fault rather than a geophysical discovery.
If you are solving for length, remember that the answer is the equivalent length to the centre of oscillation. For a light thread and a compact bob it is close enough to the pivot-to-bob-centre distance, but for a clock pendulum with a heavy rod you must account for the rod's own inertia, which typically makes the equivalent length a little shorter than the geometric one.
Pendulum length against period at standard gravity
| Length | Period (s) | Frequency (Hz) | Beats per minute |
|---|---|---|---|
| 0.0621 m | 0.5000 | 2.0000 | 240.0 |
| 0.100 m | 0.6345 | 1.5761 | 189.1 |
| 0.2484 m | 1.0000 | 1.0000 | 120.0 |
| 0.500 m | 1.4187 | 0.7049 | 84.6 |
| 0.9936 m | 2.0000 | 0.5000 | 60.0 |
| 1.000 m | 2.0064 | 0.4984 | 59.8 |
| 2.000 m | 2.8375 | 0.3524 | 42.3 |
| 3.9745 m | 4.0000 | 0.2500 | 30.0 |
The 0.9936 m row is the seconds pendulum used in longcase clocks; the 0.2484 m row is the quarter-seconds pendulum found in mantel clocks.
Pitfalls that spoil a pendulum measurement
- Counting half-swings as periods. A period is one complete there-and-back cycle. Counting each pass through the bottom doubles your count and halves your period, which quadruples the apparent gravity.
- Measuring to the top or bottom of the bob. The length runs from the pivot to the bob's centre of mass. On a 25 mm sphere that is a 12.5 mm error, worth 1.3% in g on a 1 m pendulum.
- Swinging too wide. Amplitudes above about 10° make the small-angle formula measurably wrong; use the corrected period this calculator reports, or start the swing smaller.
- Letting the bob swing in a cone. A conical or elliptical path is a different problem with a different period. Release the bob from rest in a plane and check that it stays there.
- Timing a single swing. Human reaction time is roughly 0.2 s. Over one 2 s period that is 10%; over fifty periods it is 0.2%.
- Treating a rigid rod as a simple pendulum. A uniform rod pivoted at one end has an equivalent length of two-thirds its physical length, so it swings faster than a bob on a string of the same size.
Where this fits among other oscillators
The pendulum is one of two canonical simple harmonic oscillators, and the other is the mass on a spring. Their formulas are the same shape: T = 2π√(m/k) for a spring, T = 2π√(L/g) for a pendulum. In both cases the period is two pi times the square root of an inertia divided by a restoring stiffness — a pattern you can reuse in spring problems without rederiving anything.
Because the period depends on g alone once the length is fixed, pendulums were the primary instrument of geodesy for two centuries. Kater's reversible pendulum, introduced in 1817, measured absolute gravity to a few parts per million by exploiting the fact that a rigid body has two pivot points with the same period; modern absolute gravimeters drop a corner-cube in a vacuum instead, but the pendulum method is still the one you can build on a bench.
The pendulum also shows how a linear approximation behaves as it fails. Push the amplitude towards 180° and the period diverges logarithmically — a pendulum balanced exactly at the top never comes down. That is the same class of behaviour that separates a nearly-linear system from a chaotic one, and a driven double pendulum is one of the standard demonstrations of deterministic chaos.
If you are working through an oscillation problem set, the energy bookkeeping usually comes next: at the extremes the bob holds only potential energy mgL(1 − cosθ₀), and at the bottom only kinetic energy. Timing questions then feed into wave frequency and wavelength work, while the acoustic version of a resonating system appears in sound level and Doppler problems, where the same frequency you compute here becomes the source frequency.
Key terms
- Period
- The time for one complete oscillation — out to one side, across to the other, and back. A pendulum described as beating seconds has a period of two seconds.
- Amplitude
- The maximum angular displacement from vertical, measured on one side. Because the correction depends on the square of the amplitude, halving the swing quarters the timing error.
- Equivalent length
- For a rigid pendulum, the length of a simple pendulum with the same period: I divided by the product of mass and pivot-to-centre-of-mass distance.
- Isochronism
- The property of keeping the same period regardless of amplitude. A circular-arc pendulum is only approximately isochronous; a cycloidal one is exactly so.