Simple Pendulum Calculator
Calculate the period of a simple pendulum from its length and gravity, or find the length needed for a given period. Enter the known value and get period, frequency and swings per minute.
Find the Period of a Pendulum from Its Length
Simple Pendulum Formula
Where:
- \(T\) — period of one full oscillation (s)
- \(L\) — length of the pendulum, from the pivot to the centre of the bob (m)
- \(g\) — gravitational acceleration (m/s², 9.81 on Earth)
- \(f\) — frequency in hertz (Hz), the number of swings per second
The first formula gives the period from the length; the second is the same equation rearranged to find the length required for a desired period. Notice that the mass of the bob does not appear at all — it has no effect on the period of an ideal pendulum.
Simple Pendulum – Calculation Example
Example: a pendulum with a 1 m string swings on Earth, where \(g = 9.81\ \text{m/s}^2\). What is its period?
\[ T = 2\pi\sqrt{\frac{L}{g}} = 2\pi\sqrt{\frac{1}{9.81}} = 2\pi \times 0.3193 \approx 2.006\ \text{s} \]
Result: the pendulum completes one full back-and-forth swing every 2.006 s, which corresponds to a frequency of \(f = 1/T \approx 0.4985\ \text{Hz}\), or about 29.9 swings per minute.
Working in the opposite direction: to build a pendulum with a period of exactly 2 s on Earth you would need a length of \(L = 9.81\,(2/2\pi)^2 \approx 0.994\ \text{m}\).
How to Use the Calculator
- In the Solve using dropdown, choose Pendulum length (L) if you know the length and want the period, or Oscillation period (T) if you want to find the length needed for a target period.
- Enter the known value: either Pendulum length (L) with its unit (m, cm, ft or in), or Oscillation period (T) in seconds. The calculator automatically shows only the relevant field.
- Optionally enter the Mass of the bob (m) in kg, g or lb. The mass does not change the period; it is only echoed back in the results so you have a complete record of your pendulum.
- Check Gravity (g). The default 9.81 m/s² is standard for Earth; enter 1.62 for the Moon or 3.72 for Mars to see how the same pendulum behaves elsewhere. You can also switch the unit to ft/s².
- Read the results: the period in seconds, the frequency in Hz, the angular frequency \(\omega\), swings per minute, and the length converted into m, cm, ft and in.
What Is a Simple Pendulum?
A simple pendulum is an idealized system: a point mass (the bob) hanging from a weightless, inextensible string attached to a fixed pivot. When the bob is pulled aside and released, it swings back and forth under gravity. Each complete cycle — out to one side, back through the middle, to the other side, and back again — is one oscillation, and its duration is the period \(T\).
The key insight of the pendulum formula is what it does not contain. There is no mass term and no amplitude term. A 10 g bob and a 10 kg bob on a 1 m string swing with exactly the same period, and a small swing and a slightly larger small swing take the same time. This is why pendulums became the timekeeping heart of mechanical clocks for centuries: their tick is remarkably steady as the spring winds down and the amplitude gradually shrinks.
The period grows with the square root of length, so quadrupling the length only doubles the period. It also shrinks with stronger gravity — the same 1 m pendulum that takes about 2 s on Earth would swing in a brisker 4.94 s period only under far weaker gravity: on the Moon, its period stretches to roughly 4.93 s because \(g\) there is just 1.62 m/s². If you want to explore the energy side of the motion — how potential energy at the top of the swing converts into kinetic energy at the bottom — try the Conservation of Energy and Kinetic Energy calculators.
Assumptions and Limitations
- Small amplitudes only. The formula assumes the swing angle stays below roughly 15° from vertical. At 15° the true period is already about 0.5% longer than the formula predicts; at 60° the error grows past 18%. For a rough rule, keep amplitudes small for clock-like accuracy.
- Ideal string and pivot. The string is treated as massless and perfectly flexible, and the pivot as frictionless. In reality, a heavy string, a stiff rod, or pivot friction all shift the period slightly.
- No air resistance or damping. Real pendulums lose a little energy each swing, which mainly reduces the amplitude over time; the period changes very little at first.
- Point-mass bob. \(L\) is measured to the centre of mass of the bob, not to its top or bottom edge.
- Uniform gravity. \(g\) is treated as constant over the pendulum's range of motion. On Earth, \(g\) varies slightly with latitude and altitude (from about 9.78 to 9.83 m/s²), which changes the period of a long pendulum by a measurable amount.
Typical Pendulum Periods on Earth (g = 9.81 m/s²)
| Length L | Period T | Swings per minute |
|---|---|---|
| 0.25 m | 1.00 s | 59.7 |
| 0.5 m | 1.42 s | 42.3 |
| 1 m | 2.01 s | 29.9 |
| 1.5 m | 2.46 s | 24.4 |
| 2 m | 2.84 s | 21.2 |
| 4 m | 4.01 s | 14.9 |
Practical Tips
To design a pendulum clock tick of exactly 1 s per half-swing (the classic "seconds pendulum"), aim for a full period of 2 s, which needs a length of about 0.994 m on Earth. Since the period scales with \(\sqrt{L}\), fine adjustments to the length give even finer control of the timing: shortening the string by 1% shortens the period by only 0.5%.
The pendulum is also a classic tool for measuring local gravity. Time 20 or more full swings with a stopwatch, divide to get \(T\), measure \(L\) carefully to the bob's centre, and rearrange the formula to \(g = 4\pi^2 L / T^2\). Timing many swings instead of one spreads your reaction-time error over the whole set and makes the result far more accurate. For related mechanics problems, see the Atwood Machine and Tension in Ropes calculators.
Frequently Asked Questions
How do I calculate a pendulum's period by hand?
Divide the length in metres by 9.81, take the square root, and multiply by \( 2\pi \) (about 6.283). For a 2 m pendulum: \( 2/9.81 \approx 0.204 \), \( \sqrt{0.204} \approx 0.452 \), and \( 6.283 \times 0.452 \approx 2.84 \) s. If your length is in centimetres or feet, convert to metres first: 1 ft = 0.3048 m, 1 in = 0.0254 m, 1 cm = 0.01 m.
Should I time a full swing or just one side for the period?
The period \( T \) is the time of a full oscillation: from one extreme, through the middle, to the other extreme, and all the way back. If you only time the trip from one side back to the same side, that is the full period; the trip to the opposite side is half of it. A common mistake is timing half-swings and comparing them to the formula, which makes the pendulum seem twice as fast as it is.
Why does my measured period differ slightly from the calculator's result?
Real pendulums deviate from the ideal formula for several reasons: the swing angle may exceed 15°, the string has some mass and stiffness, the pivot has friction, and air resistance slows the bob. Also check how you measured the length — it must go from the pivot to the centre of the bob, not its top or bottom. Small timing errors when using a stopwatch also add up, so time 20+ swings and divide.
Is the formula still accurate for large swing angles?
No — the basic formula assumes small amplitudes. A better approximation adds a correction term: \( T \approx T_0 \left(1 + \frac{\theta_0^2}{16}\right) \), where \( \theta_0 \) is the release angle in radians. At 10° the correction is under 0.2%, at 30° it is about 1.7%, and at 60° the true period is roughly 18% longer than the simple formula predicts. Keep swings under about 15° for clock-level accuracy.
What is the difference between a simple pendulum and a real pendulum on a rod?
A simple pendulum is an idealization: a point mass on a massless string. A real pendulum swinging on a rigid rod with distributed mass is a physical pendulum, and its period is \( T = 2\pi\sqrt{I/(mgd)} \), where \( I \) is the moment of inertia and \( d \) the distance from the pivot to the centre of mass. Grandfather clocks use physical pendulums, which is why their effective lengths differ from the simple-formula prediction.
Why does a heavier bob not swing slower?
Gravity pulls harder on a heavier bob, but a heavier bob is also harder to accelerate — the two effects cancel exactly. Doubling the mass doubles both the gravitational force and the inertia, so their ratio, and therefore the period, stays the same. This is only true in the absence of air resistance: with very light bobs like foam or paper, air drag becomes noticeable and the motion slows and dies out faster.
How do I use this pendulum to measure gravity myself?
Rearrange the formula to \( g = 4\pi^2 L / T^2 \). Hang a dense, compact bob on a thin string at least 1 m long, measure \( L \) from the pivot to the bob's centre with a tape measure, then time 30–50 full swings with a stopwatch and divide to find \( T \). Timing many swings spreads your reaction-time error, and you should get \( g \) within 1–2% of 9.81 m/s². Keep the release angle under 10°.