Pendulum Period Calculator (T = 2π√L/g)
A simple pendulum is the textbook idealisation of a small bob hanging from a massless string, swinging under gravity. Under the small-angle approximation (< 15°) the period T depends only on the pendulum length L and the local gravitational acceleration g — not on the bob mass or amplitude. This tool computes the period, frequency and angular frequency, and can reverse-solve for the length needed to hit a target period (e.g. the famous one-metre "seconds pendulum" ticks roughly every two seconds).
Please enter a valid number
Period T
—
seconds / swing
Frequency f
—
hertz (Hz)
Angular ω
—
rad / s
Inverse: length needed for a target period
Length required
—
Uses the small-angle formula T = 2π·√(L/g) (accurate to ~0.1 % for swings under 15°). Large amplitudes require an elliptic-integral correction.
Formula
T = 2π · √(L / g) // period in seconds f = 1 / T // frequency in Hz ω = √(g / L) = 2π / T // angular frequency in rad/s L = g · (T / 2π)² // inverse: required length Standard gravity g₀ = 9.80665 m/s² // BIPM SI Brochure
- · Small-angle approximation: for swings ≤ 15°, T = 2π√(L/g) is accurate to better than 0.5 %. At 60° the error grows to about 7 %; for large amplitudes you need an elliptic-integral correction with the modulus sin(θ/2).
- · Length L is measured from the pivot to the centre of mass of the bob — not to its bottom edge. Real lab pendulums use the "physical pendulum" formula T = 2π√(I/(mgd)); this tool deals only with the idealised point-mass case.
- · Mass-independent: T does not contain m, so swapping a brass bob for a lead one of the same length leaves the period unchanged. That was Galileo's 1602 observation in the Pisa cathedral.
- · Gravity presets: Earth standard g₀ = 9.80665 m/s² (BIPM 1901); the Moon ≈ 1.625 m/s² (NASA Apollo); Mars ≈ 3.711 m/s²; Jupiter's cloud tops ≈ 24.79 m/s². Hong Kong's local g ≈ 9.788 m/s² (within 0.2 % of the standard).
- · Seconds pendulum: a swing-period of T = 2 s (one tick per second on each half-swing) requires L ≈ 0.9939 m at g₀. Christiaan Huygens used it to define the second in the 17th century, and it was once a candidate definition of the metre.
- · Applications: classic grandfather clocks use ~0.9939 m pendulums as their time base; the Foucault pendulum in the Paris Panthéon is 67 m long; seismometers, balance wheels, and Kater pendulums (used to measure g) all descend from this formula.
- · Sources: Halliday, Resnick & Walker, "Fundamentals of Physics", ch. 15 (Oscillations); BIPM SI Brochure, 9th edition.
Frequently asked
Why does the pendulum period not depend on the bob's mass?
In the equation of motion, the restoring force m·g·sinθ and the inertia m·a both contain mass, so the m cancels out: a = g·sinθ. That's why a brass bob and a lead bob of the same length swing at exactly the same rate. Galileo noticed this watching the cathedral lamps swing in Pisa in 1602, and Huygens turned it into the first pendulum clock in 1656.
How long should a pendulum be to tick once per second?
A "seconds pendulum" — one that ticks once per second on each half-swing, meaning a full period T = 2 s — needs L = g·(T/2π)² ≈ 9.80665 × (2/2π)² ≈ 0.9939 m under standard gravity. That is the pendulum length of traditional grandfather clocks. The adjustment screw at the bottom of the bob fine-tunes this effective length when the clock runs fast or slow.
How would the same pendulum behave on the Moon?
Lunar surface gravity is about 1.625 m/s² — roughly one-sixth of Earth's. Since T ∝ 1/√g, the same pendulum on the Moon would swing √(9.80665 / 1.625) ≈ 2.46 times slower. A grandfather clock that ticks once per second on Earth would tick once every 2.46 seconds on the Moon — Apollo astronauts famously did a related experiment dropping a hammer and a feather to verify lunar gravity.
How accurate is the formula at large swing angles?
The error in T = 2π√(L/g) grows with amplitude: roughly 0.05 % at 5°, 0.5 % at 15°, 1.7 % at 30°, 4 % at 45°, 7.3 % at 60°, and 18 % at 90°. The exact period uses the first complete elliptic integral K(sin(θ/2)): T_exact = T_small × (2/π)·K(sin(θ/2)). Precision clocks intentionally keep the swing very small (typically 1°–3°) so the simple formula is accurate enough.
Related tools
Ohm's Law Calculator (V / I / R / P)
Enter any two of voltage, current, resistance, or power — the calculator solves for the other two using V = IR and P = VI.
Speed, Distance & Time Calculator
Enter any two of distance, time and speed to get the third — with km/h, mph, m/s, km, miles, hours and minutes supported.
Density Calculator (mass / volume)
Compute density from mass and volume (ρ = m / V), or solve for the missing variable. Built-in reference table for 19 common substances.
Projectile Motion Calculator
Enter launch speed, angle and height to compute projectile range, peak height and flight time (no air resistance). Pick from Earth, Moon, Mars and more.
Wind Chill Calculator
Compute the wind chill (feels-like temperature) from air temperature and wind speed using the 2001 Environment Canada / US NWS formula, with frostbite risk levels.
Dew Point Calculator
Compute dew point from air temperature and relative humidity using the Magnus formula — handy for HVAC, photography and weather analysis.
Kinetic Energy Calculator (KE = ½ m v²)
Compute kinetic energy KE = ½ m v² with mixed units (kg / g / lb and m/s / km/h / mph) and see the result in joules, kilojoules, food calories, foot-pounds and watt-hours.
Half-Life & Exponential Decay Calculator
Enter any three of initial amount, remaining amount, elapsed time and half-life to solve for the fourth — useful for radioactive decay, drug pharmacokinetics and radiometric dating.
Resistor Color Code Calculator (4 / 5 band)
Pick the colour bands and instantly read the resistance and tolerance — 4-band and 5-band notations supported, with Ω / kΩ / MΩ formatting and a closest E12 / E24 preferred-value check.
GPS Distance Calculator (Haversine)
Enter two latitude/longitude pairs to compute the great-circle distance using the haversine formula (km, miles, nautical miles), with bearing and midpoint.
Solution Dilution Calculator (C₁V₁ = C₂V₂)
Solve any one of C₁, V₁, C₂, V₂ from the dilution equation C₁V₁ = C₂V₂ — a daily lab essential for chemistry, biology and pharmacy work.
Decibel (dB) Sum Calculator
Two 80 dB sound sources do not equal 160 dB. Enter multiple dB values to compute the combined SPL, and subtract background noise to recover the signal alone.
Resistor Parallel / Series Calculator
Enter up to 8 resistor values to see the series total (R₁ + R₂ + …) and the parallel total (1 / Σ(1/Rᵢ)) at the same time.
Wavelength ↔ Frequency Calculator
Convert between electromagnetic wavelength and frequency via c = λf, with the matching spectrum band (radio / microwave / visible / X-ray / γ) and photon energy.
Tank Volume Calculator
Compute the capacity of vertical or horizontal cylindrical, rectangular and spherical tanks, including partial-fill volumes at a given liquid level.
Heat Index Calculator
Enter air temperature and relative humidity to get the apparent temperature (NOAA Rothfusz heat index) and the corresponding heat-stress risk band.
Vehicle Stopping Distance Calculator
Enter speed, reaction time and road friction to estimate reaction, braking and total stopping distance.
Snell's Law Refraction Calculator
Enter the refractive indices of two media and an angle of incidence — get the refraction angle and critical angle from Snell's law (n₁ sin θ₁ = n₂ sin θ₂).
Capacitor Energy Calculator
Enter capacitance (F, mF, µF, nF, pF) and voltage to compute the stored energy (E = ½CV²) and charge (Q = CV) on a capacitor.
Boiling Point at Altitude Calculator
Enter altitude to compute the boiling point of water (°C / °F) and local air pressure using the ICAO standard atmosphere and the Antoine equation — useful for hiking, cooking and high-altitude baking.
Specific Heat (Q = mcΔT) Calculator
Solve Q = m × c × ΔT for any one of heat energy, mass, specific heat capacity or temperature change — with presets for water, aluminium, iron, copper, glass, air and more.
pH and Hydrogen Ion Concentration Calculator
Convert between pH, pOH, hydrogen-ion concentration [H⁺] and hydroxide concentration [OH⁻] — with acid / neutral / alkaline classification.
Ideal Gas Law (PV = nRT) Calculator
Pick the unknown (P, V, n or T), enter the other three and PV = nRT is solved instantly — works in Pa / kPa / atm / bar / mmHg / psi, m³ / L / mL, mol / mmol / kmol and K / °C / °F.