Pendulum Calculator
Calculate the period and frequency of a simple pendulum using the formula T = 2π√(L/g). Enter the pendulum length and gravitational acceleration to find the oscillation period, frequency, and angular frequency. See also our Frequency Calculator and Spring Constant Calculator.
How to Calculate Pendulum Period
A simple pendulum consists of a mass (called a bob) suspended from a fixed point by a string or rod of negligible mass. When displaced from its equilibrium position and released, it swings back and forth in a periodic motion. The study of pendulums dates back to Galileo Galilei around 1602, who observed that the period of a pendulum is nearly independent of the amplitude of its swing — a property called isochronism.
To calculate the period of a simple pendulum, use the formula T = 2π√(L/g), where T is the period (time for one complete oscillation), L is the length from the pivot to the center of mass of the bob, and g is the local gravitational acceleration. This formula is valid for small angles of oscillation (typically less than about 15°), where the small-angle approximation sin(θ) ≈ θ holds.
The remarkable feature of this formula is that the period depends only on the length and gravity — not on the mass of the bob or the amplitude of the swing (for small angles). This made pendulums ideal for timekeeping, and pendulum clocks dominated precision timekeeping from their invention by Christiaan Huygens in 1656 until the development of quartz clocks in the 1930s. A pendulum with a period of exactly 2 seconds (1 second per swing) requires a length of approximately 0.994 meters.
For larger amplitudes, the period increases slightly. The exact period involves an elliptic integral, but a good approximation is T ≈ 2π√(L/g) × (1 + θ₀²/16), where θ₀ is the maximum angle in radians. At 30°, the period is about 1.7% longer than the small-angle prediction; at 90°, it's about 18% longer.
Pendulum Formula
Simple Pendulum Period:
T = 2π√(L/g)
Frequency:
f = 1/T = (1/2π)√(g/L)
Angular Frequency:
ω = 2πf = √(g/L)
Length for desired period:
L = g × T² / (4π²)
Large-angle correction:
T ≈ 2π√(L/g) × (1 + θ₀²/16 + 11θ₀⁴/3072 + ...)
Physical (compound) pendulum:
T = 2π√(I/(mgh))
I = moment of inertia, h = distance to center of mass
Example Calculation
A pendulum has a length of 1 meter on Earth (g = 9.81 m/s²). Calculate its period and frequency:
Given: L = 1 m, g = 9.81 m/s²
T = 2π√(L/g) = 2π√(1/9.81) = 2π√(0.10194)
T = 2π × 0.31928 = 2.0064 s
Frequency: f = 1/T = 1/2.0064 = 0.4984 Hz
Angular frequency: ω = 2πf = 3.1321 rad/s
On the Moon (g = 1.62 m/s²):
T = 2π√(1/1.62) = 2π × 0.7857 = 4.936 s
The same pendulum swings 2.46× slower on the Moon!
Pendulum Period Reference Table
| Length (m) | Period (s) | Frequency (Hz) |
|---|---|---|
| 0.1 m | 0.6347 s | 1.5756 Hz |
| 0.25 m | 1.0036 s | 0.9964 Hz |
| 0.5 m | 1.4192 s | 0.7046 Hz |
| 0.75 m | 1.7381 s | 0.5753 Hz |
| 1.0 m | 2.0064 s | 0.4984 Hz |
| 1.5 m | 2.4572 s | 0.4070 Hz |
| 2.0 m | 2.8378 s | 0.3524 Hz |
| 3.0 m | 3.4749 s | 0.2878 Hz |
| 5.0 m | 4.4880 s | 0.2228 Hz |
| 10.0 m | 6.3467 s | 0.1576 Hz |
Frequently Asked Questions
What is a simple pendulum?
A simple pendulum is an idealized model consisting of a point mass (bob) suspended from a fixed point by a massless, inextensible string. It swings freely under gravity in a vertical plane. Real pendulums approximate this model when the bob is much heavier than the string and the string length is much greater than the bob's diameter. The simple pendulum is one of the most fundamental systems in physics, demonstrating periodic motion, energy conservation, and the relationship between length and time.
Does the mass of the bob affect the period?
No. The period of a simple pendulum is independent of the mass of the bob. This is because the gravitational force (which drives the motion) and the inertia (which resists acceleration) both depend on mass in the same way — they cancel out in the equation of motion. A 1 kg bob and a 100 kg bob on strings of the same length will swing with identical periods. This is analogous to how all objects fall at the same rate in free fall, regardless of mass.
Why do pendulum clocks lose time in summer?
Pendulum clocks can lose time in warmer weather because thermal expansion lengthens the pendulum rod. Since T = 2π√(L/g), a longer pendulum has a longer period, meaning each swing takes more time and the clock runs slow. A steel rod expands by about 12 parts per million per degree Celsius. For a 1-meter pendulum, a 10°C temperature increase lengthens it by 0.12 mm, slowing the clock by about 0.5 seconds per day. Precision clocks use compensating materials like invar or temperature-controlled environments.
What is the difference between a simple and compound pendulum?
A simple pendulum has all its mass concentrated at a single point (the bob). A compound (physical) pendulum is any rigid body that can swing about a fixed pivot point. Its period is T = 2π√(I/(mgh)), where I is the moment of inertia about the pivot, m is the total mass, and h is the distance from the pivot to the center of mass. A compound pendulum always has a shorter period than a simple pendulum of the same length because its mass is distributed along its length.
How accurate is the small-angle approximation?
The small-angle formula T = 2π√(L/g) is remarkably accurate for modest amplitudes. At 5° amplitude, the error is only 0.05%. At 10°, it's 0.19%. At 15°, it's 0.43%. At 30°, the error grows to 1.7%, and at 45° it's 3.9%. For most practical purposes (classroom experiments, clock design), the formula is excellent up to about 15-20°. Beyond that, correction terms or numerical methods are needed for precision work.
How can a pendulum measure gravity?
By rearranging the formula: g = 4π²L/T². If you measure the length precisely and time many oscillations (to reduce timing errors), you can calculate g accurately. This method was historically used to measure gravitational acceleration around the world. Modern gravimeters still use free-fall and pendulum principles. Variations in g reveal underground density differences, useful for geological surveys, mineral exploration, and detecting underground structures.
Historical Significance of Pendulums
The pendulum has played a crucial role in the history of science and technology. Galileo's observations of pendulum motion led to insights about inertia and acceleration. Huygens's pendulum clock revolutionized timekeeping and navigation. Foucault's pendulum (1851) provided the first direct visual proof of Earth's rotation. The seconds pendulum (T = 2s) was even proposed as a standard of length — the meter was originally considered as the length of a seconds pendulum, though a different definition was ultimately adopted.
Today, pendulums remain important in physics education, seismology (seismometers use pendulum principles), and precision measurement. Torsion pendulums are used to measure extremely weak forces, including the gravitational constant G. Pendulum dynamics also appear in many engineering applications, from vibration dampers in skyscrapers (tuned mass dampers) to the motion of robotic legs and the analysis of human walking gait.