Calculation
T = 2π · √(L/g)
A simple pendulum swings more slowly as its length grows when the swing is small. The bob's mass cancels out.
Kleinwinkelnäherung des mathematischen Pendels · T = 2π√(L/g)
Calculate the Period of a Simple Pendulum
A simple pendulum swings more slowly as its length grows when the swing is small. The bob's mass cancels out.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Oscillation period T — Time for one complete out-and-back swing, for example from one left turning point to the next. Time several swings and divide by their count; a seconds pendulum has a period near 2 s.
- Pendulum length L — Distance from the pivot to the bob's centre of mass, not just the free string length. Measure it from the geometry; classroom pendulums are often about 0.2 to 2 m long.
- Gravitational acceleration g — Local gravitational acceleration; about 9.81 m/s² on Earth. Use the local value for precision measurements.
Example
At L = 1.00 m and g = 9.80665 m/s², one full swing takes T = 2π√(1/9.80665) = 2.006 s.
Assumptions and limits
Point-like bob on a massless, inextensible string; small swing angle, no pivot friction or air drag. At larger swings the actual period also depends on the starting angle, and this approximation underestimates it.
Understand Period of a simple pendulum
How long must a pendulum be to take two seconds for one full swing? This calculator relates its length to the time for a complete out-and-back motion.
What does this quantity describe?
The model uses a small bob on a light string. Gravity pulls it back toward its resting position. At small angles this restoring effect is nearly proportional to the angle, so the motion is approximately harmonic: T = 2π√(L/g). T is the period, L the pivot-to-centre-of-mass distance and g local gravitational acceleration. The bob mass cancels out.
Formula and variables
T = 2π · √(L/g)
Period: T = 2π√(L/g)Required length: L = g(T/2π)²Gravity from measurement: g = L(2π/T)²
| Symbol / input | Meaning |
|---|---|
| Oscillation period T | Time for one complete out-and-back swing, for example from one left turning point to the next. Time several swings and divide by their count; a seconds pendulum has a period near 2 s. |
| Pendulum length L | Distance from the pivot to the bob's centre of mass, not just the free string length. Measure it from the geometry; classroom pendulums are often about 0.2 to 2 m long. |
| Gravitational acceleration g | Local gravitational acceleration; about 9.81 m/s² on Earth. Use the local value for precision measurements. |
Choose the inputs correctly
Measure L in metres from the pivot to the bob's centre of mass; the string length alone can be too short. Lengths of 0.2 to 2 m are practical for a demonstration. On Earth g is approximately 9.81 m/s²; obtain a precise local value from a survey or measurement. To infer T, time several complete swings and divide by their number.
How to use the calculator
Enter length and gravity to calculate T. A complete swing ends at the same turning point where it started. Choose L to find the required length, or g to estimate local gravity from measured length and period.
Worked example
At L = 1.00 m and g = 9.80665 m/s², T = 2π√(1/9.80665) = 2.006 s per full swing. Ten full swings take about 20.06 s. Four times the length doubles T to 4.012 s.
Understand the result and units
The period grows with the square root of length: doubling length multiplies the period by about 1.41. This is a small-angle approximation; a large starting angle gives a longer period.
Calculations use metres, seconds and m/s² internally. Offered length units are converted to metres first. T is the time for one complete swing, not the time from one extreme to the opposite extreme.
Typical applications
Planning and checking demonstration pendulums, estimating swing times in simple experiments, and checking a gravity measurement made with a pendulum.
Assumptions, limits and common mistakes
The bob is treated as a point mass on a massless, inextensible string with a fixed pivot and small swing angle. Pivot friction and air drag are excluded. Precision clocks, large swings and extended rigid bodies require a more detailed model.
Common mistake: Common errors are measuring only the string instead of the pivot-to-centre distance, or counting half a swing as T. At large amplitudes the formula underestimates the period; bob mass is not a correction factor.
Frequently asked questions
What is “Period of a simple pendulum” used for?
Planning and checking demonstration pendulums, estimating swing times in simple experiments, and checking a gravity measurement made with a pendulum.
Where do the input values come from?
Measure L in metres from the pivot to the bob's centre of mass; the string length alone can be too short. Lengths of 0.2 to 2 m are practical for a demonstration. On Earth g is approximately 9.81 m/s²; obtain a precise local value from a survey or measurement. To infer T, time several complete swings and divide by their number.
What does the result not cover?
The bob is treated as a point mass on a massless, inextensible string with a fixed pivot and small swing angle. Pivot friction and air drag are excluded. Precision clocks, large swings and extended rigid bodies require a more detailed model.