Simple Pendulum
Find pendulum period with T = 2π√(L/g), or solve for length L or gravity g. Free calculator with frequency, angular frequency, and clear steps. Free online calculator with formulas and step-by-step solutions.
Key formula
T = 2π √(L/g)
How do you calculate pendulum period (T = 2π√(L/g))?
For a small-angle simple pendulum, T = 2π √(L/g). Rearrange to L = g T²/(4π²) or g = 4π² L / T². Frequency is f = 1/T. Use g from the question (often 9.8 m/s²) and measure L to the bob’s center.
About this tool
A simple pendulum is a small mass on a light string. For small angles the period is T = 2π√(L/g) — the formula students search as t = 2π√(l/g). This page keeps that formula visible and lets you solve for T, L, or g with substituted steps.
How to use this calculator
- Choose the variable you want to solve for and confirm which values are already known.
- Enter inputs in consistent units before calculating so the formula is applied correctly.
- Check whether the result is physically reasonable by comparing it with the size of the inputs.
- Use the step-by-step breakdown to see how the equation was rearranged and evaluated.
Worked examples
Example 1: T = 2π√(L/g) for a 1.0 m pendulum
L = 1.0 m, g = 9.8 m/s². T = 2π √(1.0/9.8) ≈ 2.007 s. Then f = 1/T ≈ 0.498 Hz.
Example 2: length for a 2.0 s period
T = 2.0 s, g = 9.8. L = g T² / (4π²) = 9.8 × 4 / (4π²) = 9.8/π² ≈ 0.993 m. A “2-second pendulum” is a little under one meter at Earth g.
Example 3: estimate g from the lab
Measured T = 2.007 s for L = 1.00 m. g = 4π² L / T² ≈ 9.80 m/s². Timing many swings and dividing by the count reduces stopwatch error.
Frequently asked questions
What is the simple pendulum period formula?
T = 2π √(L/g) for small angles (often written t = 2π√(l/g)). L is the length to the bob’s center and g is gravitational acceleration.
How do you calculate pendulum period step by step?
Convert L to meters, pick the paper’s g, then compute T = 2π √(L/g). Check that a ~1 m pendulum on Earth has T near 2 s.
Does pendulum period depend on mass?
Not in the ideal small-angle model. Mass cancels in the restoring-force derivation, just as it does in free fall.
Does period depend on amplitude?
Only weakly at small angles (about 15° or less from vertical). Large angles need a correction; this calculator uses the intro-physics formula.
How do I find length from period?
L = g T² / (4π²). Enter T and g in the calculator’s length mode.